Universal cumulants of the current in diffusive systems on a ring

Appert-Rolland, Cécile; Derrida, Bernard; Lecomte, Vivien; Wijland, Frédéric van

doi:10.1103/physreve.78.021122

Cited by 153 publications

(319 citation statements)

References 60 publications

Supporting

Mentioning

296

Contrasting

Order By: Relevance

“…A breakdown of the AP signals the onset of a dynamical phase transition. It is the purpose of this letter to formulate a necessary and sufficient condition for the validity of the AP for boundary driven systems with and without additional uniform external field E. This will extend results obtained in previous works [22][23][24] and allow to discuss the existence and the nature of such transitions. Despite the fact that out of equilibrium physics requires new approaches which are different from the familiar thermodynamics concepts, it is intuitively helpful to relate these two situations.…”

mentioning

confidence: 72%

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

Shpielberg

Akkermans

2016

Phys. Rev. Lett.

View full text Add to dashboard Cite

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...

show abstract

mentioning

confidence: 72%

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

Shpielberg

Akkermans

2016

Phys. Rev. Lett.

View full text Add to dashboard Cite

show abstract

“…The transport coefficients in (2) satisfy the local Einstein relation D(ρ) = χ(ρ) f ′′ (ρ), where f is the equilibrium free energy per unit of volume. The interaction with the external reservoirs specify the boundary conditions for the evolution defined by (1)- (2). Recalling that λ(t) is the chemical potential of the reservoirs, this boundary condition reads f ′ u(t, x) = λ(t, x), x ∈ ∂Λ.…”

mentioning

confidence: 99%

“…in which u is a generic density profile, J(t, u) is given by (2), and V λ(t),E(t) is the quasi potential relative to the state (λ(t), E(t)) with frozen t. Observe that the definition of the renormalized work involves the antisymmetric current J A (t) computed not at density profilē ρ λ(t),E(t) but at the solution u(t) of the time dependent hydrodynamic equation. That is, at time t we subtract the power the system would have dissipated if its actual state u(t) had been the stationary profile corresponding to (λ(t), E(t)).…”

mentioning

confidence: 99%

See 1 more Smart Citation

Clausius Inequality and Optimality of Quasistatic Transformations for Nonequilibrium Stationary States

et al. 2013

View full text Add to dashboard Cite

Nonequilibrium stationary states of thermodynamic systems dissipate a positive amount of energy per unit of time. If we consider transformations of such states that are realized by letting the driving depend on time, the amount of energy dissipated in an unbounded time window becomes then infinite. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work performed along any given transformation. We then show that the renormalized work satisfies a Clausius inequality and prove that equality is achieved for very slow transformations, that is in the quasi static limit. We finally connect the renormalized work to the quasi potential of the macroscopic fluctuation theory, that gives the probability of fluctuations in the stationary nonequilibrium ensemble. A main goal of nonequilibrium thermodynamics is to construct analogues of thermodynamic potentials for nonequilibrium stationary states. These potentials should describe the typical macroscopic behavior of the system as well as the asymptotic probability of fluctuations. As it has been shown in [1], this program can be implemented without the explicit knowledge of the stationary ensemble and requires as input the macroscopic dynamical behavior of systems which can be characterized by the transport coefficients. This theory, now known as macroscopic fluctuation theory, is based on an extension of Einstein equilibrium fluctuation theory to stationary nonequilibrium states combined with a dynamical point of view. It has been very powerful in studying concrete microscopic models but can be used also as a phenomenological theory. It has led to several new interesting predictions [2][3][4][5][6][7].From a thermodynamic viewpoint, the analysis of transformations from one state to another one is most relevant. This issue has been addressed by several authors in different contexts. For instance, following the basic papers [8][9][10], the case of Hamiltonian systems with finitely many degrees of freedom has been recently discussed in [11,12] while the case of Langevin dynamics is considered in [13].We here consider thermodynamic transformations for driven diffusive systems in the framework of the macroscopic fluctuation theory. With respect to the authors mentioned above, the main difference is that we deal with systems with infinitely many degrees of freedom and the spatial structure becomes relevant. For simplicity of notation, we restrict to the case of a single conservation law, e.g., the conservation of the mass. We thus consider an open system in contact with boundary reservoirs, characterized by their chemical potential λ, and under the action of an external field E. We denote by Λ ⊂ R d the bounded region occupied by the system, by x the macroscopic space coordinates and by t the macroscopic time. With respect to our previous work [1, 3, 4], we here consider the case in which λ and E depend explicitly on the time t, driving the system from a nonequilibrium state to...

show abstract

“…(3) Is there an extension to random graphs? (4) Is it possible to understand finite size corrections as in one dimension [1]? (5) Is there a generalization to systems in contact with more than 2 reservoirs at unequal densities?…”

mentioning

confidence: 99%

Universal current fluctuations in the symmetric exclusion process and other diffusive systems

Akkermans¹,

Bodineau²,

Derrida³

et al. 2013

EPL

View full text Add to dashboard Cite

-We show, using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs are the same on an arbitrary large finite domain in dimension d as in the one dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward.Introduction. -When a system is connected for a long time to two heat baths at unequal temperatures or to two reservoirs of particles at different densities, it reaches a nonequilibrium steady state, with a non vanishing current of heat or of particles. This current fluctuates with time and the study of its fluctuations and of its large deviations has become a central aspect in the theory of non equilibrium systems [4,5,12,16,19,20,23,24,26,27,29,34,35]. A quantity which characterizes these fluctuations is the probability P (Q t ) of observing an energy or a number of particles Q t flowing through the system during a time window t. A notorious property of these fluctuations is known as the fluctuation theorem [15,17,18,28] which establishes a general relation between P (Q t ) and P (−Q t ), starting from some time reversal symmetry of the microscopic dynamics. Apart from simple models, however, it is usually difficult to predict the whole shape of the distribution P (Q t ).

show abstract

Universal cumulants of the current in diffusive systems on a ring

Cited by 153 publications

References 60 publications

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

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

Clausius Inequality and Optimality of Quasistatic Transformations for Nonequilibrium Stationary States

Universal current fluctuations in the symmetric exclusion process and other diffusive systems

Contact Info

Product

Resources

About