Structure of the optimal path to a fluctuation

Abstract

Macroscopic fluctuations have become an essential tool to understand physics far from equilibrium due to the link between their statistics and nonequilibrium ensembles. The optimal path leading to a fluctuation encodes key information on this problem, shedding light on e.g. the physics behind the enhanced probability of rare events out of equilibrium, the possibility of dynamic phase transitions and new symmetries. This makes the understanding of the properties of these optimal paths a central issue. Here we derive a fundamental relation which strongly constraints the architecture of these optimal paths for general d-dimensional nonequilibrium diffusive systems, and implies a nontrivial structure for the dominant current vector fields. Interestingly, this general relation (which encompasses and explains previous results) makes manifest the spatio-temporal non-locality of the current statistics and the associated optimal trajectories.

Microscopic fluctuation theory (mFT) for interacting Poisson processes

“…large deviations at 'Level 2.5' for the joint distribution of the time-empirical-densities and the time-empirical-flows have been written in terms of explicit local-in-space rate functionals within various frameworks, namely for discretetime and discrete-space Markov chains [3,17,18], for continuous-time and discrete-space Markov jump processes [12,17,[19][20][21][22] and for continuous-time and continuous-space diffusion processes [12,22,25,26]. This 'Level 2.5' formulation allows to reconstruct any time-additive observable of the dynamical trajectory via its decomposition in terms of the empirical densities and of the empirical flows, and is thus closely related to the studies focusing on the generating functions of time-additive observables via deformed Markov operators that have attracted a lot of interest recently in various models [4,8,9,[27][28][29][30][31][32][33][34][35][36][37]. In Ref [21], this 'Level 2.5' for the joint distribution of the time-empirical-densities and the time-empirical-flows for a single Markov jump process has been extended to 'Level 2.5 in time' for the joint distribution of the ensembleempirical-occupations N t (x) and the ensemble-empirical-flows q t (y, x) for a large number of independent Markov jump processes involving the transitions rates w t (y, x) from site x to site y at time t. The output is the following measure on dynamical trajectories…”

Infinite family of universal profiles for heat current statistics in Fourier's law

“…where the average is defined with respect to the PDF P(J; t). The sCGF works as a dynamical free energy, and fully characterizes the PDF of the total current J [35,36]. The vector λ is conjugated to the averaged current J, in a similar way to the relation between temperature and energy in equilibrium.…”

“…The general solution of the spatiotemporal problem (19) remains a major challenge in most cases [25,26,86,87]. However, a powerful conjecture known as additivity principle has been put forward for systems in d = 1 [38,[88][89][90][91][92] and recently extended for d > 1 [35,36] which strongly simplifies the variational problem at hand. In brief, this additivity principle assumes that, except for initial and final transients of negligible statistical weight, the optimal path associated to a current fluctuation is time independent.…”

“…Together with the additivity principle, this can be shown to imply (see Appendix A and Refs. [35,36]) that the optimal current field exhibits a nontrivial structure of the form…”

“…In recent years a series of works have analyzed current fluctuations in a broad family of stochastic mod- * garrido@onsager.ugr.es † phurtado@onsager.ugr.es ‡ tizon@onsager.ugr.es els of transport [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37], offering a deeper comprehension of nonequilibrium fluctuating behavior. A key tool in these developments has been the Macroscopic Fluctuation Theory (MFT) of Bertini and coworkers [22-28, 38, 39], which describes dynamical fluctuations in diffusive equilibrium and nonequilibrium media starting from their fluctuating hydrodynamic description in terms of two transport coefficients, the diffusivity and the mobility.…”