*Nonequilibrium Molecular
Dynamics*
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This subject goes back to early work by Fermi, Vineyard, Alder, and Wainwright.
Bill Ashurst's 1974 dissertation and Evans and Morriss' Statistical
Mechanics of Nonequilibrium Liquids [Academic Press (1990)] are among the
early references. In the early 1970s Bill Ashurst developed algorithms
for diffusion and shear flow. Soon, Mike Gillan, and later, Denis
Evans [Physics Letters A 91 (1982)] solved the problem of simulating heat
flow with nonequilibrium molecular dynamics in such a way that the Green-Kubo
conductivity was reproduced. Although isokinetic thermostats had been in
use since the 1970's (I think Les Woodcock [Chemical Physics Letters 10 (1971)]
was the first to use them) the subject was revolutionized in 1984 by
Shuichi
Nosé (Keio University)
who showed how to reproduce Gibbs' isothermal and
isobaric ensembles with deterministic equations of motion. Nosé
reviewed the literature relevant to his discovery in a 1991 supplement volume
of Progress in Theoretical Physics. A useful paper was written by Brad
Holian and Denis Evans [Journal of Chemical Physics 83 (1985)], to show that
the type of thermostat was not important to the results. A preview of the
Holian-Evans work appears in the historically interesting Fermi School
Proceedings, "Simulation of Statistical Mechanics Systems" (1985).
I used a simpler approach than Nosé's, based on Liouville's Theorem, which
was extended and elaborated in some important papers by Bauer, Bulgac,
and Kusnezov. For references to this older work, see my paper with Holian [Physics Letters A
211, 253 (1996)].
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To me, it is a very curious thing that some mathematically-inclined people
seek out "symplectic" integrators for microscopic equations of motion. Such
integrators conserve, among other things, phase volume, in exact accord with
Hamilton's equilibrium equations of motion. And extensions to nonequilibrium
equations of motion (which lead to phase-volume changes) can be developed too.
Though conserving phase volume appears to be a good thing it comes at a price.
Trajectories are less accurate (for a given amount of computer time) than
those generated with fourth-order Runge-Kutta integrators and the rate at
which the system moves through phase space is likewise quite inaccurate.
For the simple harmonic oscillator, with
H = ½(q ^{2} + p^{2}), the symplectic "Verlet"
algorithm (well-known prior to Verlet, and used long before, for instance,
by Fermi at Los Alamos and by Vineyard at the Brookhaven Laboratory)
with a timestep of unity (dt = 1) gives the six successive (q,p)
pairs (+1,0), (+1,-1), (0,-1), (-1,0), (-1,+1), (0,+1),
which then repeat indefinitely. The period is too short, precisely six
rather than the exact value of twice pi. The symplectic
advocates like to point out that there is a "shadow Hamiltonian" for a
system which is closeby. From Yoshida's work this shadow Hamiltonian
is H(Yoshida) = ½(q^{2} + qpdt + p^{2}) for
the oscillator and it is indeed constant for the "motion" just discussed.
On the other hand, an analytic solution of the motion for Yoshida's Hamiltonian
shows that that motion is actually slower, rather than faster, than the
actual oscillator motion from Hamilton. Evidently a clock based on the
symplectic integrators is distressingly fast or slow (there are many other
higher-order symplectic integrators). The time error, together with the
inaccurate trajectories, suggest that Runge and Kutta were on the right
track after all.
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Klimenko and Dremin's shockwave simulations broke new ground. For references
see the later paper by Holian, Moran, Straub, and me [Physical Review A 22, 2798
(1980)]. Rapaport [Physical Review A 46 (1992)] and Puhl, Mansour,
and Mareschal [Physical Review A 40 (1989)] provided early simulations of
Rayleigh-Bénard instability. There is an interesting paper by
Watanabe and Kaburaki [Physical Review E 54 (1996)] on this subject. Work
on fracture, with Bill Ashurst, [Physical Review B 14, 1465 (1976)] and plastic
flow, with Tony Ladd and Bill Moran [Physical
Review Letters 48, 1818 (1982)] showed that the computer
simulations can, with difficulty, reproduce well-known macroscopic results
but are of marginal predictive value. The connection of the Least-Action
Principle to equilibrium molecular dynamics was clarified by Gililan and
Wilson [Journal of Chemical Physics 97 (1992)] and applied by me to the
thermostated case as well [Physica D 112, 225 (1998)]. With Christoph Dellago
and Harald Posch I found that under some circumstances the fractal thermostated trajectories
correspond exactly to normal Hamiltonian ones [Physical Review E 57, 4969 (1998)]. Bulk
Viscosity for Lennard-Jones systems was simulated by Hickman, Holian, Ladd, and
me[Physical Review A 21, 1756 (1980)] well before Nosé's thermostats were developed.
The (falsity) of the principle of "Material Frame Indifference" could be
demonstrated by molecular dynamics simulations [Hoover, Moran, More, and Ladd,
Physical Review A 24, 2109 (1981)]. Resolution of Loschmidt's Paradox: the
Origin of Irreversible Behavior in Atomistic Dynamics (with Brad Holian
and Harald Posch) [Physical Review Letters 59, 10 (1987)].
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*Nonequilibrium Temperature(s)*
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Very recently [2007 through 2010] Carol and I studied some unexplored
thermostats based on
standard Hamiltonian (or Lagrangian) mechanics. The thermostats constrain
configurational or kinetic temperatures using Lagrange multipliers. Though
perfectly well-suited to equilibrium systems these Hamiltonian thermostats
fail when applied to a two-temperature heat flow problem (see the January
2007 lectures on the main page). A tentative conclusion from this work is
that standard Hamiltonian mechanics is simply not useful for nonequilibrium
stationary states; Hamiltonian mechanics cannot produce the fractal structures
characterizing such states. Although the configurational temperature first
appeared (so far as I know) in the 1951 Russian Edition of Landau and Lifshitz'
Statistical Physics there has been considerable effort to apply it recently.
Igor Rychkov, working with Debra Searles Bernhardt at Griffith University,
and Owen Jepps, working with Denis Evans at the Australian National University,
have made some interesting contributions to the area, available on the LANL
arXiv and
explored in our recent publication in the Journal of Chemical Physics
Spring 2007. Karl Travis and Carlos Braga emphasized and illustrated
the usefulness of Liouville's
Theorem to deriving motion equations consistent with Gibbsian ensembles. They
derived an analog of Nosé-Hoover mechanics, consistent with the canonical
ensemble, based on Landau and Lifshitz' configurational temperature. The same
algorithm was suggested earlier in Owen Jepps' unpublished thesis work at the
Australian National University.
It is an old idea of David Jou's that the nonequilibrium temperature lies
below the "real" or "operational" or "actual" temperature. The nonequilibrium
temperature is depressed by nonequilibrium constraints. We investigated phi-4
systems in which heat is transferred from a cold Nosé-Hoover
particle to a hot one through an intervening Newtonian particle. The kinetic and
configurational temperatures of that Newtonian particle do in fact lie
significantly below the temperature of an attached thermometric chain of particles
(Physical Review E 77, 041104
2008). In this work we include a detailed dynamical demonstration of the
instantaneous and nonlocal nature of kinetic (as opposed to configurational)
temperature. This work shows why it is that kinetic temperature is a much
better choice than the alternatives, whenever spatial gradients are large.
Because the kinetic temperature is obviously anisotropic (a symmetric
second-rank tensor), there can be no straightforward extension of the Zeroth
Law of Thermodynamics to nonequilibrium systems. It was quite interesting for
me to realize (in the Fall of 2008) that an ideal gas of very small
"Hard Parallel Cubes" is a perfect thermometer for nonequilibrium temperatures,
as it can measure the {xx,yy,zz} temperatures simultaneously. A part of my PhD
dissertation (in 1961, under Andy De Rocco's direction) at Michigan dealt with
the seven-term virial expansion for a nonideal gas of such cubes. By now the
eighth, ninth, and tenth terms in the series can be evaluated for hard particles!
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