Le Chatelier Principle for Out-of-Equilibrium and Boundary-Driven Systems: Application to Dynamical Phase Transitions

Abstract

A stability analysis of out of equilibrium and boundary driven systems is presented. It is performed in the framework of the hydrodynamic macroscopic fluctuation theory and assuming the additivity principle whose interpretation is discussed with the help of a Hamiltonian description. An extension of Le Chatelier principle for out of equilibrium situations is presented which allows to formulate the conditions of validity of the additivity principle. Examples of application of these results in the realm of classical and quantum systems are provided. PACS numbers:Understanding the behaviour of out of equilibrium systems is an essential problem in physics [1] but surprisingly enough and except for few exact solutions, it still lacks both a macroscopic approach comparable to thermodynamics and a microscopic theory. However, a fruitful hydrodynamic description of driven diffusive systems far from equilibrium, the macroscopic fluctuation theory (hereafter MFT) has been proposed [2]. It is based on a variational principle which provides equations for the time evolution of the most probable density profile corresponding to a given fluctuation. The MFT was used to explore aspects of out of equilibrium systems [3][4][5][6][7][8]. The case of current fluctuations has been singled out due to its relevance to a broad range of problems generically known as full counting statistics which play an important role both in classical and quantum systems [9][10][11][12][13]. Quite often, a classical description is convenient enough to account for the behaviour of quantum systems driven out of equilibrium. Noise and current statistics in disordered quantum mesoscopic conductors or wave speckles [14,15], non equilibrium spins in superconductors [16] and thermal transport [17,18] provide important examples of such quantum systems. A great amount of effort has been devoted to the investigation of large current fluctuations since they provide a measure of the likeliness of the system to return to equilibrium. Whereas close to equilibrium, energy and density are almost uniform and can be described within linear response theory, for driving currents far enough from the steady state current, the system may preferably choose non uniform and timedependent solutions for these observables, very much like a dynamical phase transition.To make these considerations more precise, we consider a large system of size L connected for a long time to reservoirs of particles at different densities. It reaches a non-equilibrium steady state with a non vanishing and fluctuating particle current. These fluctuations are characterised by the probability P t (Q) for having a number of particles Q flowing through the medium during a time t. In the long time limit t → ∞, this probability follows a large deviation principle,where the large deviation function Φ t plays in that situation a role similar to the equilibrium free energy [19] Expression (1) is not an obvious result [20]. Moreover, finding an explicit expression for Φ t is a difficult optimisation problem. H...

Dynamical criticality in open systems: Nonperturbative physics, microscopic origin, and direct observation

“…Physically this means that, in order sustain a given mass-current long-time fluctuation, the system of interest settles after a negligible initial transient into a time-independent state (possibly followed by an equally negligible final transient). This property, known as Additivity Principle in literature [1,4,9,11,24,40,56,62,[80][81][82][83][84][85][86], strongly simplifies the variational problem at hand. In particular, the mass-current LDF now reads…”

Occupation-time statistics of a gas of interacting diffusing particles

“…Equations (39) are nonlinear, and the convenient Schrödinger analogy is lost. Still, it is not difficult to solve them.…”

Dynamical phase transitions in the current distribution of driven diffusive channels

“…In the latter case, the system is described by a small number of transport coefficients, and the LDF is obtained using the macroscopic fluctuation theory (MFT) [4,41,42]. This approach has shed light on many interesting properties of driven diffusive systems [30,[43][44][45][46][47][48][49][50][51].…”