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zFpr6asZu%MZJ#q!I(wU}ZlV8u{Adh+*mq8cKRy$1ho|9CD8jxF>wKEbLsJ1!6ZB{C z7S}c8e4GE?mPCJ9uF3jn^_OKg;KMNz{m-QlY)%kM2bP(cD(1-vGE25Vg!L`e9t{Vn rCWt%Q_|?N>Ia=10*=G>9B^P8-&cY?w{Qvn1|NlUJ|CQSCX5xPVc0JBj literal 0 HcmV?d00001 diff --git a/doc/Eqs/fix_gld1.tex b/doc/Eqs/fix_gld1.tex new file mode 100644 index 0000000000..baa0f32a5b --- /dev/null +++ b/doc/Eqs/fix_gld1.tex @@ -0,0 +1,13 @@ +\documentclass[12pt]{article} + +\usepackage{amsmath} + +\begin{document} + +\begin{align*} + &{\bf F}_{j}(t) = {\bf F}^C_j(t)-\int \limits_{0}^{t} \Gamma_j(t-s) {\bf v}_j(s)~\text{d}s + {\bf F}^R_j(t) \\ + &\Gamma_j(t-s) = \sum \limits_{k=1}^{N_k} \frac{c_k}{\tau_k} e^{-(t-s)/\tau_k} \\ + &\langle{\bf F}^R_j(t),{\bf F}^R_j(s)\rangle = \text{k$_\text{B}$T} ~\Gamma_j(t-s) +\end{align*} + +\end{document} diff --git a/doc/fix_gld.html b/doc/fix_gld.html new file mode 100644 index 0000000000..c7567078e6 --- /dev/null +++ b/doc/fix_gld.html @@ -0,0 +1,161 @@ + +
LAMMPS WWW Site - LAMMPS Documentation - LAMMPS Commands +
+ + + + + + +
+ +

fix gld command +

+

Syntax: +

+
fix ID group-ID gld Tstart Tstop N_k seed series c_1 tau_1 ... c_N_k tau_N_k keyword values ... 
+
+
  • ID, group-ID are documented in fix command + +
  • gld = style name of this fix command + +
  • Tstart,Tstop = desired temperature at start/end of run (temperature units) + +
  • N_k = number of terms in the Prony series representation of the memory kernel +seed = random number seed to use for white noise (positive integer) + +
  • series = pprony is presently the only available option + +
  • c_k = the weight of the kth term in the Prony series (mass per time units) + +
  • tau_k = the time constant of the kth term in the Prony series (time units) + +
  • zero or more keyword/value pairs may be appended + +
    keyword = frozen or zero
+  frozen value = no or yes
+    no = initialize extended variables using values drawn from equilibrium distribution at Tstart
+    yes = initialize extended variables to zero (i.e., from equilibrium distribution at zero temperature) 
+
    +
      zero value = no or yes
+    no = do not set total random force to zero
+    yes = set total random force to zero 
+
    + +
+

Examples: +

+
fix 1 all gld 1.0 1.0 2 82885 pprony 0.5 1.0 1.0 2.0 frozen yes zero yes
+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 
+
+

Description: +

+

Applies Generalized Langevin Dynamics to a group of atoms, as +described in (Baczewski). This is intended to model the +effect of an implicit solvent with a temporally non-local dissipative +force and a colored Gaussian random force, consistent with the +Fluctuation-Dissipation Theorem. The functional form of the memory +kernel associated with the temporally non-local force is constrained +to be a Prony series. +

+

With this fix active, the force on the jth atom is given as +

+
+
+

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. +

+

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). 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). +

+

While this fix bears many similarities to fix +langevin, it has one significant +difference. Namely, fix gld performs time integration, +whereas fix langevin does NOT. To this end, the +specification of another fix to perform time integration, such as fix +nve, 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. +

+
+ +

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 +momentum will not drift over time. +

+

Restart, run start/stop, minimize info: +

+

The instantaneous values of the extended variables are written to +binary restart files. 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 command. See the +run command for details of how to do this. +

+

This fix is not invoked during energy minimization. +

+

Restrictions: none +

+

Related commands: +

+

fix langevin, fix viscous, +pair_style dpd/tstat +

+

Default: +

+

The option defaults are frozen = no, zero = no. +

+
+ + + +

(Baczewski) A.D. Baczewski and S.D. Bond, J. Chem. Phys. 139, 044107 (2013). +

+ diff --git a/doc/fix_gld.txt b/doc/fix_gld.txt new file mode 100644 index 0000000000..965652a6c4 --- /dev/null +++ b/doc/fix_gld.txt @@ -0,0 +1,145 @@ +"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c + +:link(lws,http://lammps.sandia.gov) +:link(ld,Manual.html) +:link(lc,Section_commands.html#comm) + +:line + +fix gld command :h3 + +[Syntax:] + +fix ID group-ID gld Tstart Tstop N_k seed series c_1 tau_1 ... c_N_k tau_N_k keyword values ... :pre + +ID, group-ID are documented in "fix"_fix.html command :ulb,l +gld = style name of this fix command :l +Tstart,Tstop = desired temperature at start/end of run (temperature units) :l +N_k = number of terms in the Prony series representation of the memory kernel +seed = random number seed to use for white noise (positive integer) :l +series = {pprony} is presently the only available option :l +c_k = the weight of the kth term in the Prony series (mass per time units) :l +tau_k = the time constant of the kth term in the Prony series (time units) :l +zero or more keyword/value pairs may be appended :l +keyword = {frozen} or {zero} + {frozen} value = {no} or {yes} + {no} = initialize extended variables using values drawn from equilibrium distribution at Tstart + {yes} = initialize extended variables to zero (i.e., from equilibrium distribution at zero temperature) :pre + {zero} value = {no} or {yes} + {no} = do not set total random force to zero + {yes} = set total random force to zero :pre +:ule + +[Examples:] + +fix 1 all gld 1.0 1.0 2 82885 pprony 0.5 1.0 1.0 2.0 frozen yes zero yes +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 + +[Description:] + +Applies Generalized Langevin Dynamics to a group of atoms, as +described in "(Baczewski)"_#Baczewski. This is intended to model the +effect of an implicit solvent with a temporally non-local dissipative +force and a colored Gaussian random force, consistent with the +Fluctuation-Dissipation Theorem. The functional form of the memory +kernel associated with the temporally non-local force is constrained +to be a Prony series. + +With this fix active, the force on the {j}th atom is given as + +: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. + +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 + +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 +momentum will not drift over time. + +[Restart, run start/stop, minimize info:] + +The instantaneous values of the extended variables are written to +"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, +"pair_style dpd/tstat"_pair_dpd.html + +[Default:] + +The option defaults are frozen = no, zero = no. + +:line + +:link(Baczewski) +[(Baczewski)] A.D. Baczewski and S.D. Bond, J. Chem. Phys. 139, 044107 (2013).