Skip to content

Commit

Permalink
updated dynamics and notation sections
Browse files Browse the repository at this point in the history
  • Loading branch information
@ChiCheng45
ChiCheng45 committed Feb 25, 2025
1 parent e19ae19 commit 727c7c3
Show file tree
Hide file tree
Showing 5 changed files with 96 additions and 112 deletions.
1 change: 1 addition & 0 deletions Doc/index.rst
View file
Original file line number Diff line number Diff line change
Expand Up @@ -63,6 +63,7 @@ J. Chem. Inf. Model. 2017, 57, 1, 1–5 <https://doi.org/10.1021/acs.jcim.6b0057
pages/workflow
pages/trajectory
pages/correlation
pages/weights
pages/dynamics
pages/scattering
pages/structure
Expand Down
47 changes: 34 additions & 13 deletions Doc/pages/R_notation.rst
View file
Original file line number Diff line number Diff line change
Expand Up @@ -4,16 +4,37 @@
Frequently Used Symbols and Notations
=====================================

======================== ===============
**Symbol** **Description**
:math:`t` time
:math:`t_{\mathrm{cor}}` the total time of the correlation function
:math:`t_{\mathrm{tot}}` the total time of the MD simulation
:math:`\Delta t` the MD time step
:math:`T` temperature
:math:`n_{\mathrm{c}}` total number of time steps calculated for the correlation function
:math:`n_{\mathrm{t}}` total number of time steps in the MD simulation
:math:`N` total number of atoms in the unit cell, simulation box or volume :math:`V`
:math:`\omega` frequency
:math:`\Delta \omega` the frequency step
======================== ===============
============================================ ===============
**Symbol** **Description**
:math:`\alpha, \beta, \gamma, \ldots` atom types
:math:`i, j, k, \ldots` atom indices
:math:`D` diffusion constant
:math:`F(\mathbf{q}, t)` intermediate scattering function
:math:`F_{\mathrm{coh}}(\mathbf{q}, t)` coherent intermediate scattering function
:math:`F_{\mathrm{inc}}(\mathbf{q}, t)` incoherent intermediate scattering function
:math:`G(\mathbf{r}, t)` the van Hove function
:math:`G_{\mathrm{d}}(\mathbf{r}, t)` distinct part of the van Hove function
:math:`G_{\mathrm{s}}(\mathbf{r}, t)` self part of the van Hove function
:math:`t` time
:math:`t_{\mathrm{cor}}` the total time of the correlation function
:math:`t_{\mathrm{tot}}` the total time of the MD simulation
:math:`\Delta t` the MD time step
:math:`S(\mathbf{q}, \omega)` dynamic structure factor
:math:`S_{\mathrm{coh}}(\mathbf{q}, \omega)` dynamic coherent structure factor
:math:`S_{\mathrm{inc}}(\mathbf{q}, \omega)` dynamic incoherent structure factor
:math:`T` temperature
:math:`M` number of atom types in the system
:math:`n_{\mathrm{c}}` total number of time steps calculated for the correlation function
:math:`n_{\mathrm{t}}` total number of time steps in the MD simulation
:math:`N` total number of atoms in the unit cell, simulation or volume :math:`V`
:math:`N_\alpha` number of atom of type :math:`\alpha`
:math:`\rho` density
:math:`r` distance
:math:`q` wavenumber
:math:`\mathbf{q}` wavevector
:math:`\mathbf{r}(t)` the position of an atom at time :math:`t`
:math:`\mathbf{v}(t)` the velocity of an atom at time :math:`t`
:math:`W_\alpha` weighting factor for atom type :math:`\alpha`
:math:`\omega` angular frequency
:math:`\Delta \omega` the frequency step
============================================ ===============
4 changes: 2 additions & 2 deletions Doc/pages/correlation.rst
View file
Original file line number Diff line number Diff line change
@@ -1,8 +1,8 @@

.. _appendix-fca:

Analysis Theory: Time Series
============================
Time Series
===========

Correlation Functions
~~~~~~~~~~~~~~~~~~~~~
Expand Down
153 changes: 56 additions & 97 deletions Doc/pages/dynamics.rst
View file
Original file line number Diff line number Diff line change
Expand Up @@ -61,31 +61,28 @@ Density Of States

.. _theory-and-implementation-1:

MDANSE calculates the power spectrum of the VACF, which in case of
the mass-weighted VACF defines the phonon discrete DOS (see the
section on :ref:`analysis-vacf`), defined as:
MDANSE calculates the power spectrum of the VACF (see the
section on :ref:`analysis-vacf`), which in case of
the mass-weighted VACF defines the phonon discrete DOS , defined as:

.. math::
:label: pfx5
{\mathrm{DOS}\left( {\nu} \right)\doteq{\sum\limits_{\alpha}\omega_{\alpha}}{\overset{\sim}{C}}_{\mathit{vv};\mathit{\alpha\alpha}}\left(\nu \right)}
{\mathrm{DOS}( {\omega} ) = {\sum\limits_{\alpha}^{M} W_{\alpha}} {\widetilde{C}}_{\mathbf{vv};\alpha\alpha}\left(\omega \right)}
where :math:`{\overset{\sim}{C}}_{\mathit{vv};\mathit{\alpha\alpha}}\left( {\nu} \right)`
is the Fourier transform of the velocity autocorrelation function and
:math:`\omega_{\alpha}` is the weighting factor used for atom :math:`\alpha`.
The DOS
can be computed either for the isotropic case or with respect to a
where :math:`{\widetilde{C}}_{\mathbf{vv};\alpha\alpha}\left( \omega \right)`
is the Fourier transform of the velocity autocorrelation function average over atoms of type :math:`\alpha`,
:math:`W_{\alpha}` is the weighting factor of atom type :math:`\alpha`
and :math:`M` is the total number of atoms types in the system.

The DOS can be computed either for the isotropic case or with respect to a
user-defined axis. Since the DOS
is computed from the unnormalized VACF, such that :math:`\mathrm{DOS}(0)` gives an
approximate value for the diffusion constant
:math:`D = {\sum\limits_{\alpha}D_{\alpha}}`
(see Eqs. :math:numref:`pfx20` and :math:numref:`pfx21`).

The DOS
can be smoothed by, for example, a Gaussian window applied in the time domain
[Ref10]_ (see the section :ref:`appendix-fca`). Its width in the time domain
is :math:`{\sigma_{t} = \alpha}\text{/}T`
, where :math:`T` is the length of the correlation function. We remark that the diffusion
is computed from the unnormalized VACF, :math:`\mathrm{DOS}(0)` gives an
approximate value for the diffusion constant (see Eqs. :math:numref:`pfx20`)
when an equal weighting scheme is used. The DOS can be
smoothed by, for example, a Gaussian window applied in the time domain
[Ref10]_ (see the section :ref:`appendix-fca`). We remark that the diffusion
constant obtained from this DOS is biased due to the spectral smoothing
procedure since the VACF is weighted by this window Gaussian function.

Expand Down Expand Up @@ -128,93 +125,62 @@ performed on a water box of 768 water molecules. To get the diffusion
coefficient out of this plot, the slope of the linear part of the plot
should be calculated.

By defining

.. math::
:label: pfx13
{d_{\alpha}\left( {t} \right)\doteq R_{\alpha}{\left( {t} \right) - R_{\alpha}}\left( 0 \right),}
the MSD of particle :math:`\alpha` can be written as:
By defining :math:`\mathbf{d}_{i}(t) = \mathbf{r}_{i}( t ) - \mathbf{r}_{i}( 0 )`
the MSD of particle :math:`i` can be written as:

.. math::
:label: pfx14
\mathrm{\Delta}_{\alpha}^{2}{(t) = \left\langle {d_{\alpha}^{2}\left( {t} \right)} \right\rangle}
\Delta_{i}^{2}{(t) = \left\langle {d_{i}^{2}( {t} )} \right\rangle}
where :math:`R_{\alpha}(0)` and :math:`R_{\alpha}(t)` are
respectively the position of particle :math:`\alpha`
at times :math:`0` and :math:`t`. One can introduce an MSD with respect to a given axis n:
where :math:`\mathbf{r}_{i}(0)` and :math:`\mathbf{r}_{i}(t)` are
the position of particle :math:`i` at times :math:`0` and :math:`t`
and :math:`d_{i}( t ) = \vert \mathbf{d}_{i}(t) \vert`.
One can introduce an MSD with respect to a given axis :math:`\mathbf{n}`:

.. math::
:label: pfx15
{\mathrm{\Delta}_{\alpha}^{2}\left( {t;n} \right)\doteq\left\langle {d_{\alpha}^{2}\left( {t;n} \right)} \right\rangle}
with

.. math::
:label: pfx16
{d_{\alpha}^{}\left( {t;n} \right)\doteq n\cdot d_{\alpha}^{}\left( {t} \right).}
\Delta_{i}^{2}( t;\hat{\mathbf{n}} ) = \left\langle {d_{i}^{2}( {t;\hat{\mathbf{n}}} )} \right\rangle \qquad d_{i}(t;\hat{\mathbf{n}}) = \hat{\mathbf{n}} \cdot \mathbf{d}_{i}( t )
where :math:`\hat{\mathbf{n}}` is a unit vector along :math:`\mathbf{n}`.
The calculation of MSD is the standard way to obtain diffusion
coefficients from Molecular Dynamics (MD) simulations. Assuming
Einstein-diffusion in the long time limit one has for isotropic systems
coefficients from Molecular Dynamics (MD) simulations.

Assuming Einstein-diffusion in the long time limit one has for isotropic systems

.. math::
:label: pfx17
{D_{\alpha} = {\lim\limits_{t\rightarrow\infty}{\frac{1}{6t}\mathrm{\Delta}_{\alpha}^{2}(t)}}}.
{D_{i} = {\lim\limits_{t\rightarrow\infty}{\frac{1}{6t}\mathrm{\Delta}_{i}^{2}(t)}}}.
There exists also a well-known relation between the MSD and the
velocity autocorrelation function. Writing
velocity autocorrelation function. One can show (see e.g. [Ref11]_) that

.. math::
:label: pfx18
{d_{\alpha}{(t) = {\int\limits_{0}^{t}{\mathrm{d}\tau \, v_{\alpha}(\tau)}}}}
in Eq. :math:numref:`pfx14` one can show (see
e.g. [Ref11]_) that

.. math::
:label: pfx19
{\mathrm{\Delta}_{\alpha}^{2}{(t) = 6}{\int\limits_{0}^{t}{\mathrm{d}\tau \, \left( {t - \tau} \right)C_{\mathit{\upsilon\upsilon};\mathit{\alpha\alpha}}(t)}}.}
\mathbf{d}_{i}{(t) = {\int\limits_{0}^{t}{\mathrm{d}t_1 \, \mathbf{v}_{i}(t_1)}}} \qquad \text{and} \qquad \mathrm{\Delta}_{i}^{2}{(t) = 6}{\int\limits_{0}^{t}{\mathrm{d}t_1 \, ( t - t_1 )C_{\mathbf{vv};ii}(t_1)}}
where :math:`C_{\mathit{\upsilon\upsilon};\mathit{\alpha\alpha}}(t)` is the velocity autocorrelation function. Using now the definition :math:numref:`pfx17` of the diffusion
coefficient one obtains the relation
where :math:`C_{\mathbf{vv};ii}(t)` is the velocity autocorrelation function of the particle :math:`i`.
Using now the definition Eq. :math:numref:`pfx17` of the diffusion
coefficient one obtains the relations

.. math::
:label: pfx20
{{D_{\alpha} = {\int\limits_{0}^{\infty}{\mathrm{d}\tau \, C_{\mathit{\upsilon\upsilon};\mathit{\alpha\alpha}}(t)}}}.}
{{D_{i} = {\int\limits_{0}^{\infty}{\mathrm{d}t \, C_{\mathbf{vv};ii}(t)}}} = \pi \widetilde{C}_{\mathbf{vv};ii}(0).}
and can also be written as

.. math::
:label: pfx21
{{D_{\alpha} = \pi}{\overset{\sim}{C}}_{\mathit{\upsilon\upsilon};\mathit{\alpha\alpha}}(0).}
Computationally, the MSD is calculated by calculating the position autocorrelation since
from Eqs. :math:numref:`pfx13` and :math:numref:`pfx14`
from Eq. :math:numref:`pfx14`

.. math::
:label: pfx22
\mathrm{\Delta}_{\alpha}^{2}{(t) = \left\langle [R_{\alpha}{\left( t \right) - R_{\alpha}}\left( 0 \right)]^2 \right\rangle}
and therefore

.. math::
:label: pfx23
\mathrm{\Delta}_{\alpha}^{2}(t) = \left\langle R_{\alpha}^{2}\left( t \right) \right\rangle + \left\langle R_{\alpha}^{2}\left( 0 \right) \right\rangle - 2\left\langle R_{\alpha}\left(t \right)R_{\alpha}\left( 0 \right) \right\rangle
\Delta_{i}^{2}(t) = \left\langle [\mathbf{r}_{i}( t ) - \mathbf{r}_{i}(0)]^2 \right\rangle = \left\langle \mathbf{r}_{i}^{2}(t) \right\rangle + \left\langle \mathbf{r}_{i}^{2}( 0 ) \right\rangle - 2\left\langle \mathbf{r}_{i}(t )\mathbf{r}_{i}(0) \right\rangle
where the last part on the right side Eq. :math:numref:`pfx23` is the position autocorrelation of the atom :math:`\alpha`.
where the last part on the right side Eq. :math:numref:`pfx22` is the position autocorrelation of the particle :math:`i`.

.. _analysis-op:

Expand Down Expand Up @@ -317,19 +283,19 @@ Position AutoCorrelation Function
The position autocorrelation function (PACF) is similar to the
velocity autocorrelation function described below. In MDANSE the PACF
is calculated relative to the atoms average position over the entire
trajectory
trajectory. The PACF of atom type :math:`\alpha` is

.. math::
:label: pfx44a
{\mathrm{PACF}_{\alpha}(t)\doteq\frac{1}{3}\left\langle {\Delta R_{\alpha}\left( 0 \right)\cdot \Delta R_{\alpha}\left( t \right)} \right\rangle}
\mathrm{PACF}_{\alpha}(t) = \frac{1}{3}\frac{W_\alpha}{N_{\alpha}} \sum_{i}^{N_{\alpha}} \left\langle {\Delta \mathbf{r}_{i}(0)\cdot \Delta \mathbf{r}_{i}(t)} \right\rangle
where

.. math::
:label: pfx44b
\Delta R_{\alpha}\left( t \right) = R_{\alpha}\left( t \right) - \langle R_{\alpha} \rangle
\Delta \mathbf{r}_{i}\left( t \right) = \mathbf{r}_{i}(t) - \langle \mathbf{r}_{i}(t_0) \rangle_{t_0}
so that the origin dependence of the PACF function is removed.

Expand All @@ -338,44 +304,37 @@ so that the origin dependence of the PACF function is removed.
Van Hove Function
'''''''''''''''''
The van Hove function describes the probability of finding a particle
:math:`j` at a distance :math:`R` at time :math:`t` from a
:math:`j` at time :math:`t` with a displacement of :math:`\mathbf{r}` from a
particle :math:`i` at a time :math:`0`

.. math::
G(R, t) = \frac{1}{N} \sum_{i}^{N}\sum_{j}^{N} \left\langle \delta [R - R_{j}(t) + R_{i}(0)] \right\rangle
G(\mathbf{r}, t) = \frac{1}{N} \sum_{i}^{N}\sum_{j}^{N} \left\langle \delta [\mathbf{r} - \mathbf{r}_{j}(t) + \mathbf{r}_{i}(0)] \right\rangle.
and can be rewritten so that

.. math::
G(R, t) &= \frac{1}{N} \int \sum_{i}^{N}\sum_{j}^{N} \left\langle \delta [R' + R - R_{j}(t)]\delta [R' - R_{i}(0)] \right\rangle \, \mathrm{d} R' \\
&= \frac{1}{\rho} \left\langle \rho (R, t) \rho (0, 0) \right\rangle
so that the van Hove function is a density-density time-correlation
function. The van Hove function is related to the intermediate scattering
The van Hove function is related to the intermediate scattering
function via a Fourier transform and the dynamic structure factor
via a double Fourier transform

.. math::
F(q, t) &= \int G(R,t) \exp(i q \cdot R) \, \mathrm{d}R \\
S(q,\omega) &= \int\int G(R,t) \exp(i q \cdot R - i \omega t) \, \mathrm{d}R\mathrm{d}t
F(\mathbf{q}, t) &= \int \mathrm{d}\mathbf{r} \, G(\mathbf{r},t) \exp(i \mathbf{q} \cdot \mathbf{r}) \\
S(\mathbf{q}, \omega) &= \int \mathrm{d}t \int \mathrm{d}\mathbf{r} \, G(\mathbf{r},t) \exp(i \mathbf{q} \cdot \mathbf{r} - i \omega t)
and can be split into distinct and self parts where

.. math::
G_{\mathrm{d}}(R, t) &= \frac{1}{N} \sum_{i}^{N}\sum_{j \neq i}^{N} \left\langle \delta [R - R_{j}(t) + R_{i}(0)] \right\rangle \\
G_{\mathrm{s}}(R, t) &= \frac{1}{N} \sum_{i} \left\langle \delta [R - R_{i}(t) + R_{i}(0)] \right\rangle
G_{\mathrm{d}}(\mathbf{r}, t) &= \frac{1}{N} \sum_{i}^{N}\sum_{j \neq i}^{N} \left\langle \delta [\mathbf{r} - \mathbf{r}_{j}(t) + \mathbf{r}_{i}(0)] \right\rangle \\
G_{\mathrm{s}}(\mathbf{r}, t) &= \frac{1}{N} \sum_{i} \left\langle \delta [\mathbf{r} - \mathbf{r}_{i}(t) + \mathbf{r}_{i}(0)] \right\rangle
In MDANSE the distinct part of the van Hove function is spherically
averaged and normalised so that for liquid or gaseous systems
In MDANSE the distinct and self parts of the van Hove function are
spherically averaged and normalised so that
:math:`\mathcal{G}_{\mathrm{d}}(r, t) = G_{\mathrm{d}}(r, t) / \rho` and
:math:`\mathcal{G}_{\mathrm{s}}(r, t) = G_{\mathrm{s}}(r, t) / \rho`.
For liquid or gaseous systems

.. math::
\lim_{r \rightarrow \infty } G_{\mathrm{d}}(r, t) = \lim_{t \rightarrow \infty } G_{\mathrm{d}}(r, t) = 1
and the self part of the van Hove function in MDANSE is summed so
that it only depends on the distance :math:`r`
&\lim_{r \rightarrow \infty } \mathcal{G}_{\mathrm{d}}(r, t) = \lim_{t \rightarrow \infty } \mathcal{G}_{\mathrm{d}}(r, t) = 1 \\
&\lim_{t \rightarrow \infty } \mathcal{G}_{\mathrm{s}}(r, t) = N^{-1}
.. math::
G_{\mathrm{s}}(r, t) = \frac{1}{N} \sum_{i} \left\langle \delta \left[r - \vert R_{i}(t) + R_{i}(0) \vert \right] \right\rangle
where in the thermodynamic limit :math:`N \rightarrow \infty`.

.. _analysis-vacf:

Expand Down
3 changes: 3 additions & 0 deletions Doc/pages/weights.rst
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,3 @@
Weighting Scheme
================

0 comments on commit 727c7c3

Please sign in to comment.