From 3784a6278ce441e516f12a929f4a4e4fccbd4708 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 2 Feb 2024 07:04:58 +0100 Subject: [PATCH] update week3 --- doc/pub/week3/html/._week3-bs000.html | 6 +- doc/pub/week3/html/._week3-bs001.html | 6 +- doc/pub/week3/html/._week3-bs002.html | 6 +- doc/pub/week3/html/._week3-bs003.html | 6 +- doc/pub/week3/html/._week3-bs004.html | 6 +- doc/pub/week3/html/._week3-bs005.html | 6 +- doc/pub/week3/html/._week3-bs006.html | 6 +- doc/pub/week3/html/._week3-bs007.html | 14 +- doc/pub/week3/html/._week3-bs008.html | 17 +- doc/pub/week3/html/._week3-bs009.html | 12 +- doc/pub/week3/html/._week3-bs010.html | 14 +- doc/pub/week3/html/._week3-bs011.html | 14 +- doc/pub/week3/html/._week3-bs012.html | 12 +- doc/pub/week3/html/._week3-bs013.html | 12 +- doc/pub/week3/html/._week3-bs014.html | 12 +- doc/pub/week3/html/._week3-bs015.html | 12 +- doc/pub/week3/html/._week3-bs016.html | 13 +- doc/pub/week3/html/._week3-bs017.html | 11 +- doc/pub/week3/html/._week3-bs018.html | 12 +- doc/pub/week3/html/._week3-bs019.html | 13 +- doc/pub/week3/html/._week3-bs020.html | 12 +- doc/pub/week3/html/._week3-bs021.html | 12 +- doc/pub/week3/html/._week3-bs022.html | 32 +- doc/pub/week3/html/._week3-bs023.html | 44 +- doc/pub/week3/html/._week3-bs024.html | 34 +- doc/pub/week3/html/._week3-bs025.html | 20 +- doc/pub/week3/html/._week3-bs026.html | 13 +- doc/pub/week3/html/._week3-bs027.html | 13 +- doc/pub/week3/html/._week3-bs028.html | 12 +- doc/pub/week3/html/._week3-bs029.html | 12 +- doc/pub/week3/html/._week3-bs030.html | 12 +- doc/pub/week3/html/._week3-bs031.html | 13 +- doc/pub/week3/html/._week3-bs032.html | 12 +- doc/pub/week3/html/._week3-bs033.html | 12 +- doc/pub/week3/html/._week3-bs034.html | 12 +- doc/pub/week3/html/._week3-bs035.html | 13 +- doc/pub/week3/html/._week3-bs036.html | 12 +- doc/pub/week3/html/._week3-bs037.html | 13 +- doc/pub/week3/html/._week3-bs038.html | 12 +- doc/pub/week3/html/._week3-bs039.html | 6 +- doc/pub/week3/html/._week3-bs040.html | 6 +- doc/pub/week3/html/._week3-bs041.html | 6 +- doc/pub/week3/html/._week3-bs042.html | 6 +- doc/pub/week3/html/._week3-bs043.html | 6 +- doc/pub/week3/html/._week3-bs044.html | 6 +- doc/pub/week3/html/._week3-bs045.html | 6 +- doc/pub/week3/html/._week3-bs046.html | 6 +- doc/pub/week3/html/._week3-bs047.html | 6 +- doc/pub/week3/html/._week3-bs048.html | 6 +- doc/pub/week3/html/week3-bs.html | 6 +- doc/pub/week3/html/week3-reveal.html | 248 +++------ doc/pub/week3/html/week3-solarized.html | 264 +++------ doc/pub/week3/html/week3.html | 264 +++------ .../Results/FigureFiles/QdotImportance.png | Bin 67363 -> 66600 bytes .../VMCQdotImportance/VMCQdotImportance.dat | 200 +++---- doc/pub/week3/ipynb/ipynb-week3-src.tar.gz | Bin 191 -> 191 bytes doc/pub/week3/ipynb/week3.ipynb | 522 ++++++++++-------- doc/pub/week3/pdf/week3-beamer.pdf | Bin 361741 -> 362674 bytes doc/pub/week3/pdf/week3.pdf | Bin 425122 -> 424755 bytes doc/src/week3/week3.do.txt | 222 ++++---- 60 files changed, 947 insertions(+), 1364 deletions(-) diff --git a/doc/pub/week3/html/._week3-bs000.html b/doc/pub/week3/html/._week3-bs000.html index ba242581..df1a8285 100644 --- a/doc/pub/week3/html/._week3-bs000.html +++ b/doc/pub/week3/html/._week3-bs000.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs001.html b/doc/pub/week3/html/._week3-bs001.html index b065dd83..cf1a12bd 100644 --- a/doc/pub/week3/html/._week3-bs001.html +++ b/doc/pub/week3/html/._week3-bs001.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs002.html b/doc/pub/week3/html/._week3-bs002.html index 507f2c89..635151c5 100644 --- a/doc/pub/week3/html/._week3-bs002.html +++ b/doc/pub/week3/html/._week3-bs002.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs003.html b/doc/pub/week3/html/._week3-bs003.html index 97975656..36da776d 100644 --- a/doc/pub/week3/html/._week3-bs003.html +++ b/doc/pub/week3/html/._week3-bs003.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs004.html b/doc/pub/week3/html/._week3-bs004.html index 9a11a696..6320ff79 100644 --- a/doc/pub/week3/html/._week3-bs004.html +++ b/doc/pub/week3/html/._week3-bs004.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs005.html b/doc/pub/week3/html/._week3-bs005.html index 81571ba2..5fb4ad27 100644 --- a/doc/pub/week3/html/._week3-bs005.html +++ b/doc/pub/week3/html/._week3-bs005.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs006.html b/doc/pub/week3/html/._week3-bs006.html index 5eb2a99d..fb3aec38 100644 --- a/doc/pub/week3/html/._week3-bs006.html +++ b/doc/pub/week3/html/._week3-bs006.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs007.html b/doc/pub/week3/html/._week3-bs007.html index 9a701576..e2928882 100644 --- a/doc/pub/week3/html/._week3-bs007.html +++ b/doc/pub/week3/html/._week3-bs007.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -275,7 +275,13 @@

    Code example for the interacting case with importance sampling

    -

    We are now ready to implement importance sampling. This is done here for the two-electron case with the Coulomb interaction, as in the previous example. We have two variational parameters \( \alpha \) and \( \beta \). After the set up of files

    +Note: this is best seen using the Jupyter-notebook. + +

    We are now ready to implement importance sampling. This is done here +for the two-electron case with the Coulomb interaction, as in the +previous example. We have two variational parameters \( \alpha \) and +\( \beta \). After the set up of files +

    diff --git a/doc/pub/week3/html/._week3-bs008.html b/doc/pub/week3/html/._week3-bs008.html index 894753bd..3f9b64ab 100644 --- a/doc/pub/week3/html/._week3-bs008.html +++ b/doc/pub/week3/html/._week3-bs008.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -273,11 +273,9 @@

     

     

     

    -

    Importance sampling, program elements

    -
    -
    - -

    The general derivative formula of the Jastrow factor is (the subscript \( C \) stands for Correlation)

    +

    Importance sampling, programming elements

    + +

    The general derivative formula of the Jastrow factor (or the ansatz for the correlated part of the wave function) is (the subscript \( C \) stands for Correlation)

    $$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} @@ -295,9 +293,6 @@

    Importance sampling

    the gradient needed for the quantum force and local energy is easy to compute. The function \( f(r_{ij}) \) will depends on the system under study. In the equations below we will keep this general form.

    -

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs009.html b/doc/pub/week3/html/._week3-bs009.html index ee9ce811..a336c251 100644 --- a/doc/pub/week3/html/._week3-bs009.html +++ b/doc/pub/week3/html/._week3-bs009.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, program elements

    -
    -
    - +

    In the Metropolis/Hasting algorithm, the acceptance ratio determines the probability for a particle to be accepted at a new position. The ratio of the trial wave functions evaluated at the new and current positions is given by (\( OB \) for the onebody part)

    $$ R \equiv \frac{\Psi_{T}^{new}}{\Psi_{T}^{old}} = @@ -291,8 +289,6 @@

    Importance sampling $$ \frac{\mathbf{\mathbf{\nabla}} \Psi}{\Psi} = \frac{\mathbf{\nabla} (\Psi_{OB} \, \Psi_{C})}{\Psi_{OB} \, \Psi_{C}} = \frac{ \Psi_C \mathbf{\nabla} \Psi_{OB} + \Psi_{OB} \mathbf{\nabla} \Psi_{C}}{\Psi_{OB} \Psi_{C}} = \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} $$ -

    -

    diff --git a/doc/pub/week3/html/._week3-bs010.html b/doc/pub/week3/html/._week3-bs010.html index 1d71147f..b6fced42 100644 --- a/doc/pub/week3/html/._week3-bs010.html +++ b/doc/pub/week3/html/._week3-bs010.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,10 +274,8 @@

    Importance sampling

    -
    -
    - -

    The expectation value of the kinetic energy expressed in atomic units for electron \( i \) is

    + +

    The expectation value of the kinetic energy expressed in scaled units for particle \( i \) is

    $$ \langle \hat{K}_i \rangle = -\frac{1}{2}\frac{\langle\Psi|\mathbf{\nabla}_{i}^2|\Psi \rangle}{\langle\Psi|\Psi \rangle}, $$ @@ -285,8 +283,6 @@

    Importance sampling

    $$ \hat{K}_i = -\frac{1}{2}\frac{\mathbf{\nabla}_{i}^{2} \Psi}{\Psi}. $$ -
    -

    diff --git a/doc/pub/week3/html/._week3-bs011.html b/doc/pub/week3/html/._week3-bs011.html index 923163f7..72ab5c15 100644 --- a/doc/pub/week3/html/._week3-bs011.html +++ b/doc/pub/week3/html/._week3-bs011.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,17 +274,13 @@

    Importance sampling

    -
    -
    - +

    The second derivative which enters the definition of the local energy is

    $$ \frac{\mathbf{\nabla}^2 \Psi}{\Psi}=\frac{\mathbf{\nabla}^2 \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla}^2 \Psi_C}{ \Psi_C} + 2 \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}}\cdot\frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} $$ -

    We discuss here how to calculate these quantities in an optimal way,

    -
    -
    +

    We discuss here how to calculate these quantities in an optimal way.

    diff --git a/doc/pub/week3/html/._week3-bs012.html b/doc/pub/week3/html/._week3-bs012.html index 3a486bff..d4e6aad5 100644 --- a/doc/pub/week3/html/._week3-bs012.html +++ b/doc/pub/week3/html/._week3-bs012.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    We have defined the correlated function as

    $$ \Psi_C=\prod_{i < j}g(r_{ij})=\prod_{i < j}^Ng(r_{ij})= \prod_{i=1}^N\prod_{j=i+1}^Ng(r_{ij}), @@ -291,8 +289,6 @@

    Importance sampling

    $$ \Psi_C=\prod_{i < j}g(r_{ij})=\exp{\left\{\sum_{i < j}f(r_{ij})\right\}}. $$ -
    -

    diff --git a/doc/pub/week3/html/._week3-bs013.html b/doc/pub/week3/html/._week3-bs013.html index 871395ce..594e7ece 100644 --- a/doc/pub/week3/html/._week3-bs013.html +++ b/doc/pub/week3/html/._week3-bs013.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    The total number of different relative distances \( r_{ij} \) is \( N(N-1)/2 \). In a matrix storage format, the relative distances form a strictly upper triangular matrix

    $$ \mathbf{r} \equiv \begin{pmatrix} @@ -291,8 +289,6 @@

    Importance sampling

    This applies to \( \mathbf{g} = \mathbf{g}(r_{ij}) \) as well.

    In our algorithm we will move one particle at the time, say the \( kth \)-particle. This sampling will be seen to be particularly efficient when we are going to compute a Slater determinant.

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs014.html b/doc/pub/week3/html/._week3-bs014.html index 4a53b98f..0df62d6b 100644 --- a/doc/pub/week3/html/._week3-bs014.html +++ b/doc/pub/week3/html/._week3-bs014.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    We have that the ratio between Jastrow factors \( R_C \) is given by

    $$ R_{C} = \frac{\Psi_{C}^\mathrm{new}}{\Psi_{C}^\mathrm{cur}} = @@ -297,8 +295,6 @@

    Importance sampling

    + \sum_{i=k+1}^{N}\big(f_{ki}^\mathrm{new}-f_{ki}^\mathrm{cur}\big) $$ -
    -

    diff --git a/doc/pub/week3/html/._week3-bs015.html b/doc/pub/week3/html/._week3-bs015.html index 9b095f02..b30b46f8 100644 --- a/doc/pub/week3/html/._week3-bs015.html +++ b/doc/pub/week3/html/._week3-bs015.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    One needs to develop a special algorithm that runs only through the elements of the upper triangular matrix \( \mathbf{g} \) and have \( k \) as an index. @@ -288,8 +286,6 @@

    Importance sampling

    $$

    for all dimensions and with \( i \) running over all particles.

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs016.html b/doc/pub/week3/html/._week3-bs016.html index 37ff6430..555af7cf 100644 --- a/doc/pub/week3/html/._week3-bs016.html +++ b/doc/pub/week3/html/._week3-bs016.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    For the first derivative only \( N-1 \) terms survive the ratio because the \( g \)-terms that are not differentiated cancel with their corresponding ones in the denominator. Then,

    $$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = @@ -294,9 +292,6 @@

    Importance sampling

    $$

    with both expressions scaling as \( \mathcal{O}(N) \).

    -
    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs017.html b/doc/pub/week3/html/._week3-bs017.html index f8efa40e..5b172d39 100644 --- a/doc/pub/week3/html/._week3-bs017.html +++ b/doc/pub/week3/html/._week3-bs017.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,6 @@

    Importance sampling

    -
    -
    -

    Using the identity

    $$ @@ -296,8 +293,6 @@

    Importance sampling

    \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} -\sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i}. $$ -
    -

    diff --git a/doc/pub/week3/html/._week3-bs018.html b/doc/pub/week3/html/._week3-bs018.html index d218b689..87a5e202 100644 --- a/doc/pub/week3/html/._week3-bs018.html +++ b/doc/pub/week3/html/._week3-bs018.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    For correlation forms depending only on the scalar distances \( r_{ij} \) we can use the chain rule. Noting that

    $$ \frac{\partial g_{ij}}{\partial x_j} = \frac{\partial g_{ij}}{\partial r_{ij}} \frac{\partial r_{ij}}{\partial x_j} = \frac{x_j - x_i}{r_{ij}} \frac{\partial g_{ij}}{\partial r_{ij}}, @@ -288,8 +286,6 @@

    Importance sampling

    \sum_{i=1}^{k-1}\frac{1}{g_{ik}} \frac{\mathbf{r_{ik}}}{r_{ik}} \frac{\partial g_{ik}}{\partial r_{ik}} -\sum_{i=k+1}^{N}\frac{1}{g_{ki}}\frac{\mathbf{r_{ki}}}{r_{ki}}\frac{\partial g_{ki}}{\partial r_{ki}}. $$ -
    -

    diff --git a/doc/pub/week3/html/._week3-bs019.html b/doc/pub/week3/html/._week3-bs019.html index 58b12125..c4a4ba08 100644 --- a/doc/pub/week3/html/._week3-bs019.html +++ b/doc/pub/week3/html/._week3-bs019.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    Note that for the Pade-Jastrow form we can set \( g_{ij} \equiv g(r_{ij}) = e^{f(r_{ij})} = e^{f_{ij}} \) and

    $$ \frac{\partial g_{ij}}{\partial r_{ij}} = g_{ij} \frac{\partial f_{ij}}{\partial r_{ij}}. @@ -295,9 +293,6 @@

    Importance sampling

    $$

    is the relative distance.

    -
    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs020.html b/doc/pub/week3/html/._week3-bs020.html index e4c8d802..2a25777f 100644 --- a/doc/pub/week3/html/._week3-bs020.html +++ b/doc/pub/week3/html/._week3-bs020.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    The second derivative of the Jastrow factor divided by the Jastrow factor (the way it enters the kinetic energy) is

    $$ \left[\frac{\mathbf{\nabla}^2 \Psi_C}{\Psi_C}\right]_x =\ @@ -288,8 +286,6 @@

    Importance sampling

    \sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i} \right)^2 $$ -
    -

    diff --git a/doc/pub/week3/html/._week3-bs021.html b/doc/pub/week3/html/._week3-bs021.html index 7a53d2c6..bc64cb8f 100644 --- a/doc/pub/week3/html/._week3-bs021.html +++ b/doc/pub/week3/html/._week3-bs021.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling

    -
    -
    - +

    But we have a simple form for the function, namely

    $$ \Psi_{C}=\prod_{i < j}\exp{f(r_{ij})}, @@ -290,8 +288,6 @@

    Importance sampling

    \sum_{ij\ne k}\frac{(\mathbf{r}_k-\mathbf{r}_i)(\mathbf{r}_k-\mathbf{r}_j)}{r_{ki}r_{kj}}f'(r_{ki})f'(r_{kj})+ \sum_{j\ne k}\left( f''(r_{kj})+\frac{2}{r_{kj}}f'(r_{kj})\right) $$ -
    -

    diff --git a/doc/pub/week3/html/._week3-bs022.html b/doc/pub/week3/html/._week3-bs022.html index 3fa3e00f..41c01c75 100644 --- a/doc/pub/week3/html/._week3-bs022.html +++ b/doc/pub/week3/html/._week3-bs022.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,26 +274,20 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - -

    A stochastic process is simply a function of two variables, one is the time, -the other is a stochastic variable \( X \), defined by specifying + +

    A stochastic process is simply a function of two variables, one is the +time, the other is a stochastic variable \( X \), defined by specifying

    -
      + +
      1. the set \( \left\{x\right\} \) of possible values for \( X \);
      2. the probability distribution, \( w_X(x) \), over this set, or briefly \( w(x) \)
        3. -
    -

    The set of values \( \left\{x\right\} \) for \( X \) -may be discrete, or continuous. If the set of -values is continuous, then \( w_X (x) \) is a probability density so that -\( w_X (x)dx \) -is the probability that one finds the stochastic variable \( X \) to have values -in the range \( [x, x + dx] \) . + +

    The set of values \( \left\{x\right\} \) for \( X \) may be discrete, or +continuous. If the set of values is continuous, then \( w_X (x) \) is a +probability density so that \( w_X (x)dx \) is the probability that one +finds the stochastic variable \( X \) to have values in the range \( [x, x +dx] \) .

    -
    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs023.html b/doc/pub/week3/html/._week3-bs023.html index a21e5fab..0ad469bd 100644 --- a/doc/pub/week3/html/._week3-bs023.html +++ b/doc/pub/week3/html/._week3-bs023.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,34 +274,28 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - -

    An arbitrary number of other stochastic variables may be derived from -\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a stochastic -variable. The mapping may also be time-dependent, that is, the mapping -depends on an additional variable \( t \) -

    -$$ - Y_X (t) = f (X, t) . -$$ -

    The quantity \( Y_X (t) \) is called a random function, or, since \( t \) often is time, -a stochastic process. A stochastic process is a function of two variables, -one is the time, the other is a stochastic variable \( X \). Let \( x \) be one of the -possible values of \( X \) then +

    An arbitrary number of other stochastic variables may be derived from +\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a +stochastic variable. The mapping may also be time-dependent, that is, +the mapping depends on an additional variable \( t \)

    -$$ - y(t) = f (x, t), +$$ Y_X (t) = f(X, t). $$ -

    is a function of \( t \), called a sample function or realization of the process. -In physics one considers the stochastic process to be an ensemble of such -sample functions. +

    The quantity \( Y_X (t) \) is called a random function, +or, since \( t \) often is time, a stochastic process. A stochastic +process is a function of two variables, one is the time, the other is +a stochastic variable \( X \). Let \( x \) be one of the possible values of +\( X \) then

    -
    -
    +$$ y(t) = f (x, t), $$ +

    is a function of \( t \), called a +sample function or realization of the process. In physics one +considers the stochastic process to be an ensemble of such sample +functions. +

    diff --git a/doc/pub/week3/html/._week3-bs024.html b/doc/pub/week3/html/._week3-bs024.html index efe14835..9dfa2d7a 100644 --- a/doc/pub/week3/html/._week3-bs024.html +++ b/doc/pub/week3/html/._week3-bs024.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,31 +274,25 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - -

    For many physical systems initial distributions of a stochastic -variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow w_0(y) \) -as \( t\rightarrow\infty \). In -equilibrium detailed balance constrains the transition rates + +

    For many physical systems initial distributions of a stochastic +variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow +w_0(y) \) as \( t\rightarrow\infty \). In equilibrium detailed balance +constrains the transition rates

    $$ W(y\rightarrow y')w(y ) = W(y'\rightarrow y)w_0 (y), $$ -

    where \( W(y'\rightarrow y) \) -is the probability, per unit time, that the system changes -from a state \( |y\rangle \) , characterized by the value \( y \) -for the stochastic variable \( Y \) , to a state \( |y'\rangle \). +

    where \( W(y'\rightarrow y) \) is the probability, per unit time, that the +system changes from a state \( |y\rangle \) , characterized by the value +\( y \) for the stochastic variable \( Y \) , to a state \( |y'\rangle \).

    -

    Note that for a system in equilibrium the transition rate -\( W(y'\rightarrow y) \) and -the reverse \( W(y\rightarrow y') \) may be very different. +

    Note that for a system in equilibrium the transition rate +\( W(y'\rightarrow y) \) and the reverse \( W(y\rightarrow y') \) may be very +different.

    -
    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs025.html b/doc/pub/week3/html/._week3-bs025.html index 9fb312a8..b91958e1 100644 --- a/doc/pub/week3/html/._week3-bs025.html +++ b/doc/pub/week3/html/._week3-bs025.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,17 +274,14 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - -

    Consider, for instance, a simple -system that has only two energy levels \( \epsilon_0 = 0 \) and -\( \epsilon_1 = \Delta E \). + +

    Consider, for instance, a simple system that has only two energy +levels \( \epsilon_0 = 0 \) and \( \epsilon_1 = \Delta E \).

    For a system governed by the Boltzmann distribution we find (the partition function has been taken out)

    $$ - W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)} + W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)}. $$

    We get then

    @@ -293,9 +290,6 @@

    which goes to zero when \( T \) tends to zero.

    -

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs026.html b/doc/pub/week3/html/._week3-bs026.html index 4a8f3a88..5d2873e1 100644 --- a/doc/pub/week3/html/._week3-bs026.html +++ b/doc/pub/week3/html/._week3-bs026.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    If we assume a discrete set of events, our initial probability distribution function can be given by @@ -299,9 +297,6 @@

    -

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs027.html b/doc/pub/week3/html/._week3-bs027.html index dcb621c9..fdac3496 100644 --- a/doc/pub/week3/html/._week3-bs027.html +++ b/doc/pub/week3/html/._week3-bs027.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    The transition from a state \( j \) to a state \( i \) is now replaced by a transition to a state with position \( \mathbf{y} \) from a state with position \( \mathbf{x} \). The discrete sum of transition probabilities can then be replaced by an integral @@ -297,9 +295,6 @@

    that is no time-dependence. Note our change of notation for \( W \)

    -

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs028.html b/doc/pub/week3/html/._week3-bs028.html index 671c308f..f45c8421 100644 --- a/doc/pub/week3/html/._week3-bs028.html +++ b/doc/pub/week3/html/._week3-bs028.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    We can solve the equation for \( w(\mathbf{y},t) \) by making a Fourier transform to momentum space. The PDF \( w(\mathbf{x},t) \) is related to its Fourier transform @@ -297,8 +295,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs029.html b/doc/pub/week3/html/._week3-bs029.html index 1d9d0ee8..8349cfcb 100644 --- a/doc/pub/week3/html/._week3-bs029.html +++ b/doc/pub/week3/html/._week3-bs029.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    We can then use the Fourier-transformed diffusion equation

    $$ \frac{\partial \tilde{w}(\mathbf{k},t)}{\partial t} = -D\mathbf{k}^2\tilde{w}(\mathbf{k},t), @@ -287,8 +285,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs030.html b/doc/pub/week3/html/._week3-bs030.html index 844bfb6e..851df226 100644 --- a/doc/pub/week3/html/._week3-bs030.html +++ b/doc/pub/week3/html/._week3-bs030.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    With the Fourier transform we obtain

    $$ w(\mathbf{x},t)=\int_{-\infty}^{\infty}d\mathbf{k} \exp{\left[i\mathbf{kx}\right]}\frac{1}{2\pi}\exp{\left[-(D\mathbf{k}^2t)\right]}= @@ -287,8 +285,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs031.html b/doc/pub/week3/html/._week3-bs031.html index fb247c06..7994bd65 100644 --- a/doc/pub/week3/html/._week3-bs031.html +++ b/doc/pub/week3/html/._week3-bs031.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    The solution represents the probability of finding our random walker at position \( \mathbf{x} \) at time \( t \) if the initial distribution was placed at \( \mathbf{x}=0 \) at \( t=0 \). @@ -296,9 +294,6 @@

    and that it satisfies the normalization condition and is itself a solution to the diffusion equation.

    -

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs032.html b/doc/pub/week3/html/._week3-bs032.html index 9f26754f..858bb491 100644 --- a/doc/pub/week3/html/._week3-bs032.html +++ b/doc/pub/week3/html/._week3-bs032.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    Let us now assume that we have three PDFs for times \( t_0 < t' < t \), that is \( w(\mathbf{x}_0,t_0) \), \( w(\mathbf{x}',t') \) and \( w(\mathbf{x},t) \). We have then @@ -294,8 +292,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs033.html b/doc/pub/week3/html/._week3-bs033.html index 2d6ff856..d2c1e680 100644 --- a/doc/pub/week3/html/._week3-bs033.html +++ b/doc/pub/week3/html/._week3-bs033.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    We can combine these equations and arrive at the famous Einstein-Smoluchenski-Kolmogorov-Chapman (ESKC) relation

    $$ W(\mathbf{x}t|\mathbf{x}_0t_0) = \int_{-\infty}^{\infty} W(\mathbf{x},t|\mathbf{x}',t')W(\mathbf{x}',t'|\mathbf{x}_0,t_0)d\mathbf{x}'. @@ -288,8 +286,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs034.html b/doc/pub/week3/html/._week3-bs034.html index 81646da5..082e3c3c 100644 --- a/doc/pub/week3/html/._week3-bs034.html +++ b/doc/pub/week3/html/._week3-bs034.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    We will now derive the Fokker-Planck equation. We start from the ESKC equation

    @@ -288,8 +286,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs035.html b/doc/pub/week3/html/._week3-bs035.html index cd2b8e76..229fdd5f 100644 --- a/doc/pub/week3/html/._week3-bs035.html +++ b/doc/pub/week3/html/._week3-bs035.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,18 +274,13 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    Assume now that \( \tau \) is very small so that we can make an expansion in terms of a small step \( xi \), with \( \mathbf{x}'=\mathbf{x}-\xi \), that is

    $$ W(\mathbf{x},s|\mathbf{x}_0)+\frac{\partial W}{\partial s}\tau +O(\tau^2) = \int_{-\infty}^{\infty} W(\mathbf{x},\tau|\mathbf{x}-\xi)W(\mathbf{x}-\xi,s|\mathbf{x}_0)d\mathbf{x}'. $$

    We assume that \( W(\mathbf{x},\tau|\mathbf{x}-\xi) \) takes non-negligible values only when \( \xi \) is small. This is just another way of stating the Master equation!!

    -
    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs036.html b/doc/pub/week3/html/._week3-bs036.html index daa68a6e..ba430cd2 100644 --- a/doc/pub/week3/html/._week3-bs036.html +++ b/doc/pub/week3/html/._week3-bs036.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -286,8 +284,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs037.html b/doc/pub/week3/html/._week3-bs037.html index 60758ede..6e7d98f1 100644 --- a/doc/pub/week3/html/._week3-bs037.html +++ b/doc/pub/week3/html/._week3-bs037.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    We can then rewrite the ESKC equation as

    $$ \frac{\partial W}{\partial s}\tau=-W(\mathbf{x},s|\mathbf{x}_0)+ @@ -287,9 +285,6 @@

    We have neglected higher powers of \( \tau \) and have used that for \( n=0 \) we get simply \( W(\mathbf{x},s|\mathbf{x}_0) \) due to normalization.

    -

    -
    -

    diff --git a/doc/pub/week3/html/._week3-bs038.html b/doc/pub/week3/html/._week3-bs038.html index 5f4a911a..cf256dfb 100644 --- a/doc/pub/week3/html/._week3-bs038.html +++ b/doc/pub/week3/html/._week3-bs038.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • @@ -274,9 +274,7 @@

    Importance sampling, Fokker-Planck and Langevin equations

    -
    -
    - +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -286,8 +284,6 @@

    -

    diff --git a/doc/pub/week3/html/._week3-bs039.html b/doc/pub/week3/html/._week3-bs039.html index 5f807427..52c25fae 100644 --- a/doc/pub/week3/html/._week3-bs039.html +++ b/doc/pub/week3/html/._week3-bs039.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@

  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs040.html b/doc/pub/week3/html/._week3-bs040.html index 1fe51d09..0947921a 100644 --- a/doc/pub/week3/html/._week3-bs040.html +++ b/doc/pub/week3/html/._week3-bs040.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs041.html b/doc/pub/week3/html/._week3-bs041.html index b12f99f2..bbd617c0 100644 --- a/doc/pub/week3/html/._week3-bs041.html +++ b/doc/pub/week3/html/._week3-bs041.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs042.html b/doc/pub/week3/html/._week3-bs042.html index 1599b9eb..70c012ca 100644 --- a/doc/pub/week3/html/._week3-bs042.html +++ b/doc/pub/week3/html/._week3-bs042.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs043.html b/doc/pub/week3/html/._week3-bs043.html index e962aa3e..3522cb0e 100644 --- a/doc/pub/week3/html/._week3-bs043.html +++ b/doc/pub/week3/html/._week3-bs043.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs044.html b/doc/pub/week3/html/._week3-bs044.html index a78480b0..a3e9a3f0 100644 --- a/doc/pub/week3/html/._week3-bs044.html +++ b/doc/pub/week3/html/._week3-bs044.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs045.html b/doc/pub/week3/html/._week3-bs045.html index c52e83eb..3db1d895 100644 --- a/doc/pub/week3/html/._week3-bs045.html +++ b/doc/pub/week3/html/._week3-bs045.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs046.html b/doc/pub/week3/html/._week3-bs046.html index 5c259d51..e0790701 100644 --- a/doc/pub/week3/html/._week3-bs046.html +++ b/doc/pub/week3/html/._week3-bs046.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs047.html b/doc/pub/week3/html/._week3-bs047.html index 7c2cdb59..dd07279c 100644 --- a/doc/pub/week3/html/._week3-bs047.html +++ b/doc/pub/week3/html/._week3-bs047.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/._week3-bs048.html b/doc/pub/week3/html/._week3-bs048.html index 77fa06e0..4665816a 100644 --- a/doc/pub/week3/html/._week3-bs048.html +++ b/doc/pub/week3/html/._week3-bs048.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/week3-bs.html b/doc/pub/week3/html/week3-bs.html index ba242581..df1a8285 100644 --- a/doc/pub/week3/html/week3-bs.html +++ b/doc/pub/week3/html/week3-bs.html @@ -52,10 +52,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -221,7 +221,7 @@
  • Importance sampling
  • Importance sampling
  • Code example for the interacting case with importance sampling
    • -
  • Importance sampling, program elements
    • +
  • Importance sampling, programming elements
  • Importance sampling, program elements
  • Importance sampling
  • Importance sampling
    • diff --git a/doc/pub/week3/html/week3-reveal.html b/doc/pub/week3/html/week3-reveal.html index 7edc3666..fd740c1e 100644 --- a/doc/pub/week3/html/week3-reveal.html +++ b/doc/pub/week3/html/week3-reveal.html @@ -341,7 +341,13 @@

    Importance sampling

    Code example for the interacting case with importance sampling

    -

    We are now ready to implement importance sampling. This is done here for the two-electron case with the Coulomb interaction, as in the previous example. We have two variational parameters \( \alpha \) and \( \beta \). After the set up of files

    +Note: this is best seen using the Jupyter-notebook. + +

    We are now ready to implement importance sampling. This is done here +for the two-electron case with the Coulomb interaction, as in the +previous example. We have two variational parameters \( \alpha \) and +\( \beta \). After the set up of files +

    @@ -600,11 +606,9 @@

    Code exa

    -

    Importance sampling, program elements

    -
    - -

    -

    The general derivative formula of the Jastrow factor is (the subscript \( C \) stands for Correlation)

    +

    Importance sampling, programming elements

    + +

    The general derivative formula of the Jastrow factor (or the ansatz for the correlated part of the wave function) is (the subscript \( C \) stands for Correlation)

     
    $$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = @@ -626,14 +630,11 @@

    Importance sampling, program eleme

    the gradient needed for the quantum force and local energy is easy to compute. The function \( f(r_{ij}) \) will depends on the system under study. In the equations below we will keep this general form.

    -

    Importance sampling, program elements

    -
    - -

    +

    In the Metropolis/Hasting algorithm, the acceptance ratio determines the probability for a particle to be accepted at a new position. The ratio of the trial wave functions evaluated at the new and current positions is given by (\( OB \) for the onebody part)

     
    $$ @@ -652,15 +653,12 @@

    Importance sampling, program eleme \frac{\mathbf{\mathbf{\nabla}} \Psi}{\Psi} = \frac{\mathbf{\nabla} (\Psi_{OB} \, \Psi_{C})}{\Psi_{OB} \, \Psi_{C}} = \frac{ \Psi_C \mathbf{\nabla} \Psi_{OB} + \Psi_{OB} \mathbf{\nabla} \Psi_{C}}{\Psi_{OB} \Psi_{C}} = \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} $$

     
    -

    Importance sampling

    -
    - -

    -

    The expectation value of the kinetic energy expressed in atomic units for electron \( i \) is

    + +

    The expectation value of the kinetic energy expressed in scaled units for particle \( i \) is

     
    $$ \langle \hat{K}_i \rangle = -\frac{1}{2}\frac{\langle\Psi|\mathbf{\nabla}_{i}^2|\Psi \rangle}{\langle\Psi|\Psi \rangle}, @@ -672,14 +670,11 @@

    Importance sampling

    \hat{K}_i = -\frac{1}{2}\frac{\mathbf{\nabla}_{i}^{2} \Psi}{\Psi}. $$

     
    -

    Importance sampling

    -
    - -

    +

    The second derivative which enters the definition of the local energy is

     
    $$ @@ -687,15 +682,12 @@

    Importance sampling

    $$

     
    -

    We discuss here how to calculate these quantities in an optimal way,

    -
    +

    We discuss here how to calculate these quantities in an optimal way.

    Importance sampling

    -
    - -

    +

    We have defined the correlated function as

     
    $$ @@ -714,14 +706,11 @@

    Importance sampling

    \Psi_C=\prod_{i < j}g(r_{ij})=\exp{\left\{\sum_{i < j}f(r_{ij})\right\}}. $$

     
    -

    Importance sampling

    -
    - -

    +

    The total number of different relative distances \( r_{ij} \) is \( N(N-1)/2 \). In a matrix storage format, the relative distances form a strictly upper triangular matrix

     
    $$ @@ -738,14 +727,11 @@

    Importance sampling

    This applies to \( \mathbf{g} = \mathbf{g}(r_{ij}) \) as well.

    In our algorithm we will move one particle at the time, say the \( kth \)-particle. This sampling will be seen to be particularly efficient when we are going to compute a Slater determinant.

    -

    Importance sampling

    -
    - -

    +

    We have that the ratio between Jastrow factors \( R_C \) is given by

     
    $$ @@ -772,14 +758,11 @@

    Importance sampling

    \sum_{i=k+1}^{N}\big(f_{ki}^\mathrm{new}-f_{ki}^\mathrm{cur}\big) $$

     
    -

    Importance sampling

    -
    - -

    +

    One needs to develop a special algorithm that runs only through the elements of the upper triangular matrix \( \mathbf{g} \) and have \( k \) as an index. @@ -793,14 +776,11 @@

    Importance sampling

     

    for all dimensions and with \( i \) running over all particles.

    -

    Importance sampling

    -
    - -

    +

    For the first derivative only \( N-1 \) terms survive the ratio because the \( g \)-terms that are not differentiated cancel with their corresponding ones in the denominator. Then,

     
    $$ @@ -822,14 +802,10 @@

    Importance sampling

     

    with both expressions scaling as \( \mathcal{O}(N) \).

    -

    Importance sampling

    -
    - -

    Using the identity

     
    @@ -855,14 +831,11 @@

    Importance sampling

    -\sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i}. $$

     
    -

    Importance sampling

    -
    - -

    +

    For correlation forms depending only on the scalar distances \( r_{ij} \) we can use the chain rule. Noting that

     
    $$ @@ -878,14 +851,11 @@

    Importance sampling

    -\sum_{i=k+1}^{N}\frac{1}{g_{ki}}\frac{\mathbf{r_{ki}}}{r_{ki}}\frac{\partial g_{ki}}{\partial r_{ki}}. $$

     
    -

    Importance sampling

    -
    - -

    +

    Note that for the Pade-Jastrow form we can set \( g_{ij} \equiv g(r_{ij}) = e^{f(r_{ij})} = e^{f_{ij}} \) and

     
    $$ @@ -910,14 +880,11 @@

    Importance sampling

     

    is the relative distance.

    -

    Importance sampling

    -
    - -

    +

    The second derivative of the Jastrow factor divided by the Jastrow factor (the way it enters the kinetic energy) is

     
    $$ @@ -931,14 +898,11 @@

    Importance sampling

    \right)^2 $$

     
    -

    Importance sampling

    -
    - -

    +

    But we have a simple form for the function, namely

     
    $$ @@ -956,75 +920,64 @@

    Importance sampling

    \sum_{j\ne k}\left( f''(r_{kj})+\frac{2}{r_{kj}}f'(r_{kj})\right) $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    A stochastic process is simply a function of two variables, one is the time, -the other is a stochastic variable \( X \), defined by specifying + +

    A stochastic process is simply a function of two variables, one is the +time, the other is a stochastic variable \( X \), defined by specifying

    -
      + +

      1. the set \( \left\{x\right\} \) of possible values for \( X \);

      2. the probability distribution, \( w_X(x) \), over this set, or briefly \( w(x) \)
        3. -
    +

    -

    The set of values \( \left\{x\right\} \) for \( X \) -may be discrete, or continuous. If the set of -values is continuous, then \( w_X (x) \) is a probability density so that -\( w_X (x)dx \) -is the probability that one finds the stochastic variable \( X \) to have values -in the range \( [x, x + dx] \) . +

    The set of values \( \left\{x\right\} \) for \( X \) may be discrete, or +continuous. If the set of values is continuous, then \( w_X (x) \) is a +probability density so that \( w_X (x)dx \) is the probability that one +finds the stochastic variable \( X \) to have values in the range \( [x, x +dx] \) .

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    An arbitrary number of other stochastic variables may be derived from -\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a stochastic -variable. The mapping may also be time-dependent, that is, the mapping -depends on an additional variable \( t \) + +

    An arbitrary number of other stochastic variables may be derived from +\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a +stochastic variable. The mapping may also be time-dependent, that is, +the mapping depends on an additional variable \( t \)

     
    -$$ - Y_X (t) = f (X, t) . +$$ Y_X (t) = f(X, t). $$

     
    -

    The quantity \( Y_X (t) \) is called a random function, or, since \( t \) often is time, -a stochastic process. A stochastic process is a function of two variables, -one is the time, the other is a stochastic variable \( X \). Let \( x \) be one of the -possible values of \( X \) then +

    The quantity \( Y_X (t) \) is called a random function, +or, since \( t \) often is time, a stochastic process. A stochastic +process is a function of two variables, one is the time, the other is +a stochastic variable \( X \). Let \( x \) be one of the possible values of +\( X \) then

     
    -$$ - y(t) = f (x, t), -$$ +$$ y(t) = f (x, t), $$

     
    -

    is a function of \( t \), called a sample function or realization of the process. -In physics one considers the stochastic process to be an ensemble of such -sample functions. +

    is a function of \( t \), called a +sample function or realization of the process. In physics one +considers the stochastic process to be an ensemble of such sample +functions.

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    For many physical systems initial distributions of a stochastic -variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow w_0(y) \) -as \( t\rightarrow\infty \). In -equilibrium detailed balance constrains the transition rates + +

    For many physical systems initial distributions of a stochastic +variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow +w_0(y) \) as \( t\rightarrow\infty \). In equilibrium detailed balance +constrains the transition rates

     
    $$ @@ -1032,33 +985,28 @@

    Importance sam $$

     
    -

    where \( W(y'\rightarrow y) \) -is the probability, per unit time, that the system changes -from a state \( |y\rangle \) , characterized by the value \( y \) -for the stochastic variable \( Y \) , to a state \( |y'\rangle \). +

    where \( W(y'\rightarrow y) \) is the probability, per unit time, that the +system changes from a state \( |y\rangle \) , characterized by the value +\( y \) for the stochastic variable \( Y \) , to a state \( |y'\rangle \).

    -

    Note that for a system in equilibrium the transition rate -\( W(y'\rightarrow y) \) and -the reverse \( W(y\rightarrow y') \) may be very different. +

    Note that for a system in equilibrium the transition rate +\( W(y'\rightarrow y) \) and the reverse \( W(y\rightarrow y') \) may be very +different.

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    Consider, for instance, a simple -system that has only two energy levels \( \epsilon_0 = 0 \) and -\( \epsilon_1 = \Delta E \). + +

    Consider, for instance, a simple system that has only two energy +levels \( \epsilon_0 = 0 \) and \( \epsilon_1 = \Delta E \).

    For a system governed by the Boltzmann distribution we find (the partition function has been taken out)

     
    $$ - W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)} + W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)}. $$

     
    @@ -1070,14 +1018,11 @@

    Importance sam

     

    which goes to zero when \( T \) tends to zero.

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    If we assume a discrete set of events, our initial probability distribution function can be given by @@ -1106,14 +1051,11 @@

    Importance sam vector \( \mathbf{x} \). The Kroenecker \( \delta \) function is replaced by the \( \delta \) distribution function \( \delta(\mathbf{x}) \) at \( t=0 \).

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    The transition from a state \( j \) to a state \( i \) is now replaced by a transition to a state with position \( \mathbf{y} \) from a state with position \( \mathbf{x} \). The discrete sum of transition probabilities can then be replaced by an integral @@ -1140,14 +1082,11 @@

    Importance sam

     

    that is no time-dependence. Note our change of notation for \( W \)

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can solve the equation for \( w(\mathbf{y},t) \) by making a Fourier transform to momentum space. The PDF \( w(\mathbf{x},t) \) is related to its Fourier transform @@ -1174,14 +1113,11 @@

    Importance sam \tilde{w}(\mathbf{k},0)=1/2\pi. $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can then use the Fourier-transformed diffusion equation

     
    $$ @@ -1196,14 +1132,11 @@

    Importance sam \frac{1}{2\pi}\exp{\left[-(D\mathbf{k}^2t)\right]}. $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    With the Fourier transform we obtain

     
    $$ @@ -1218,14 +1151,11 @@

    Importance sam \int_{-\infty}^{\infty}w(\mathbf{x},t)d\mathbf{x}=1. $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    The solution represents the probability of finding our random walker at position \( \mathbf{x} \) at time \( t \) if the initial distribution was placed at \( \mathbf{x}=0 \) at \( t=0 \). @@ -1247,14 +1177,11 @@

    Importance sam

    and that it satisfies the normalization condition and is itself a solution to the diffusion equation.

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    Let us now assume that we have three PDFs for times \( t_0 < t' < t \), that is \( w(\mathbf{x}_0,t_0) \), \( w(\mathbf{x}',t') \) and \( w(\mathbf{x},t) \). We have then @@ -1278,14 +1205,11 @@

    Importance sam w(\mathbf{x}',t')= \int_{-\infty}^{\infty} W(\mathbf{x}'.t'|\mathbf{x}_0,t_0)w(\mathbf{x}_0,t_0)d\mathbf{x}_0. $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can combine these equations and arrive at the famous Einstein-Smoluchenski-Kolmogorov-Chapman (ESKC) relation

     
    $$ @@ -1301,14 +1225,11 @@

    Importance sam W(\mathbf{v},t|\mathbf{v}_0,t_0) = \int_{-\infty}^{\infty} W(\mathbf{v},t|\mathbf{v}',t')W(\mathbf{v}',t'|\mathbf{v}_0,t_0)d\mathbf{x}'. $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We will now derive the Fokker-Planck equation. We start from the ESKC equation

    @@ -1324,14 +1245,11 @@

    Importance sam W(\mathbf{x},s+\tau|\mathbf{x}_0) = \int_{-\infty}^{\infty} W(\mathbf{x},\tau|\mathbf{x}')W(\mathbf{x}',s|\mathbf{x}_0)d\mathbf{x}'. $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    Assume now that \( \tau \) is very small so that we can make an expansion in terms of a small step \( xi \), with \( \mathbf{x}'=\mathbf{x}-\xi \), that is

     
    $$ @@ -1340,14 +1258,11 @@

    Importance sam

     

    We assume that \( W(\mathbf{x},\tau|\mathbf{x}-\xi) \) takes non-negligible values only when \( \xi \) is small. This is just another way of stating the Master equation!!

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -1359,14 +1274,11 @@

    Importance sam \right]. $$

     
    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can then rewrite the ESKC equation as

     
    $$ @@ -1379,14 +1291,11 @@

    Importance sam

    We have neglected higher powers of \( \tau \) and have used that for \( n=0 \) we get simply \( W(\mathbf{x},s|\mathbf{x}_0) \) due to normalization.

    -

    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -1398,7 +1307,6 @@

    Importance sam \right]. $$

     
    -

    diff --git a/doc/pub/week3/html/week3-solarized.html b/doc/pub/week3/html/week3-solarized.html index c785d65f..4154629a 100644 --- a/doc/pub/week3/html/week3-solarized.html +++ b/doc/pub/week3/html/week3-solarized.html @@ -79,10 +79,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -375,7 +375,13 @@

    Importance sampling











    Code example for the interacting case with importance sampling

    -

    We are now ready to implement importance sampling. This is done here for the two-electron case with the Coulomb interaction, as in the previous example. We have two variational parameters \( \alpha \) and \( \beta \). After the set up of files

    +Note: this is best seen using the Jupyter-notebook. + +

    We are now ready to implement importance sampling. This is done here +for the two-electron case with the Coulomb interaction, as in the +previous example. We have two variational parameters \( \alpha \) and +\( \beta \). After the set up of files +

    @@ -634,11 +640,9 @@

    Code exa









    -

    Importance sampling, program elements

    -
    - -

    -

    The general derivative formula of the Jastrow factor is (the subscript \( C \) stands for Correlation)

    +

    Importance sampling, programming elements

    + +

    The general derivative formula of the Jastrow factor (or the ansatz for the correlated part of the wave function) is (the subscript \( C \) stands for Correlation)

    $$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} @@ -656,14 +660,10 @@

    Importance sampling, program eleme

    the gradient needed for the quantum force and local energy is easy to compute. The function \( f(r_{ij}) \) will depends on the system under study. In the equations below we will keep this general form.

    -

    -









    Importance sampling, program elements

    -
    - -

    +

    In the Metropolis/Hasting algorithm, the acceptance ratio determines the probability for a particle to be accepted at a new position. The ratio of the trial wave functions evaluated at the new and current positions is given by (\( OB \) for the onebody part)

    $$ R \equiv \frac{\Psi_{T}^{new}}{\Psi_{T}^{old}} = @@ -678,15 +678,12 @@

    Importance sampling, program eleme $$ \frac{\mathbf{\mathbf{\nabla}} \Psi}{\Psi} = \frac{\mathbf{\nabla} (\Psi_{OB} \, \Psi_{C})}{\Psi_{OB} \, \Psi_{C}} = \frac{ \Psi_C \mathbf{\nabla} \Psi_{OB} + \Psi_{OB} \mathbf{\nabla} \Psi_{C}}{\Psi_{OB} \Psi_{C}} = \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} $$ -











    Importance sampling

    -
    - -

    -

    The expectation value of the kinetic energy expressed in atomic units for electron \( i \) is

    + +

    The expectation value of the kinetic energy expressed in scaled units for particle \( i \) is

    $$ \langle \hat{K}_i \rangle = -\frac{1}{2}\frac{\langle\Psi|\mathbf{\nabla}_{i}^2|\Psi \rangle}{\langle\Psi|\Psi \rangle}, $$ @@ -694,27 +691,21 @@

    Importance sampling

    $$ \hat{K}_i = -\frac{1}{2}\frac{\mathbf{\nabla}_{i}^{2} \Psi}{\Psi}. $$ -










    Importance sampling

    -
    - -

    +

    The second derivative which enters the definition of the local energy is

    $$ \frac{\mathbf{\nabla}^2 \Psi}{\Psi}=\frac{\mathbf{\nabla}^2 \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla}^2 \Psi_C}{ \Psi_C} + 2 \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}}\cdot\frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} $$ -

    We discuss here how to calculate these quantities in an optimal way,

    -
    +

    We discuss here how to calculate these quantities in an optimal way.











    Importance sampling

    -
    - -

    +

    We have defined the correlated function as

    $$ \Psi_C=\prod_{i < j}g(r_{ij})=\prod_{i < j}^Ng(r_{ij})= \prod_{i=1}^N\prod_{j=i+1}^Ng(r_{ij}), @@ -729,14 +720,11 @@

    Importance sampling

    $$ \Psi_C=\prod_{i < j}g(r_{ij})=\exp{\left\{\sum_{i < j}f(r_{ij})\right\}}. $$ -










    Importance sampling

    -
    - -

    +

    The total number of different relative distances \( r_{ij} \) is \( N(N-1)/2 \). In a matrix storage format, the relative distances form a strictly upper triangular matrix

    $$ \mathbf{r} \equiv \begin{pmatrix} @@ -751,13 +739,10 @@

    Importance sampling

    This applies to \( \mathbf{g} = \mathbf{g}(r_{ij}) \) as well.

    In our algorithm we will move one particle at the time, say the \( kth \)-particle. This sampling will be seen to be particularly efficient when we are going to compute a Slater determinant.

    -










    Importance sampling

    -
    - -

    +

    We have that the ratio between Jastrow factors \( R_C \) is given by

    $$ R_{C} = \frac{\Psi_{C}^\mathrm{new}}{\Psi_{C}^\mathrm{cur}} = @@ -778,14 +763,11 @@

    Importance sampling

    + \sum_{i=k+1}^{N}\big(f_{ki}^\mathrm{new}-f_{ki}^\mathrm{cur}\big) $$ -










    Importance sampling

    -
    - -

    +

    One needs to develop a special algorithm that runs only through the elements of the upper triangular matrix \( \mathbf{g} \) and have \( k \) as an index. @@ -797,13 +779,10 @@

    Importance sampling

    $$

    for all dimensions and with \( i \) running over all particles.

    -










    Importance sampling

    -
    - -

    +

    For the first derivative only \( N-1 \) terms survive the ratio because the \( g \)-terms that are not differentiated cancel with their corresponding ones in the denominator. Then,

    $$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = @@ -821,14 +800,9 @@

    Importance sampling

    $$

    with both expressions scaling as \( \mathcal{O}(N) \).

    -
    -









    Importance sampling

    -
    - -

    Using the identity

    $$ @@ -848,14 +822,11 @@

    Importance sampling

    \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} -\sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i}. $$ -










    Importance sampling

    -
    - -

    +

    For correlation forms depending only on the scalar distances \( r_{ij} \) we can use the chain rule. Noting that

    $$ \frac{\partial g_{ij}}{\partial x_j} = \frac{\partial g_{ij}}{\partial r_{ij}} \frac{\partial r_{ij}}{\partial x_j} = \frac{x_j - x_i}{r_{ij}} \frac{\partial g_{ij}}{\partial r_{ij}}, @@ -867,14 +838,11 @@

    Importance sampling

    \sum_{i=1}^{k-1}\frac{1}{g_{ik}} \frac{\mathbf{r_{ik}}}{r_{ik}} \frac{\partial g_{ik}}{\partial r_{ik}} -\sum_{i=k+1}^{N}\frac{1}{g_{ki}}\frac{\mathbf{r_{ki}}}{r_{ki}}\frac{\partial g_{ki}}{\partial r_{ki}}. $$ -










    Importance sampling

    -
    - -

    +

    Note that for the Pade-Jastrow form we can set \( g_{ij} \equiv g(r_{ij}) = e^{f(r_{ij})} = e^{f_{ij}} \) and

    $$ \frac{\partial g_{ij}}{\partial r_{ij}} = g_{ij} \frac{\partial f_{ij}}{\partial r_{ij}}. @@ -893,14 +861,10 @@

    Importance sampling

    $$

    is the relative distance.

    -
    -









    Importance sampling

    -
    - -

    +

    The second derivative of the Jastrow factor divided by the Jastrow factor (the way it enters the kinetic energy) is

    $$ \left[\frac{\mathbf{\nabla}^2 \Psi_C}{\Psi_C}\right]_x =\ @@ -912,14 +876,11 @@

    Importance sampling

    \sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i} \right)^2 $$ -










    Importance sampling

    -
    - -

    +

    But we have a simple form for the function, namely

    $$ \Psi_{C}=\prod_{i < j}\exp{f(r_{ij})}, @@ -933,101 +894,82 @@

    Importance sampling

    \sum_{ij\ne k}\frac{(\mathbf{r}_k-\mathbf{r}_i)(\mathbf{r}_k-\mathbf{r}_j)}{r_{ki}r_{kj}}f'(r_{ki})f'(r_{kj})+ \sum_{j\ne k}\left( f''(r_{kj})+\frac{2}{r_{kj}}f'(r_{kj})\right) $$ -










    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    A stochastic process is simply a function of two variables, one is the time, -the other is a stochastic variable \( X \), defined by specifying + +

    A stochastic process is simply a function of two variables, one is the +time, the other is a stochastic variable \( X \), defined by specifying

    -
      + +
      1. the set \( \left\{x\right\} \) of possible values for \( X \);
      2. the probability distribution, \( w_X(x) \), over this set, or briefly \( w(x) \)
        3. -
    -

    The set of values \( \left\{x\right\} \) for \( X \) -may be discrete, or continuous. If the set of -values is continuous, then \( w_X (x) \) is a probability density so that -\( w_X (x)dx \) -is the probability that one finds the stochastic variable \( X \) to have values -in the range \( [x, x + dx] \) . + +

    The set of values \( \left\{x\right\} \) for \( X \) may be discrete, or +continuous. If the set of values is continuous, then \( w_X (x) \) is a +probability density so that \( w_X (x)dx \) is the probability that one +finds the stochastic variable \( X \) to have values in the range \( [x, x +dx] \) .

    -
    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    An arbitrary number of other stochastic variables may be derived from -\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a stochastic -variable. The mapping may also be time-dependent, that is, the mapping -depends on an additional variable \( t \) -

    -$$ - Y_X (t) = f (X, t) . -$$ -

    The quantity \( Y_X (t) \) is called a random function, or, since \( t \) often is time, -a stochastic process. A stochastic process is a function of two variables, -one is the time, the other is a stochastic variable \( X \). Let \( x \) be one of the -possible values of \( X \) then +

    An arbitrary number of other stochastic variables may be derived from +\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a +stochastic variable. The mapping may also be time-dependent, that is, +the mapping depends on an additional variable \( t \)

    -$$ - y(t) = f (x, t), +$$ Y_X (t) = f(X, t). $$ -

    is a function of \( t \), called a sample function or realization of the process. -In physics one considers the stochastic process to be an ensemble of such -sample functions. +

    The quantity \( Y_X (t) \) is called a random function, +or, since \( t \) often is time, a stochastic process. A stochastic +process is a function of two variables, one is the time, the other is +a stochastic variable \( X \). Let \( x \) be one of the possible values of +\( X \) then

    -
    +$$ y(t) = f (x, t), $$ +

    is a function of \( t \), called a +sample function or realization of the process. In physics one +considers the stochastic process to be an ensemble of such sample +functions. +











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    For many physical systems initial distributions of a stochastic -variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow w_0(y) \) -as \( t\rightarrow\infty \). In -equilibrium detailed balance constrains the transition rates + +

    For many physical systems initial distributions of a stochastic +variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow +w_0(y) \) as \( t\rightarrow\infty \). In equilibrium detailed balance +constrains the transition rates

    $$ W(y\rightarrow y')w(y ) = W(y'\rightarrow y)w_0 (y), $$ -

    where \( W(y'\rightarrow y) \) -is the probability, per unit time, that the system changes -from a state \( |y\rangle \) , characterized by the value \( y \) -for the stochastic variable \( Y \) , to a state \( |y'\rangle \). +

    where \( W(y'\rightarrow y) \) is the probability, per unit time, that the +system changes from a state \( |y\rangle \) , characterized by the value +\( y \) for the stochastic variable \( Y \) , to a state \( |y'\rangle \).

    -

    Note that for a system in equilibrium the transition rate -\( W(y'\rightarrow y) \) and -the reverse \( W(y\rightarrow y') \) may be very different. +

    Note that for a system in equilibrium the transition rate +\( W(y'\rightarrow y) \) and the reverse \( W(y\rightarrow y') \) may be very +different.

    -
    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    Consider, for instance, a simple -system that has only two energy levels \( \epsilon_0 = 0 \) and -\( \epsilon_1 = \Delta E \). + +

    Consider, for instance, a simple system that has only two energy +levels \( \epsilon_0 = 0 \) and \( \epsilon_1 = \Delta E \).

    For a system governed by the Boltzmann distribution we find (the partition function has been taken out)

    $$ - W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)} + W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)}. $$

    We get then

    @@ -1036,14 +978,10 @@

    Importance sam $$

    which goes to zero when \( T \) tends to zero.

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    If we assume a discrete set of events, our initial probability distribution function can be given by @@ -1066,14 +1004,10 @@

    Importance sam vector \( \mathbf{x} \). The Kroenecker \( \delta \) function is replaced by the \( \delta \) distribution function \( \delta(\mathbf{x}) \) at \( t=0 \).

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    The transition from a state \( j \) to a state \( i \) is now replaced by a transition to a state with position \( \mathbf{y} \) from a state with position \( \mathbf{x} \). The discrete sum of transition probabilities can then be replaced by an integral @@ -1094,14 +1028,10 @@

    Importance sam $$

    that is no time-dependence. Note our change of notation for \( W \)

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can solve the equation for \( w(\mathbf{y},t) \) by making a Fourier transform to momentum space. The PDF \( w(\mathbf{x},t) \) is related to its Fourier transform @@ -1122,14 +1052,11 @@

    Importance sam $$ \tilde{w}(\mathbf{k},0)=1/2\pi. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can then use the Fourier-transformed diffusion equation

    $$ \frac{\partial \tilde{w}(\mathbf{k},t)}{\partial t} = -D\mathbf{k}^2\tilde{w}(\mathbf{k},t), @@ -1140,14 +1067,11 @@

    Importance sam \tilde{w}(\mathbf{k},t)=\tilde{w}(\mathbf{k},0)\exp{\left[-(D\mathbf{k}^2t)\right)}= \frac{1}{2\pi}\exp{\left[-(D\mathbf{k}^2t)\right]}. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    With the Fourier transform we obtain

    $$ w(\mathbf{x},t)=\int_{-\infty}^{\infty}d\mathbf{k} \exp{\left[i\mathbf{kx}\right]}\frac{1}{2\pi}\exp{\left[-(D\mathbf{k}^2t)\right]}= @@ -1158,14 +1082,11 @@

    Importance sam $$ \int_{-\infty}^{\infty}w(\mathbf{x},t)d\mathbf{x}=1. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    The solution represents the probability of finding our random walker at position \( \mathbf{x} \) at time \( t \) if the initial distribution was placed at \( \mathbf{x}=0 \) at \( t=0 \). @@ -1185,14 +1106,10 @@

    Importance sam

    and that it satisfies the normalization condition and is itself a solution to the diffusion equation.

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    Let us now assume that we have three PDFs for times \( t_0 < t' < t \), that is \( w(\mathbf{x}_0,t_0) \), \( w(\mathbf{x}',t') \) and \( w(\mathbf{x},t) \). We have then @@ -1210,14 +1127,11 @@

    Importance sam $$ w(\mathbf{x}',t')= \int_{-\infty}^{\infty} W(\mathbf{x}'.t'|\mathbf{x}_0,t_0)w(\mathbf{x}_0,t_0)d\mathbf{x}_0. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can combine these equations and arrive at the famous Einstein-Smoluchenski-Kolmogorov-Chapman (ESKC) relation

    $$ W(\mathbf{x}t|\mathbf{x}_0t_0) = \int_{-\infty}^{\infty} W(\mathbf{x},t|\mathbf{x}',t')W(\mathbf{x}',t'|\mathbf{x}_0,t_0)d\mathbf{x}'. @@ -1229,14 +1143,11 @@

    Importance sam $$ W(\mathbf{v},t|\mathbf{v}_0,t_0) = \int_{-\infty}^{\infty} W(\mathbf{v},t|\mathbf{v}',t')W(\mathbf{v}',t'|\mathbf{v}_0,t_0)d\mathbf{x}'. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We will now derive the Fokker-Planck equation. We start from the ESKC equation

    @@ -1248,28 +1159,21 @@

    Importance sam $$ W(\mathbf{x},s+\tau|\mathbf{x}_0) = \int_{-\infty}^{\infty} W(\mathbf{x},\tau|\mathbf{x}')W(\mathbf{x}',s|\mathbf{x}_0)d\mathbf{x}'. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    Assume now that \( \tau \) is very small so that we can make an expansion in terms of a small step \( xi \), with \( \mathbf{x}'=\mathbf{x}-\xi \), that is

    $$ W(\mathbf{x},s|\mathbf{x}_0)+\frac{\partial W}{\partial s}\tau +O(\tau^2) = \int_{-\infty}^{\infty} W(\mathbf{x},\tau|\mathbf{x}-\xi)W(\mathbf{x}-\xi,s|\mathbf{x}_0)d\mathbf{x}'. $$

    We assume that \( W(\mathbf{x},\tau|\mathbf{x}-\xi) \) takes non-negligible values only when \( \xi \) is small. This is just another way of stating the Master equation!!

    -
    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -1279,14 +1183,11 @@

    Importance sam \sum_{n=0}^{\infty}\frac{(-\xi)^n}{n!}\frac{\partial^n}{\partial x^n}\left[W(\mathbf{x}+\xi,\tau|\mathbf{x})W(\mathbf{x},s|\mathbf{x}_0) \right]. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can then rewrite the ESKC equation as

    $$ \frac{\partial W}{\partial s}\tau=-W(\mathbf{x},s|\mathbf{x}_0)+ @@ -1297,14 +1198,10 @@

    Importance sam

    We have neglected higher powers of \( \tau \) and have used that for \( n=0 \) we get simply \( W(\mathbf{x},s|\mathbf{x}_0) \) due to normalization.

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -1314,7 +1211,6 @@

    Importance sam \sum_{n=0}^{\infty}\frac{(-\xi)^n}{n!}\frac{\partial^n}{\partial x^n}\left[W(\mathbf{x}+\xi,\tau|\mathbf{x})W(\mathbf{x},s|\mathbf{x}_0) \right]. $$ -











    diff --git a/doc/pub/week3/html/week3.html b/doc/pub/week3/html/week3.html index 8277d863..b5e1af69 100644 --- a/doc/pub/week3/html/week3.html +++ b/doc/pub/week3/html/week3.html @@ -156,10 +156,10 @@ 2, None, 'code-example-for-the-interacting-case-with-importance-sampling'), - ('Importance sampling, program elements', + ('Importance sampling, programming elements', 2, None, - 'importance-sampling-program-elements'), + 'importance-sampling-programming-elements'), ('Importance sampling, program elements', 2, None, @@ -452,7 +452,13 @@

    Importance sampling











    Code example for the interacting case with importance sampling

    -

    We are now ready to implement importance sampling. This is done here for the two-electron case with the Coulomb interaction, as in the previous example. We have two variational parameters \( \alpha \) and \( \beta \). After the set up of files

    +Note: this is best seen using the Jupyter-notebook. + +

    We are now ready to implement importance sampling. This is done here +for the two-electron case with the Coulomb interaction, as in the +previous example. We have two variational parameters \( \alpha \) and +\( \beta \). After the set up of files +

    @@ -711,11 +717,9 @@

    Code exa









    -

    Importance sampling, program elements

    -
    - -

    -

    The general derivative formula of the Jastrow factor is (the subscript \( C \) stands for Correlation)

    +

    Importance sampling, programming elements

    + +

    The general derivative formula of the Jastrow factor (or the ansatz for the correlated part of the wave function) is (the subscript \( C \) stands for Correlation)

    $$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} @@ -733,14 +737,10 @@

    Importance sampling, program eleme

    the gradient needed for the quantum force and local energy is easy to compute. The function \( f(r_{ij}) \) will depends on the system under study. In the equations below we will keep this general form.

    -

    -









    Importance sampling, program elements

    -
    - -

    +

    In the Metropolis/Hasting algorithm, the acceptance ratio determines the probability for a particle to be accepted at a new position. The ratio of the trial wave functions evaluated at the new and current positions is given by (\( OB \) for the onebody part)

    $$ R \equiv \frac{\Psi_{T}^{new}}{\Psi_{T}^{old}} = @@ -755,15 +755,12 @@

    Importance sampling, program eleme $$ \frac{\mathbf{\mathbf{\nabla}} \Psi}{\Psi} = \frac{\mathbf{\nabla} (\Psi_{OB} \, \Psi_{C})}{\Psi_{OB} \, \Psi_{C}} = \frac{ \Psi_C \mathbf{\nabla} \Psi_{OB} + \Psi_{OB} \mathbf{\nabla} \Psi_{C}}{\Psi_{OB} \Psi_{C}} = \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} $$ -











    Importance sampling

    -
    - -

    -

    The expectation value of the kinetic energy expressed in atomic units for electron \( i \) is

    + +

    The expectation value of the kinetic energy expressed in scaled units for particle \( i \) is

    $$ \langle \hat{K}_i \rangle = -\frac{1}{2}\frac{\langle\Psi|\mathbf{\nabla}_{i}^2|\Psi \rangle}{\langle\Psi|\Psi \rangle}, $$ @@ -771,27 +768,21 @@

    Importance sampling

    $$ \hat{K}_i = -\frac{1}{2}\frac{\mathbf{\nabla}_{i}^{2} \Psi}{\Psi}. $$ -










    Importance sampling

    -
    - -

    +

    The second derivative which enters the definition of the local energy is

    $$ \frac{\mathbf{\nabla}^2 \Psi}{\Psi}=\frac{\mathbf{\nabla}^2 \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla}^2 \Psi_C}{ \Psi_C} + 2 \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}}\cdot\frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} $$ -

    We discuss here how to calculate these quantities in an optimal way,

    -
    +

    We discuss here how to calculate these quantities in an optimal way.











    Importance sampling

    -
    - -

    +

    We have defined the correlated function as

    $$ \Psi_C=\prod_{i < j}g(r_{ij})=\prod_{i < j}^Ng(r_{ij})= \prod_{i=1}^N\prod_{j=i+1}^Ng(r_{ij}), @@ -806,14 +797,11 @@

    Importance sampling

    $$ \Psi_C=\prod_{i < j}g(r_{ij})=\exp{\left\{\sum_{i < j}f(r_{ij})\right\}}. $$ -










    Importance sampling

    -
    - -

    +

    The total number of different relative distances \( r_{ij} \) is \( N(N-1)/2 \). In a matrix storage format, the relative distances form a strictly upper triangular matrix

    $$ \mathbf{r} \equiv \begin{pmatrix} @@ -828,13 +816,10 @@

    Importance sampling

    This applies to \( \mathbf{g} = \mathbf{g}(r_{ij}) \) as well.

    In our algorithm we will move one particle at the time, say the \( kth \)-particle. This sampling will be seen to be particularly efficient when we are going to compute a Slater determinant.

    -










    Importance sampling

    -
    - -

    +

    We have that the ratio between Jastrow factors \( R_C \) is given by

    $$ R_{C} = \frac{\Psi_{C}^\mathrm{new}}{\Psi_{C}^\mathrm{cur}} = @@ -855,14 +840,11 @@

    Importance sampling

    + \sum_{i=k+1}^{N}\big(f_{ki}^\mathrm{new}-f_{ki}^\mathrm{cur}\big) $$ -










    Importance sampling

    -
    - -

    +

    One needs to develop a special algorithm that runs only through the elements of the upper triangular matrix \( \mathbf{g} \) and have \( k \) as an index. @@ -874,13 +856,10 @@

    Importance sampling

    $$

    for all dimensions and with \( i \) running over all particles.

    -










    Importance sampling

    -
    - -

    +

    For the first derivative only \( N-1 \) terms survive the ratio because the \( g \)-terms that are not differentiated cancel with their corresponding ones in the denominator. Then,

    $$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = @@ -898,14 +877,9 @@

    Importance sampling

    $$

    with both expressions scaling as \( \mathcal{O}(N) \).

    -
    -









    Importance sampling

    -
    - -

    Using the identity

    $$ @@ -925,14 +899,11 @@

    Importance sampling

    \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} -\sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i}. $$ -










    Importance sampling

    -
    - -

    +

    For correlation forms depending only on the scalar distances \( r_{ij} \) we can use the chain rule. Noting that

    $$ \frac{\partial g_{ij}}{\partial x_j} = \frac{\partial g_{ij}}{\partial r_{ij}} \frac{\partial r_{ij}}{\partial x_j} = \frac{x_j - x_i}{r_{ij}} \frac{\partial g_{ij}}{\partial r_{ij}}, @@ -944,14 +915,11 @@

    Importance sampling

    \sum_{i=1}^{k-1}\frac{1}{g_{ik}} \frac{\mathbf{r_{ik}}}{r_{ik}} \frac{\partial g_{ik}}{\partial r_{ik}} -\sum_{i=k+1}^{N}\frac{1}{g_{ki}}\frac{\mathbf{r_{ki}}}{r_{ki}}\frac{\partial g_{ki}}{\partial r_{ki}}. $$ -










    Importance sampling

    -
    - -

    +

    Note that for the Pade-Jastrow form we can set \( g_{ij} \equiv g(r_{ij}) = e^{f(r_{ij})} = e^{f_{ij}} \) and

    $$ \frac{\partial g_{ij}}{\partial r_{ij}} = g_{ij} \frac{\partial f_{ij}}{\partial r_{ij}}. @@ -970,14 +938,10 @@

    Importance sampling

    $$

    is the relative distance.

    -
    -









    Importance sampling

    -
    - -

    +

    The second derivative of the Jastrow factor divided by the Jastrow factor (the way it enters the kinetic energy) is

    $$ \left[\frac{\mathbf{\nabla}^2 \Psi_C}{\Psi_C}\right]_x =\ @@ -989,14 +953,11 @@

    Importance sampling

    \sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i} \right)^2 $$ -










    Importance sampling

    -
    - -

    +

    But we have a simple form for the function, namely

    $$ \Psi_{C}=\prod_{i < j}\exp{f(r_{ij})}, @@ -1010,101 +971,82 @@

    Importance sampling

    \sum_{ij\ne k}\frac{(\mathbf{r}_k-\mathbf{r}_i)(\mathbf{r}_k-\mathbf{r}_j)}{r_{ki}r_{kj}}f'(r_{ki})f'(r_{kj})+ \sum_{j\ne k}\left( f''(r_{kj})+\frac{2}{r_{kj}}f'(r_{kj})\right) $$ -










    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    A stochastic process is simply a function of two variables, one is the time, -the other is a stochastic variable \( X \), defined by specifying + +

    A stochastic process is simply a function of two variables, one is the +time, the other is a stochastic variable \( X \), defined by specifying

    -
      + +
      1. the set \( \left\{x\right\} \) of possible values for \( X \);
      2. the probability distribution, \( w_X(x) \), over this set, or briefly \( w(x) \)
        3. -
    -

    The set of values \( \left\{x\right\} \) for \( X \) -may be discrete, or continuous. If the set of -values is continuous, then \( w_X (x) \) is a probability density so that -\( w_X (x)dx \) -is the probability that one finds the stochastic variable \( X \) to have values -in the range \( [x, x + dx] \) . + +

    The set of values \( \left\{x\right\} \) for \( X \) may be discrete, or +continuous. If the set of values is continuous, then \( w_X (x) \) is a +probability density so that \( w_X (x)dx \) is the probability that one +finds the stochastic variable \( X \) to have values in the range \( [x, x +dx] \) .

    -
    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    An arbitrary number of other stochastic variables may be derived from -\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a stochastic -variable. The mapping may also be time-dependent, that is, the mapping -depends on an additional variable \( t \) -

    -$$ - Y_X (t) = f (X, t) . -$$ -

    The quantity \( Y_X (t) \) is called a random function, or, since \( t \) often is time, -a stochastic process. A stochastic process is a function of two variables, -one is the time, the other is a stochastic variable \( X \). Let \( x \) be one of the -possible values of \( X \) then +

    An arbitrary number of other stochastic variables may be derived from +\( X \). For example, any \( Y \) given by a mapping of \( X \), is also a +stochastic variable. The mapping may also be time-dependent, that is, +the mapping depends on an additional variable \( t \)

    -$$ - y(t) = f (x, t), +$$ Y_X (t) = f(X, t). $$ -

    is a function of \( t \), called a sample function or realization of the process. -In physics one considers the stochastic process to be an ensemble of such -sample functions. +

    The quantity \( Y_X (t) \) is called a random function, +or, since \( t \) often is time, a stochastic process. A stochastic +process is a function of two variables, one is the time, the other is +a stochastic variable \( X \). Let \( x \) be one of the possible values of +\( X \) then

    -
    +$$ y(t) = f (x, t), $$ +

    is a function of \( t \), called a +sample function or realization of the process. In physics one +considers the stochastic process to be an ensemble of such sample +functions. +











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    For many physical systems initial distributions of a stochastic -variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow w_0(y) \) -as \( t\rightarrow\infty \). In -equilibrium detailed balance constrains the transition rates + +

    For many physical systems initial distributions of a stochastic +variable \( y \) tend to equilibrium distributions: \( w(y, t)\rightarrow +w_0(y) \) as \( t\rightarrow\infty \). In equilibrium detailed balance +constrains the transition rates

    $$ W(y\rightarrow y')w(y ) = W(y'\rightarrow y)w_0 (y), $$ -

    where \( W(y'\rightarrow y) \) -is the probability, per unit time, that the system changes -from a state \( |y\rangle \) , characterized by the value \( y \) -for the stochastic variable \( Y \) , to a state \( |y'\rangle \). +

    where \( W(y'\rightarrow y) \) is the probability, per unit time, that the +system changes from a state \( |y\rangle \) , characterized by the value +\( y \) for the stochastic variable \( Y \) , to a state \( |y'\rangle \).

    -

    Note that for a system in equilibrium the transition rate -\( W(y'\rightarrow y) \) and -the reverse \( W(y\rightarrow y') \) may be very different. +

    Note that for a system in equilibrium the transition rate +\( W(y'\rightarrow y) \) and the reverse \( W(y\rightarrow y') \) may be very +different.

    -
    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    -

    Consider, for instance, a simple -system that has only two energy levels \( \epsilon_0 = 0 \) and -\( \epsilon_1 = \Delta E \). + +

    Consider, for instance, a simple system that has only two energy +levels \( \epsilon_0 = 0 \) and \( \epsilon_1 = \Delta E \).

    For a system governed by the Boltzmann distribution we find (the partition function has been taken out)

    $$ - W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)} + W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)}. $$

    We get then

    @@ -1113,14 +1055,10 @@

    Importance sam $$

    which goes to zero when \( T \) tends to zero.

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    If we assume a discrete set of events, our initial probability distribution function can be given by @@ -1143,14 +1081,10 @@

    Importance sam vector \( \mathbf{x} \). The Kroenecker \( \delta \) function is replaced by the \( \delta \) distribution function \( \delta(\mathbf{x}) \) at \( t=0 \).

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    The transition from a state \( j \) to a state \( i \) is now replaced by a transition to a state with position \( \mathbf{y} \) from a state with position \( \mathbf{x} \). The discrete sum of transition probabilities can then be replaced by an integral @@ -1171,14 +1105,10 @@

    Importance sam $$

    that is no time-dependence. Note our change of notation for \( W \)

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can solve the equation for \( w(\mathbf{y},t) \) by making a Fourier transform to momentum space. The PDF \( w(\mathbf{x},t) \) is related to its Fourier transform @@ -1199,14 +1129,11 @@

    Importance sam $$ \tilde{w}(\mathbf{k},0)=1/2\pi. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can then use the Fourier-transformed diffusion equation

    $$ \frac{\partial \tilde{w}(\mathbf{k},t)}{\partial t} = -D\mathbf{k}^2\tilde{w}(\mathbf{k},t), @@ -1217,14 +1144,11 @@

    Importance sam \tilde{w}(\mathbf{k},t)=\tilde{w}(\mathbf{k},0)\exp{\left[-(D\mathbf{k}^2t)\right)}= \frac{1}{2\pi}\exp{\left[-(D\mathbf{k}^2t)\right]}. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    With the Fourier transform we obtain

    $$ w(\mathbf{x},t)=\int_{-\infty}^{\infty}d\mathbf{k} \exp{\left[i\mathbf{kx}\right]}\frac{1}{2\pi}\exp{\left[-(D\mathbf{k}^2t)\right]}= @@ -1235,14 +1159,11 @@

    Importance sam $$ \int_{-\infty}^{\infty}w(\mathbf{x},t)d\mathbf{x}=1. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    The solution represents the probability of finding our random walker at position \( \mathbf{x} \) at time \( t \) if the initial distribution was placed at \( \mathbf{x}=0 \) at \( t=0 \). @@ -1262,14 +1183,10 @@

    Importance sam

    and that it satisfies the normalization condition and is itself a solution to the diffusion equation.

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    Let us now assume that we have three PDFs for times \( t_0 < t' < t \), that is \( w(\mathbf{x}_0,t_0) \), \( w(\mathbf{x}',t') \) and \( w(\mathbf{x},t) \). We have then @@ -1287,14 +1204,11 @@

    Importance sam $$ w(\mathbf{x}',t')= \int_{-\infty}^{\infty} W(\mathbf{x}'.t'|\mathbf{x}_0,t_0)w(\mathbf{x}_0,t_0)d\mathbf{x}_0. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can combine these equations and arrive at the famous Einstein-Smoluchenski-Kolmogorov-Chapman (ESKC) relation

    $$ W(\mathbf{x}t|\mathbf{x}_0t_0) = \int_{-\infty}^{\infty} W(\mathbf{x},t|\mathbf{x}',t')W(\mathbf{x}',t'|\mathbf{x}_0,t_0)d\mathbf{x}'. @@ -1306,14 +1220,11 @@

    Importance sam $$ W(\mathbf{v},t|\mathbf{v}_0,t_0) = \int_{-\infty}^{\infty} W(\mathbf{v},t|\mathbf{v}',t')W(\mathbf{v}',t'|\mathbf{v}_0,t_0)d\mathbf{x}'. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We will now derive the Fokker-Planck equation. We start from the ESKC equation

    @@ -1325,28 +1236,21 @@

    Importance sam $$ W(\mathbf{x},s+\tau|\mathbf{x}_0) = \int_{-\infty}^{\infty} W(\mathbf{x},\tau|\mathbf{x}')W(\mathbf{x}',s|\mathbf{x}_0)d\mathbf{x}'. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    Assume now that \( \tau \) is very small so that we can make an expansion in terms of a small step \( xi \), with \( \mathbf{x}'=\mathbf{x}-\xi \), that is

    $$ W(\mathbf{x},s|\mathbf{x}_0)+\frac{\partial W}{\partial s}\tau +O(\tau^2) = \int_{-\infty}^{\infty} W(\mathbf{x},\tau|\mathbf{x}-\xi)W(\mathbf{x}-\xi,s|\mathbf{x}_0)d\mathbf{x}'. $$

    We assume that \( W(\mathbf{x},\tau|\mathbf{x}-\xi) \) takes non-negligible values only when \( \xi \) is small. This is just another way of stating the Master equation!!

    -
    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -1356,14 +1260,11 @@

    Importance sam \sum_{n=0}^{\infty}\frac{(-\xi)^n}{n!}\frac{\partial^n}{\partial x^n}\left[W(\mathbf{x}+\xi,\tau|\mathbf{x})W(\mathbf{x},s|\mathbf{x}_0) \right]. $$ -











    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We can then rewrite the ESKC equation as

    $$ \frac{\partial W}{\partial s}\tau=-W(\mathbf{x},s|\mathbf{x}_0)+ @@ -1374,14 +1275,10 @@

    Importance sam

    We have neglected higher powers of \( \tau \) and have used that for \( n=0 \) we get simply \( W(\mathbf{x},s|\mathbf{x}_0) \) due to normalization.

    -

    -









    Importance sampling, Fokker-Planck and Langevin equations

    -
    - -

    +

    We say thus that \( \mathbf{x} \) changes only by a small amount in the time interval \( \tau \). This means that we can make a Taylor expansion in terms of \( \xi \), that is we expand @@ -1391,7 +1288,6 @@

    Importance sam \sum_{n=0}^{\infty}\frac{(-\xi)^n}{n!}\frac{\partial^n}{\partial x^n}\left[W(\mathbf{x}+\xi,\tau|\mathbf{x})W(\mathbf{x},s|\mathbf{x}_0) \right]. $$ -











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This is done here for the two-electron case with the Coulomb interaction, as in the previous example. We have two variational parameters $\alpha$ and $\beta$. After the set up of files +_Note: this is best seen using the Jupyter-notebook_. + +We are now ready to implement importance sampling. This is done here +for the two-electron case with the Coulomb interaction, as in the +previous example. We have two variational parameters $\alpha$ and +$\beta$. After the set up of files !bc pycod # Common imports @@ -308,9 +313,9 @@ plt.show() !split -===== Importance sampling, program elements ===== -!bblock -The general derivative formula of the Jastrow factor is (the subscript $C$ stands for Correlation) +===== Importance sampling, programming elements ===== + +The general derivative formula of the Jastrow factor (or the ansatz for the correlated part of the wave function) is (the subscript $C$ stands for Correlation) !bt \[ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = @@ -329,12 +334,12 @@ with our written in way which can be reused later as the gradient needed for the quantum force and local energy is easy to compute. The function $f(r_{ij})$ will depends on the system under study. In the equations below we will keep this general form. -!eblock + !split ===== Importance sampling, program elements ===== -!bblock + In the Metropolis/Hasting algorithm, the *acceptance ratio* determines the probability for a particle to be accepted at a new position. The ratio of the trial wave functions evaluated at the new and current positions is given by ($OB$ for the onebody part) !bt \[ @@ -351,12 +356,12 @@ The second is needed when we compute the kinetic energy term of the local energy \frac{\mathbf{\mathbf{\nabla}} \Psi}{\Psi} = \frac{\mathbf{\nabla} (\Psi_{OB} \, \Psi_{C})}{\Psi_{OB} \, \Psi_{C}} = \frac{ \Psi_C \mathbf{\nabla} \Psi_{OB} + \Psi_{OB} \mathbf{\nabla} \Psi_{C}}{\Psi_{OB} \Psi_{C}} = \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} \] !et -!eblock + !split ===== Importance sampling ===== -!bblock -The expectation value of the kinetic energy expressed in atomic units for electron $i$ is + +The expectation value of the kinetic energy expressed in scaled units for particle $i$ is !bt \[ \langle \hat{K}_i \rangle = -\frac{1}{2}\frac{\langle\Psi|\mathbf{\nabla}_{i}^2|\Psi \rangle}{\langle\Psi|\Psi \rangle}, @@ -368,23 +373,24 @@ The expectation value of the kinetic energy expressed in atomic units for electr \] !et -!eblock + !split ===== Importance sampling ===== -!bblock + The second derivative which enters the definition of the local energy is !bt \[ \frac{\mathbf{\nabla}^2 \Psi}{\Psi}=\frac{\mathbf{\nabla}^2 \Psi_{OB}}{\Psi_{OB}} + \frac{\mathbf{\nabla}^2 \Psi_C}{ \Psi_C} + 2 \frac{\mathbf{\nabla} \Psi_{OB}}{\Psi_{OB}}\cdot\frac{\mathbf{\nabla} \Psi_C}{ \Psi_C} \] !et -We discuss here how to calculate these quantities in an optimal way, -!eblock +We discuss here how to calculate these quantities in an optimal way. + + !split ===== Importance sampling ===== -!bblock + We have defined the correlated function as !bt \[ @@ -401,12 +407,12 @@ In our particular case we have \Psi_C=\prod_{i< j}g(r_{ij})=\exp{\left\{\sum_{i