Non-equilibrium Diffusive Systems Research Articles

Correlations in nonequilibrium diffusive systems.

We study the behavior of stationary nonequilibrium two-body correlation functions for diffusive systems with equilibrium reference states (DSe). We describe a DSe at the mesoscopic level by M locally conserved continuum fields that evolve through coupled Langevin equationswith white noises. The dynamic is designed such that the system may reach equilibrium states for a set of boundary conditions. In this form, we make the system driven to a nonequilibrium stationary state by changing the equilibrium boundary conditions. We decompose the correlations in a known local equilibrium part and another one that contains the nonequilibrium behavior and that we call correlation's excess C[over ¯](x,z). We formally derive the differential equationsfor C[over ¯]. To solve them order by order, we define a perturbative expansion around the equilibrium state. We show that the C[over ¯]'s first-order expansion, C[over ¯]^{(1)}, is always zero for the unique field case, M=1. Moreover, C[over ¯]^{(1)} is always long range or zero when M>1. We obtain the surprising result that their associated fluctuations, the space integrals of C[over ¯]^{(1)}, are always zero. Therefore, fluctuations are dominated by local equilibrium up to second order in the perturbative expansion around the equilibrium. We derive the behaviors of C[over ¯]^{(1)} in real space for dimensions d=1 and 2 explicitly. Finally, we derive the two first perturbative orders of the correlation's excess for a generic M=2 case and a hydrodynamic model.

Structure of the optimal path to a fluctuation

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 constrains 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 spatiotemporal nonlocality of the current statistics and the associated optimal trajectories.

Correlations of the density and of the current in non-equilibrium diffusive systems

We use fluctuating hydrodynamics to analyze the dynamical properties in the non-equilibrium steady state of a diffusive system coupled with reservoirs. We derive the two-time correlations of the density and of the current in the hydrodynamic limit in terms of the diffusivity and the mobility. Within this hydrodynamic framework we discuss a generalization of the fluctuation dissipation relation in a non-equilibrium steady state where the response function is expressed in terms of the two-time correlations. We compare our results to an exact solution of the symmetric exclusion process. This exact solution also allows one to directly verify the fluctuating hydrodynamics equation.

Thermodynamics of stationary states

In the study of nonequilibrium phenomena, improving on near-equilibrium linear approximations has long been a major goal. Over the past ten years, in collaboration with Bertini, De Sole, Gabrielli and Landim, we developed a general approach to nonequilibrium diffusive systems known as Macroscopic Fluctuation Theory, making some progress in this direction. Our theory has been inspired by stochastic models of interacting particles (stochastic lattice gases). It is based on the study of rare fluctuations of macroscopic variables in stationary states and leads to interesting new results and predictions. This paper gives a compact presentation of the theory with some applications.

Thermodynamics of Currents in Nonequilibrium Diffusive Systems: Theory and Simulation

Understanding the physics of nonequilibrium systems remains as one of the major challenges of theoretical physics. This problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics and associated structures. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new 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.

Anomalous long-range correlations at a non-equilibrium phase transition

Non-equilibrium diffusive systems are known to exhibit long-range correlations, which decay like the inverse 1/L of the system size L in one dimension. Here, taking the example of the ABC model, we show that this size dependence becomes anomalous (the decay becomes a non-integer power of L) when the diffusive system approaches a second-order phase transition. This power-law decay as well as the L-dependence of the time–time correlations can be understood in terms of the dynamics of the amplitude of the first Fourier mode of the particle densities. This amplitude evolves according to a Langevin equation in a quartic potential, which was introduced in a previous work to explain the anomalous behavior of the cumulants of the current near this second-order phase transition. Here we also compute some of these cumulants away from the transition and show that they become singular as the transition is approached, matching with what we already knew in the critical regime.

Distribution of current in nonequilibrium diffusive systems and phase transitions

We consider diffusive lattice gases on a ring and analyze the stability of their density profiles conditionally to a current deviation. Depending on the current, one observes a phase transition between a regime where the density remains constant and another regime where the density becomes time dependent. Numerical data confirm this phase transition. This time dependent profile persists in the large drift limit and allows one to understand on physical grounds the results obtained earlier for the totally asymmetric exclusion process on a ring.

Current Fluctuations in Nonequilibrium Diffusive Systems: An Additivity Principle

We formulate a simple additivity principle allowing one to calculate the whole distribution of current fluctuations through a large one dimensional system in contact with two reservoirs at unequal densities from the knowledge of its first two cumulants. This distribution (which in general is non-Gaussian) satisfies the Gallavotti-Cohen symmetry and generalizes the one predicted recently for the symmetric simple exclusion process. The additivity principle can be used to study more complex diffusive networks including loops.