Nonequilibrium Monte Carlo simulation for a driven Brownian particle

Abstract

The author's nonequilibrium probability distribution is tested for time-varying mechanical work. Nonequilibrium Monte Carlo (NEMC) is used to simulate a Brownian particle in a soft-sphere solvent, driven by a moving external potential. Results are obtained for the phase lag and amplitude for drive frequencies ranging from the steady state to the transient regime. This now extends the application of the NEMC algorithm to a time-varying nonequilibrium system. The results are shown to agree with those obtained by nonequilibrium stochastic molecular dynamics and by Nosé-Hoover molecular dynamics, from which it is concluded that the nonequilibrium probability distribution correctly describes time-varying mechanical work and that it provides a fundamental basis for nonequilibrium statistical mechanics.

Simplified Derivation of the Non-Equilibrium Probability Distribution

“…The third justification follows from the resulting equations of motion and is given in the conclusion. This approximation has been found to be accurate in computer simulation tests for both mechanical [13][14][15] and thermodynamic 12,14 non-equilibrium systems.…”

“…In obtaining the final equality, the approximation S ′′ r (t) ≈ S ′′ st (t) has been used for the reservoir gradient, Eqs ( 14), (15), and (16). Further, by evaluating the first three terms at Γ(t 1 ) in the forward direction, the reservoir contribution to the evolution of the most likely trajectory has been identified as…”

“…12 Although still lacking broad consensus, it is true to say that it is the only non-equilibrium probability distribution that respects the second law of thermodynamics and that has been successfully tested analytically and numerically with computer simulation. [12][13][14][15] The most complete account of the theory and the various tests is given in Ref. [9].…”

Shear Thinning in Lennard-Jones Fluids by Stochastic Dissipative Molecular Dynamics Simulation

“…Alternatively, one could perform Monte Carlo simulations based on the non-equilibrium probability distribution, as has previously been done for heat flow 17 and for driven Brownian motion. 28 The non-equilibrium Monte Carlo (NEMC) method is somewhat inefficient computationally, but it has the advantage that versions can be formulated without invoking the fluctuation-dissipation parameter. However NEMC implicitly relies upon the reservoir formalism, so it is not at all clear that it would resolve all the ambiguities discussed here in the non-linear nonequilibrium regime.…”

Quantum Statistical Mechanics. IV. Non-Equilibrium Probability Operator and Stochastic, Dissipative Schrodinger Equation

“…There is an abundance of computer simulation data for classical system that show that the fluctuations in a non-equilibrium system are identical to those in the corresponding local equilibrium system. 1,2,45,46 Further, this replacement appears necessary on physical grounds, namely that in the dissipative Schrödinger equation, this term comes from the reservoir-sub-system interactions and it is the static part of the entropy operator that fully reflects such interactions (c.f. the discussion in the conclusion of Ref.…”