Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

Abstract

Abstract:We describe some general results that constrain the dynamical fluctuations that can occur in non-equilibrium steady states, with a focus on molecular dynamics. That is, we consider Hamiltonian systems, coupled to external heat baths, and driven out of equilibrium by non-conservative forces. We focus on the probabilities of rare events (large deviations). First, we discuss a PT (parity-time) symmetry that appears in ensembles of trajectories where a current is constrained to have a large (non-typical) value. We analyse the heat flow in such ensembles, and compare it with non-equilibrium steady states. Second, we consider pathwise large deviations that are defined by considering many copies of a system. We show how the probability currents in such systems can be decomposed into orthogonal contributions that are related to convergence to equilibrium and to dissipation. We discuss the implications of these results for modelling non-equilibrium steady states.

Ergodicity and large deviations in physical systems with stochastic dynamics

“…A short calculation shows that the extremum occurs for λ = 0, and is characterised by − ∇φ · v − ∇ 2 φ + 1 2 |∇φ| 2 − 2sb = 2γ (A6) 23 The integral τ 0 |∂tCt| 2 dt that appears in (A1) is not mathematically well-defined for processes like (30). This may be resolved by defining Pτ (C) as a density with respect to the path-measure for a Brownian motion (see for example [98]), or by considering P (C) to be the probability density of an explicitly time-discretised trajectory. In either case one arrives at the same result in (A2), which is well-defined and unambiguous.…”

“…We write C R for the trajectory that is obtained by reversing the arrow of time in trajectory C. In the simplest case, this means that C R τ −t = C t . More generally the time-reversal operation might involve a change in some system variables, such as reversal of molecular velocities, as in [97,98]. Then, a (time-integrated) measure of irreversibility for trajectory C within a given model can be identified as…”

Large deviation function for a driven underdamped particle in a periodic potential

“…So far, however, much less is known about the properties of large deviation functions for currents in the underdamped regime compared to the overdamped regime. Only very recently, large deviation theory has been considered for underdamped Langevin dynamics in a rather general, mathematical setting [21]. In this paper, we derive an explicit expression for the level 2 large deviation function for underdamped dynamics in a periodic potential.…”

On decompositions of non-reversible processes

“…In the rest of this section, combining results from [17] and an adaption of arguments in [5], we provide a mathematical interpretation of iso-dissipation forces. The key idea is to link them to the dual force (associated to a dual process of the original one).…”