Statistical mechanical theory for steady state systems. VIII. General theory for a Brownian particle driven by a time- and space-varying force

Abstract

A Brownian particle subject to a time- and space-varying force is studied with the second entropy theory for nonequilibrium statistical mechanics. A fluctuation expression is obtained for the second entropy of the path, and this is maximized to obtain the most likely path of the particle. Two approaches are used, one based on the velocity correlation function and one based on the position correlation function. The approaches are a perturbation about the free particle result and are exact for weak external forces. They provide a particularly simple way of including memory effects in time-varying driven diffusion. The theories are tested against computer simulation data for a Brownian particle trapped in an oscillating parabolic well. They accurately predict the phase lag and amplitude as a function of drive frequency, and they account quantitatively for the memory effects that are important at high frequencies and that are missing in the simplest Langevin equation.

Statistical mechanical theory for non-equilibrium systems. IX. Stochastic molecular dynamics

“…The position autocorrelation function of the solute in the solvent in the absence of the trap, which is required as input into the perturbation theory, was obtained using the equilibrium Nosé-Hoover chain thermostat simulations. 26 The stochastic thermostat performed well. A time step of ⌬ t =10 −3 and a variance =10 −3 were used.…”

“…26 Figure 6 shows that solvent memory effects, which causes the breakdown of the Langevin approximation at higher frequencies, are well accounted for second entropy perturbation theory. 26 Keeping terms to fifth order describes with reasonable accuracy all transient effects in the present system. The position autocorrelation function of the solute in the solvent in the absence of the trap, which is required as input into the perturbation theory, was obtained using the equilibrium Nosé-Hoover chain thermostat simulations.…”

“…The diffusion constant used in the Langevin equation, D = 0.105, was obtained from equilibrium Nosé-Hoover chain thermostat simulations, ͑the mean square displacement in the absence of a trap͒. 26 Figure 6 shows that solvent memory effects, which causes the breakdown of the Langevin approximation at higher frequencies, are well accounted for second entropy perturbation theory. 26 Keeping terms to fifth order describes with reasonable accuracy all transient effects in the present system.…”

“…The results of the present SMD algorithm are compared with conventional Nosé-Hoover simulations that employ a chain thermostat. 26 As discussed in Sec. II C, the most likely trajectory for this type of intensive mechanical work has the same functional form as in the equilibrium case.…”

Statistical mechanical theory for nonequilibrium systems. X. Nonequilibrium phase transitions

“…No doubt the regime of application of the present results will be similarly stretched, although in the case of the second entropy there is a formally exact path for treating transient phenomena and including memory effects. 2,22 This section shows that the integral form of the second entropy, Eq. ͑23͒, provides the variational principle for single component nonequilibrium systems.…”