A large deviation theory perspective on nanoscale transport phenomena

Abstract

Understanding transport processes in complex nanoscale systems, like ionic conductivities in nanofluidic devices or heat conduction in low dimensional solids, poses the problem of examining fluctuations of currents within nonequilibrium steady states and relating those fluctuations to nonlinear or anomalous responses. We have developed a systematic framework for computing distributions of time integrated currents in molecular models and relating cumulants of those distributions to nonlinear transport coefficients. The approach elaborated upon in this perspective follows from the theory of dynamical large deviations, benefits from substantial previous formal development, and has been illustrated in several applications. The framework provides a microscopic basis for going beyond traditional hydrodynamics in instances where local equilibrium assumptions break down, which are ubiquitous at the nanoscale.

Reinforcement learning of rare diffusive dynamics

“…In molecular systems, rare events determine the rates by which chemical reactions occur and phases interconvert, 5 and they also encode the response of systems driven to flow or unfold. [6][7][8][9][10] Strategies that afford a means of studying rare dynamical events in statistically unbiased ways are particularly desired, in order to deduce the intrinsic pathways by which they occur and to evaluate their likelihoods. Borrowing notions from reinforcement learning, 11 we have developed a method to generate rare dynamical trajectories directly through the optimization of an auxiliary dynamics that generates an ensemble of trajectories with the correct relative statistical weights.…”

Topological phases in the dynamics of the simple exclusion process

“…Refs. [1][2][3][4][5][6][7]). In the long time limit, the statistics of a time-extensive function of a stochastic trajectory (such as the dynamical activity [8,9] or a time-integrated current [10]) often obeys a large deviation (LD) principle, whereby its distribution and moment generating function (MGF) scale exponentially in time [1][2][3][4][5][6][7].…”

“…[1][2][3][4][5][6][7]). In the long time limit, the statistics of a time-extensive function of a stochastic trajectory (such as the dynamical activity [8,9] or a time-integrated current [10]) often obeys a large deviation (LD) principle, whereby its distribution and moment generating function (MGF) scale exponentially in time [1][2][3][4][5][6][7]. In the LD regime, all relevant information is contained in the functions in the exponent, known as the rate function for the probability and the scaled cumulant generating function (SCGF) for the MGF, with rate function and SCGF related by a Legendre transform [1][2][3][4][5][6][7].…”

“…In the long time limit, the statistics of a time-extensive function of a stochastic trajectory (such as the dynamical activity [8,9] or a time-integrated current [10]) often obeys a large deviation (LD) principle, whereby its distribution and moment generating function (MGF) scale exponentially in time [1][2][3][4][5][6][7]. In the LD regime, all relevant information is contained in the functions in the exponent, known as the rate function for the probability and the scaled cumulant generating function (SCGF) for the MGF, with rate function and SCGF related by a Legendre transform [1][2][3][4][5][6][7]. This is the generalization of the ensemble method of statistical mechanics to dynamics [11][12][13], with trajectories being the microstates, the long-time limit the thermodynamic limit, the MGF the partition sum, and rate function and SCGF the entropy density and free energy density, respectively.…”

“…In particular, the SCGF can be obtained [1][2][3][4][5][6][7] as the largest eigenvalue of a deformation, or tilting, of the Markov generator of the dynamics [14]. The problem of calculating dynamical LDs is thus equivalent to finding the ground state of a stoquastic Hamiltonian [15].…”