Thermodynamics of Currents in Nonequilibrium Diffusive Systems: Theory and Simulation

Abstract

Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics, associated structures and microscopic origin. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. In this work we study the statistics of current fluctuations in nonequilibrium diffusive systems, using macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. Our quest will bring us from (1) the confirmation of an additivity conjecture in one and two dimensions, which considerably simplifies the MFT complex variational problem to compute the thermodynamics of currents, to (2) the discovery of novel isometric fluctuation relations, which opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations, and to (3) the observation of coherent structures in fluctuations, which appear via dynamic phase transitions involving a spontaneous symmetry breaking event at the fluctuating level. The clear-cut observation, measurement and characterization of these unexpected phenomena, well described by MFT, strongly support this theoretical scheme as the natural theory to understand the thermodynamics of currents in nonequilibrium diffusive media, opening new avenues of research in nonequilibrium physics.

Symmetry and the thermodynamics of currents in open quantum systems

“…As we show below, distinct symmetry eigenspaces may dominate different fluctuation regimes, separated by first-order-type dynamic phase transitions. Note that a different type of spontaneous symmetry breaking scenario at the fluctuating level has been recently reported in classical diffusive systems [6,8,[40][41][42] .…”

“…Our aim now is to study the implications of such strong symmetry for the statistics of the current flowing through a given reservoir, a main observable out of equilibrium [4,[6][7][8][34][35][36][37][38]. For that, we first introduce the reduced density matrix ρ Q (t), which is the projection of the full density matrix to the space of Q events, Q being the total (energy, spin, exciton, ...) current flowing from a reservoir to the system in a time t. This current can be appropriately defined in the quantum realm via the unraveling of the master equation (1) [36].…”

“…For that, we first introduce the reduced density matrix ρ Q (t), which is the projection of the full density matrix to the space of Q events, Q being the total (energy, spin, exciton, ...) current flowing from a reservoir to the system in a time t. This current can be appropriately defined in the quantum realm via the unraveling of the master equation (1) [36]. The probability of observing a given current fluctuation, typical or rare, is thus P t (Q) = Tr[ρ Q (t)], and scales in a large deviation form for long times, P t (Q) exp[+tG(Q/t)], where G(q) 0 is the current large deviation function [3][4][5][6][7][8]. This scaling shows that the probability of observing a significant current fluctuation away from its average is exponentially small in time.…”

“…With this idea in mind, we introduce the Laplace transform ρ λ (t) = Q ρ Q (t) exp(−λQ) with λ a counting field conjugated to the current, such that Z λ (t) ≡ Tr[ρ λ (t)] corresponds to the moment generating function of the current distribution, which also obeys a large deviation principle of the form Z λ (t) exp[+tμ(λ)] for long times. Here μ(λ) = max q [G(q) − λq] is the Legendre transform of the current LDF, in a way equivalent to the thermodynamic relation between the canonical and microcanonical potentials [3][4][5][6][7][8]. Interestingly, ρ Q (t) obeys a complex hierarchy of equations which is however disentangled by the Laplace transform [4,39], yielding a closed evolution equation for ρ λ (t)…”

“…Advancing this line of research is both of fundamental and practical importance. On one hand, the current LDF plays in nonequilibrium a role equivalent to the equilibrium free energy, governing the thermodynamics of currents and hence the transport and collective behavior out of equilibrium [6][7][8].…”