Numerical study of the steady state fluctuation relations far from equilibrium

Equilibrium distribution functions: connection with microscopic dynamics

Standard textbook derivations of the equilibrium distribution function rely on assumptions that may not satisfy all readers. Here, we present a straightforward approach to derive the equilibrium distribution function from the microscopic dynamics, and review how it can be used to obtain the expected expressions. In molecular dynamics simulations the equations of motion are often modified to simulate different ensembles or phenomena. We show that in some cases these equations will sample an equilibrium ensemble whereas in other cases they will not. For example, we find that for charged particles driven by a field, an equilibrium distribution is only possible when the system is confined. Furthermore, the approach correctly predicts that neither SLLOD shear flow dynamics nor constant temperature dynamics with a Berendsen thermostat sample any time-independent phase space distributions.

Viscuit and the fluctuation theorem investigation of shear viscosity by molecular dynamics simulations: The information and the noise

The shear viscosity, η, of model liquids and solids is investigated within the framework of the viscuit and Fluctuation Theorem (FT) probability distribution function (PDF) theories, following Heyes et al. [J. Chem. Phys. 152, 194504 (2020)] using equilibrium molecular dynamics (MD) simulations on Lennard-Jones and Weeks-Chandler-Andersen model systems. The viscosity can be obtained in equilibrium MD simulation from the first moment of the viscuit PDF, which is shown for finite simulation lengths to give a less noisy plateau region than the Green-Kubo method. Two other formulas for the shear viscosity in terms of the viscuit and PDF analysis are also derived. A separation of the time-dependent average negative and positive viscuits extrapolated from the noise dominated region to zero time provides another route to η. The third method involves the relative number of positive and negative viscuits and their PDF standard deviations on the two sides for an equilibrium system. For the FT and finite shear rates, accurate analytic expressions for the relative number of positive to negative block average shear stresses is derived assuming a shifted Gaussian PDF, which is shown to agree well with non-equilibrium molecular dynamics simulations. A similar treatment of the positive and negative block average contributions to the viscosity is also shown to match the simulation data very well.

Fluctuation theorems

Fluctuation theorems, developed over the past 15 years, have resulted in fundamental breakthroughs in our understanding of how irreversibility emerges from reversible dynamics and have provided new statistical mechanical relationships for free-energy changes. They describe the statistical fluctuations in time-averaged properties of many-particle systems such as fluids driven to nonequilibrium states and provide some of the few analytical expressions that describe nonequilibrium states. Quantitative predictions on fluctuations in small systems that are monitored over short periods can also be made, and therefore the fluctuation theorems allow thermodynamic concepts to be extended to apply to finite systems. For this reason, we anticipate an important role for fluctuation theorems in the design of nanotechnological devices and in the understanding of biological processes. This review discusses these theorems, their physical significance, and results for experimental and model systems.