Nonequilibrium Molecular Dynamics
   

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)].

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(q² + p²), 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(q² + qpdt + p²) 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.

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)].

Nonequilibrium Temperature(s)
   

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!