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\documentclass[12pt]{article}
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\usepackage{amsmath}
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\begin{document}
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\begin{align*}
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&{\bf F}_{j}(t) = {\bf F}^C_j(t)-\int \limits_{0}^{t} \Gamma_j(t-s) {\bf v}_j(s)~\text{d}s + {\bf F}^R_j(t) \\
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&\Gamma_j(t-s) = \sum \limits_{k=1}^{N_k} \frac{c_k}{\tau_k} e^{-(t-s)/\tau_k} \\
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&\langle{\bf F}^R_j(t),{\bf F}^R_j(s)\rangle = \text{k$_\text{B}$T} ~\Gamma_j(t-s)
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\end{align*}
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\end{document}
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<HTML>
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<CENTER><A HREF = "http://lammps.sandia.gov">LAMMPS WWW Site</A> - <A HREF = "Manual.html">LAMMPS Documentation</A> - <A HREF = "Section_commands.html#comm">LAMMPS Commands</A>
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</CENTER>
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<HR>
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<H3>fix gld command
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</H3>
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<P><B>Syntax:</B>
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</P>
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<PRE>fix ID group-ID gld Tstart Tstop N_k seed series c_1 tau_1 ... c_N_k tau_N_k keyword values ...
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</PRE>
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<UL><LI>ID, group-ID are documented in <A HREF = "fix.html">fix</A> command
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<LI>gld = style name of this fix command
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<LI>Tstart,Tstop = desired temperature at start/end of run (temperature units)
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<LI>N_k = number of terms in the Prony series representation of the memory kernel
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seed = random number seed to use for white noise (positive integer)
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<LI>series = <I>pprony</I> is presently the only available option
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<LI>c_k = the weight of the kth term in the Prony series (mass per time units)
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<LI>tau_k = the time constant of the kth term in the Prony series (time units)
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<LI>zero or more keyword/value pairs may be appended
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<PRE>keyword = <I>frozen</I> or <I>zero</I>
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<I>frozen</I> value = <I>no</I> or <I>yes</I>
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<I>no</I> = initialize extended variables using values drawn from equilibrium distribution at Tstart
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<I>yes</I> = initialize extended variables to zero (i.e., from equilibrium distribution at zero temperature)
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</PRE>
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<PRE> <I>zero</I> value = <I>no</I> or <I>yes</I>
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<I>no</I> = do not set total random force to zero
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<I>yes</I> = set total random force to zero
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</PRE>
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</UL>
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<P><B>Examples:</B>
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</P>
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<PRE>fix 1 all gld 1.0 1.0 2 82885 pprony 0.5 1.0 1.0 2.0 frozen yes zero yes
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fix 3 rouse gld 7.355 7.355 4 48823 pprony 107.1 0.02415 186.0 0.04294 428.6 0.09661 1714 0.38643
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</PRE>
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<P><B>Description:</B>
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</P>
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<P>Applies Generalized Langevin Dynamics to a group of atoms, as
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described in <A HREF = "#Baczewski">(Baczewski)</A>. This is intended to model the
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effect of an implicit solvent with a temporally non-local dissipative
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force and a colored Gaussian random force, consistent with the
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Fluctuation-Dissipation Theorem. The functional form of the memory
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kernel associated with the temporally non-local force is constrained
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to be a Prony series.
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</P>
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<P>With this fix active, the force on the <I>j</I>th atom is given as
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</P>
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<CENTER><IMG SRC = "Eqs/fix_gld1.jpg">
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</CENTER>
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<P>Here, the first term is representative of all conservative (pairwise,
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bonded, etc) forces external to this fix, the second is the temporally
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non-local dissipative force given as a Prony series, and the third is
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the colored Gaussian random force.
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</P>
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<P>The Prony series form of the memory kernel is chosen to enable an
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extended variable formalism, with a number of exemplary mathematical
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features discussed in <A HREF = "#Baczewski">(Baczewski)</A>. In particular, 3N_k
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extended variables are added to each atom, which effect the action of
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the memory kernel without having to explicitly evaluate the integral
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over time in the second term of the force. This also has the benefit
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of requiring the generation of uncorrelated random forces, rather than
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correlated random forces as specified in the third term of the force.
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</P>
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<P>Presently, the Prony series coefficients are limited to being greater
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than or equal to zero, and the time constants are limited to being
|
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greater than zero. To this end, the value of series MUST be set to
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<I>pprony</I>, for now. Future updates will allow for negative coefficients
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and other representations of the memory kernel. It is with these
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updates in mind that the series option was included.
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</P>
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<P>The units of the Prony series coefficients are chosen to be mass per
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time to ensure that the numerical integration scheme stably approaches
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the Newtonian and Langevin limits. Details of these limits, and the
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associated numerical concerns are discussed in
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<A HREF = "#Baczewski">(Baczewski)</A>.
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</P>
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<P>While this fix bears many similarities to <A HREF = "fix_langevin.html">fix
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langevin</A>, it has one significant
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difference. Namely, <A HREF = "fix_gld.html">fix gld</A> performs time integration,
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whereas <A HREF = "fix_langevin.html">fix langevin</A> does NOT. To this end, the
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specification of another fix to perform time integration, such as <A HREF = "fix_nve.html">fix
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nve</A>, is NOT necessary.
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</P>
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<P>The desired temperature at each timestep is ramped from <I>Tstart</I> to
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<I>Tstop</I> over the course of the next run.
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</P>
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<P>The random # <I>seed</I> must be a positive integer. A Marsaglia random
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number generator is used. Each processor uses the input seed to
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generate its own unique seed and its own stream of random
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numbers. Thus the dynamics of the system will not be identical on two
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runs on different numbers of processors.
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</P>
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<HR>
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<P>The keyword/value option pairs are used in the following ways.
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</P>
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<P>The keyword <I>frozen</I> can be used to specify how the extended variables
|
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associated with the GLD memory kernel are initialized. Specifying no
|
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(the default), the initial values are drawn at random from an
|
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equilibrium distribution at <I>Tstart</I>, consistent with the
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Fluctuation-Dissipation Theorem. Specifying yes, initializes the
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extended variables to zero.
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</P>
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<P>The keyword <I>zero</I> can be used to eliminate drift due to the
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thermostat. Because the random forces on different atoms are
|
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independent, they do not sum exactly to zero. As a result, this fix
|
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applies a small random force to the entire system, and the
|
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center-of-mass of the system undergoes a slow random walk. If the
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keyword <I>zero</I> is set to <I>yes</I>, the total random force is set exactly
|
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to zero by subtracting off an equal part of it from each atom in the
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group. As a result, the center-of-mass of a system with zero initial
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momentum will not drift over time.
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</P>
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<P><B>Restart, run start/stop, minimize info:</B>
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</P>
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<P>The instantaneous values of the extended variables are written to
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<A HREF = "restart.html">binary restart files</A>. Because the state of the random
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number generator is not saved in restart files, this means you cannot
|
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do "exact" restarts with this fix, where the simulation continues on
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the same as if no restart had taken place. However, in a statistical
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sense, a restarted simulation should produce the same behavior.
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</P>
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<P>This fix can ramp its target temperature over multiple runs, using the
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<I>start</I> and <I>stop</I> keywords of the <A HREF = "run.html">run</A> command. See the
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<A HREF = "run.html">run</A> command for details of how to do this.
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</P>
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<P>This fix is not invoked during <A HREF = "minimize.html">energy minimization</A>.
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</P>
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<P><B>Restrictions:</B> none
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</P>
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<P><B>Related commands:</B>
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</P>
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<P><A HREF = "fix_langevin.html">fix langevin</A>, <A HREF = "fix_viscous.html">fix viscous</A>,
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<A HREF = "pair_dpd.html">pair_style dpd/tstat</A>
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</P>
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<P><B>Default:</B>
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</P>
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<P>The option defaults are frozen = no, zero = no.
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</P>
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<HR>
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<A NAME = "Baczewski"></A>
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<P><B>(Baczewski)</B> A.D. Baczewski and S.D. Bond, J. Chem. Phys. 139, 044107 (2013).
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</P>
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</HTML>
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|
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@ -0,0 +1,145 @@
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"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
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:link(lws,http://lammps.sandia.gov)
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:link(ld,Manual.html)
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:link(lc,Section_commands.html#comm)
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:line
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fix gld command :h3
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[Syntax:]
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fix ID group-ID gld Tstart Tstop N_k seed series c_1 tau_1 ... c_N_k tau_N_k keyword values ... :pre
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ID, group-ID are documented in "fix"_fix.html command :ulb,l
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gld = style name of this fix command :l
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Tstart,Tstop = desired temperature at start/end of run (temperature units) :l
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N_k = number of terms in the Prony series representation of the memory kernel
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seed = random number seed to use for white noise (positive integer) :l
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series = {pprony} is presently the only available option :l
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c_k = the weight of the kth term in the Prony series (mass per time units) :l
|
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tau_k = the time constant of the kth term in the Prony series (time units) :l
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zero or more keyword/value pairs may be appended :l
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keyword = {frozen} or {zero}
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{frozen} value = {no} or {yes}
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{no} = initialize extended variables using values drawn from equilibrium distribution at Tstart
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{yes} = initialize extended variables to zero (i.e., from equilibrium distribution at zero temperature) :pre
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{zero} value = {no} or {yes}
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{no} = do not set total random force to zero
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{yes} = set total random force to zero :pre
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:ule
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[Examples:]
|
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|
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fix 1 all gld 1.0 1.0 2 82885 pprony 0.5 1.0 1.0 2.0 frozen yes zero yes
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fix 3 rouse gld 7.355 7.355 4 48823 pprony 107.1 0.02415 186.0 0.04294 428.6 0.09661 1714 0.38643 :pre
|
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|
||||
[Description:]
|
||||
|
||||
Applies Generalized Langevin Dynamics to a group of atoms, as
|
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described in "(Baczewski)"_#Baczewski. This is intended to model the
|
||||
effect of an implicit solvent with a temporally non-local dissipative
|
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force and a colored Gaussian random force, consistent with the
|
||||
Fluctuation-Dissipation Theorem. The functional form of the memory
|
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kernel associated with the temporally non-local force is constrained
|
||||
to be a Prony series.
|
||||
|
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With this fix active, the force on the {j}th atom is given as
|
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|
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:c,image(Eqs/fix_gld1.jpg)
|
||||
|
||||
Here, the first term is representative of all conservative (pairwise,
|
||||
bonded, etc) forces external to this fix, the second is the temporally
|
||||
non-local dissipative force given as a Prony series, and the third is
|
||||
the colored Gaussian random force.
|
||||
|
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The Prony series form of the memory kernel is chosen to enable an
|
||||
extended variable formalism, with a number of exemplary mathematical
|
||||
features discussed in "(Baczewski)"_#Baczewski. In particular, 3N_k
|
||||
extended variables are added to each atom, which effect the action of
|
||||
the memory kernel without having to explicitly evaluate the integral
|
||||
over time in the second term of the force. This also has the benefit
|
||||
of requiring the generation of uncorrelated random forces, rather than
|
||||
correlated random forces as specified in the third term of the force.
|
||||
|
||||
Presently, the Prony series coefficients are limited to being greater
|
||||
than or equal to zero, and the time constants are limited to being
|
||||
greater than zero. To this end, the value of series MUST be set to
|
||||
{pprony}, for now. Future updates will allow for negative coefficients
|
||||
and other representations of the memory kernel. It is with these
|
||||
updates in mind that the series option was included.
|
||||
|
||||
The units of the Prony series coefficients are chosen to be mass per
|
||||
time to ensure that the numerical integration scheme stably approaches
|
||||
the Newtonian and Langevin limits. Details of these limits, and the
|
||||
associated numerical concerns are discussed in
|
||||
"(Baczewski)"_#Baczewski.
|
||||
|
||||
While this fix bears many similarities to "fix
|
||||
langevin"_fix_langevin.html, it has one significant
|
||||
difference. Namely, "fix gld"_fix_gld.html performs time integration,
|
||||
whereas "fix langevin"_fix_langevin.html does NOT. To this end, the
|
||||
specification of another fix to perform time integration, such as "fix
|
||||
nve"_fix_nve.html, is NOT necessary.
|
||||
|
||||
The desired temperature at each timestep is ramped from {Tstart} to
|
||||
{Tstop} over the course of the next run.
|
||||
|
||||
The random # {seed} must be a positive integer. A Marsaglia random
|
||||
number generator is used. Each processor uses the input seed to
|
||||
generate its own unique seed and its own stream of random
|
||||
numbers. Thus the dynamics of the system will not be identical on two
|
||||
runs on different numbers of processors.
|
||||
|
||||
:line
|
||||
|
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The keyword/value option pairs are used in the following ways.
|
||||
|
||||
The keyword {frozen} can be used to specify how the extended variables
|
||||
associated with the GLD memory kernel are initialized. Specifying no
|
||||
(the default), the initial values are drawn at random from an
|
||||
equilibrium distribution at {Tstart}, consistent with the
|
||||
Fluctuation-Dissipation Theorem. Specifying yes, initializes the
|
||||
extended variables to zero.
|
||||
|
||||
The keyword {zero} can be used to eliminate drift due to the
|
||||
thermostat. Because the random forces on different atoms are
|
||||
independent, they do not sum exactly to zero. As a result, this fix
|
||||
applies a small random force to the entire system, and the
|
||||
center-of-mass of the system undergoes a slow random walk. If the
|
||||
keyword {zero} is set to {yes}, the total random force is set exactly
|
||||
to zero by subtracting off an equal part of it from each atom in the
|
||||
group. As a result, the center-of-mass of a system with zero initial
|
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momentum will not drift over time.
|
||||
|
||||
[Restart, run start/stop, minimize info:]
|
||||
|
||||
The instantaneous values of the extended variables are written to
|
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"binary restart files"_restart.html. Because the state of the random
|
||||
number generator is not saved in restart files, this means you cannot
|
||||
do "exact" restarts with this fix, where the simulation continues on
|
||||
the same as if no restart had taken place. However, in a statistical
|
||||
sense, a restarted simulation should produce the same behavior.
|
||||
|
||||
This fix can ramp its target temperature over multiple runs, using the
|
||||
{start} and {stop} keywords of the "run"_run.html command. See the
|
||||
"run"_run.html command for details of how to do this.
|
||||
|
||||
This fix is not invoked during "energy minimization"_minimize.html.
|
||||
|
||||
[Restrictions:] none
|
||||
|
||||
[Related commands:]
|
||||
|
||||
"fix langevin"_fix_langevin.html, "fix viscous"_fix_viscous.html,
|
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"pair_style dpd/tstat"_pair_dpd.html
|
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|
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[Default:]
|
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The option defaults are frozen = no, zero = no.
|
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:line
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|
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:link(Baczewski)
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[(Baczewski)] A.D. Baczewski and S.D. Bond, J. Chem. Phys. 139, 044107 (2013).
|
||||
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