Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States

By Lorenzo Bertini, A. Sole, D. Gabrielli, Giovanni Jona-Lasinio, C. Landim. 2001

398 citations recorded by the source corpus.

We formulate a dynamical fluctuation theory for stationary non-equilibrium states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Within this theory we derive the following results: the modification of the Onsager–Machlup theory in the SNS; a general Hamilton–Jacobi equation for the macroscopic entropy; a non-equilibrium, nonlinear fluctuation dissipation relation valid for a wide class of systems; an H theorem for the entropy. We discuss in detail two models of stochastic boundary driven lattice gases: the zero range and the simple exclusion processes. In the first model the invariant measure is explicitly known and we verify the predictions of the general theory. For the one dimensional simple exclusion process, as recently shown by Derrida, Lebowitz, and Speer, it is possible to express the macroscopic entropy in terms of the solution of a nonlinear ordinary differential equation; by using the Hamilton–Jacobi equation, we obtain a logically independent derivation of this result.

Subfields: Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c9fac2a6d34bdd1e71d4b946511f77e9711b4764

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In Statistical and Nonlinear Physics

Citations in the corpus

Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process (2001, neutral): “3Also Department of Physics, Rutgers.”
Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport (2011, neutral): “2434 [83] Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G, Landim C.”
On a class of first order Hamilton–Jacobi equations in metric spaces (2014, background): “This includes examples in statistical mechanics [4, 18], optimal control and probability [15, 16], classical game theory [8, 9], fluid mechanics [19, 20, 14, 17, 21, 22], and mean-field games (we refer to [7] for a compilation of references).”
Stochastic interacting particle systems out of equilibrium (2007, background): “Finally, as discussed in [3], for generic nonequilibrium models the quasi-potential V is a non-local functional of ρ.”
The large deviation approach to statistical mechanics (2008, background): “[140] give an interesting derivation of the above formula by studying the rate function associated with the time-reverse image of the trajectories x(t) determined by Eq.”
Non Equilibrium Current Fluctuations in Stochastic Lattice Gases (2005, background): “(4), we can integrate by parts the left hand side above and get J (ρ) + J (ρ) = −χ (ρ)∇ δF δρ (5.”
Temperature in and out of equilibrium: A review of concepts, tools and attempts (2017, background): “Despite some important recent results, such as the Fluctuation Relations or the extensions of the fluctuation–dissipation theorem [165,71,318], general methods to treat nonequilibriumdynamics are not available.”
Macroscopic fluctuation theory (2014, methodology): “Here we just show, in a specific example, how the basic principles of the MFT need to be modified to cover these processes.”
Metric viscosity solutions of Hamilton–Jacobi equations depending on local slopes (2015, background): “Such equations arise for instance in statistical mechanics and large deviations [9, 10, 11, 12, 13, 14, 32, 33, 34], fluid mechanics [30, 33, 34, 35, 36, 38, 39, 40], Mean Field Games [16, 51, 52], optimal control [31, 33, 34], study of functional inequalities [6, 42, 43, 53] and other related areas”
Lattice models of nonequilibrium bacterial dynamics (2010, background): “This is the usual fluctuating hydrodynamics, as considered, for instance, in [32].”
Lecture notes on Generalised Hydrodynamics (2019, background): “encode, in system-independent symmetry relations, properties of driving potentials into ﬂuctuations of quantities transported over long times. It is also at the basis of macroscopic ﬂuctuation theory [23,24,25,26], which establishes a deep connection between the hydrodynamics of diffusive transport,”
Nonequilibrium steady states of matrix-product form: a solver's guide (2007, background): “11) has been obtained in a purely macroscopic formalism [135].”
Fluctuation-dissipation: Response theory in statistical physics (2008, methodology): “[272,167] has a nonlocal nature, as easily understood in the works of Derrida and collaborators [273,166], who independently worked out the explicit solutions of what can now be seen as specific examples of the theory of [272,167].”
Frenesy: Time-symmetric dynamical activity in nonequilibria (2019, background): “In a macroscopic dynamical fluctuation theory, [52–54] we examine the probability of empirically possible trajectories.”