Commit 89e27803 authored by Thomas Ple's avatar Thomas Ple

Fixed bugs in description of the method

parent 3fb673b6
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......@@ -58,8 +58,7 @@ equation :
where :math:`p` is the momentum vector of the set of
atoms, interacting via the potential :math:`U` , :math:`\gamma` is the damping coefficient
and :math:`F(t)` is the stochastic force. The random force :math:`F(t)`
is gaussian and delta-correlated.
To enforce the classical fluctuation-dissipation theorem,
is a Gaussian white noise: to enforce the classical fluctuation-dissipation theorem,
its autocorrelation spectrum is given by :
.. math:: C_{FF}(\omega)= \int \limits_{-\infty}^{+\infty} dt \langle F(t) F(t+\tau) \rangle e^{-i \omega t} = 2m \gamma k_B T
......@@ -71,8 +70,8 @@ Quantum Thermal Bath (QTB)
--------------------------
The Quantum Thermal Bath uses a generalized Langevin equation in order
to approximate nuclear quantum effects[Dam]_ . In QTB dynamics, the
stochastic force is no longer delta-correlated but is colored according to the
to approximate nuclear quantum effects [Dam]_ . In QTB dynamics, the
stochastic force is no longer a white noise but is colored according to the
following formula :
.. math:: C_{FF}(\omega)=2m \gamma \theta(\omega,T)
......@@ -85,8 +84,8 @@ with
where :math:`\beta = \frac{1}{k_BT}` and :math:`2 \pi \hbar` is the Planck
constant. The function :math:`\theta(\omega,T)` describes the energy
of a quantum harmonic oscillators of angular frequency
:math:`\omega` at a temperature :math:`T` . The colored random force allows to
approximate zero-point energy contributions to the equilibrium properties of the system.
:math:`\omega` at a temperature :math:`T` . The colored random force allows
approximating zero-point energy contributions to the equilibrium properties of the system.
The QTB method is known to lead to qualitatively
good results [Bri]_ but as many semi-classical methods, it suffers from zero-point energy leakage
(ZPEL) [Hern]_.
......@@ -104,6 +103,7 @@ In practice, this is done by minimizing the fluctuation-dissipation spectrum
.. math:: \Delta_{FDT} (\omega) = {\rm{Re}} \left[C_{vF}(\omega)\right] - m \gamma_{r} (\omega) C_{vv} (\omega)
:label: eqDFDT
where :math:`C_{vF}` is the velocity random force cross-correlation
spectrum, :math:`C_{vv}` the velocity autocorrelation spectrum and
......@@ -128,14 +128,14 @@ adjusted with a first-order differential equation and an adaptation coefficient
during a preliminary “adaptation time” until they reach convergence.
Observables are then computed while the adaptive process is kept active.
Further informations and precise implementation details can be found in ref.[Man]_.
Further informations and precise implementation details can be found in ref. [Man]_.
Two implementations are currently available in PaPIM:
#. Random force adaptive QTB (adQTB-r)
In this variant, the dissipation
kernel is left unchanged, i.e. :math:`\gamma_{f}(\omega) = \gamma`
#. Random force adaptive QTB (adQTB-r):
In this variant, the dissipation kernel is left unchanged, i.e. :math:`\gamma_{f}(\omega) = \gamma`
while the random force is modified according to a frequency-dependent
set of damping coefficients :math:`\gamma_r(\omega)` to satisfy
:math:`\Delta_{FDT} = 0` (eq. :eq:`eqDFDT`):
......@@ -143,14 +143,14 @@ In this variant, the dissipation
.. math:: C_{FF}(\omega)=2m \gamma_r(\omega) \theta(\omega,T)
:label: eqadQTBr
This method is applicable only if the initial damping coefficient
:math:`\gamma` is large enough to compensate effects of a possible
zero-point energy leakage.
This method is applicable only if the initial damping coefficient
:math:`\gamma` is large enough to compensate effects of a possible
zero-point energy leakage.
#. Dissipative kernel adaptive QTB (adQTB-f)
In this approach, the
random force is not modified, i.e.
In this approach, the random force is not modified, i.e.
:math:`\gamma_{r} (\omega) = \gamma` and remains the same as in the standard QTB
method (eq. :eq:`eqQTB`)) but the dissipation term is not
described by a viscous damping term anymore (:math:`-m \gamma v`) but
......@@ -160,15 +160,16 @@ In this approach, the
.. math:: \dot p = -\nabla U - \int_0^\infty \ \gamma_f(\tau) p(t-\tau) \ d\tau + F(t)
:label: eqgenlgv
In order to avoid solving with brute force this integro-differential
equation, the dissipative memory kernel is expressed as a sum of
equally spaced (:math:`\Delta \omega`) lorentzian functions of width
:math:`\alpha` :
In order to avoid solving with brute force this integro-differential
equation, the dissipative memory kernel is expressed as a sum of
equally spaced (:math:`\Delta \omega`) lorentzian functions of width
:math:`\alpha` :
.. math:: \gamma_f(\omega) = \frac{\Delta \omega}{\pi}\sum_{j=0}^{n_\omega}
\frac{ \gamma_{f,j} }{\alpha + i(\omega-\omega_j)} +\frac{ \gamma_{f,j}}{\alpha + i(\omega+\omega_j)}
:label: eqlorentzgenlgv
The parameter :math:`\gamma_{f,j}` are then modified to satisfy
:math:`\Delta_{FDT} = 0` (eq. :eq:`eqDFDT`).
......
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