Currents of particles or energy in driven non-equilibrium steady states are known to satisfy certain symmetries, referred to as fluctuation relations, determining the ratio of the probabilities of positive fluctuations to negative ones. A generalization of these fluctuation relations has been proposed recently for extended non-equilibrium systems of dimension greater than one, assuming, crucially, that they are isotropic [P. I. Hurtado, C. Pérez-Espigares, J. J. del Pozo, and P. L. Garrido, Proc. Nat. Acad. Sci. (USA) 108, 7704 (2011)]. Here we relax this assumption and derive a fluctuation relation for d-dimensional systems having anisotropic bulk driving rates. We test the validity of this anisotropic fluctuation relation by calculating the particle current fluctuations in the 2-d anisotropic zero-range process, using both exact and fluctuating hydrodynamic approaches.Here P (A, t) denotes the probability density function 1 (pdf) of the time-averaged observable A and c is a time-2 independent constant. This relation has been derived for 3 many non-equilibrium observables, including the entropy 4 production of chaotic systems, integrated currents in inter-5 acting particle models, and work-or heat-like quantities 6 defined in the context of driven Langevin equations [10-13]. 7The GCFR has also been verified experimentally, e.g., in 8 turbulent fluids [14] and for manipulated Brownian parti-9 cles [15, 16]. 10 Recently, Hurtado et al. [17] have proposed a generaliza-R. Villavicencio-Sanchez1 R. J. Harris1 H. Touchette2,3 hydrodynamic fluctuation theory as well as exactly from 27 the microscopic definition of the process for system sizes 28 up to 10 5 × 10 5 sites, which is much larger than currently 29 accessible in numerical simulations. This allows us to de-30 termine in a precise way the regime of current fluctuations 31 for which the AFR effectively describes the fluctuation 32 symmetries of extended systems. 33 function. A similar calculation can be done for w n = n, 101 which corresponds to non-interacting particles. In both 102 cases, the results confirm that the currents satisfying the 103 AFR (13) are located on ellipses verifying (14). This is 104 shown for the interacting case w n = 1 in Fig. 2 with hop-105 ping rates p x = 1 and p y = 1/2. Moreover, the optimal 106 density profile associated with currents on each ellipse is 107 invariant, a non-trivial result which follows again from (11). 108The specific shape of the optimal density profile depends 109 on the current fluctuation considered, as shown in Fig. 3, 110 again for the case w n = 1. We observe that the non-111 linearity of Eq. (20) results in two different kinds of density 112 profile: for small fluctuations, the density is maximal at 113 the left boundary and decreases monotonically, whereas for 114 large fluctuations, the maximum density occurs at a point 115 x max between the two boundaries. For w n = w, the density 116 at x max diverges at a critical current given by J T c ΛJ c = K 117 where K can be explicitly calculated. This is the ...
We study numerically the two-dimensional Ising model with non-conserved dynamics quenched from an initial equilibrium state at the temperature Ti ≥ Tc to a final temperature T f below the critical one. By considering processes initiating both from a disordered state at infinite temperature Ti = ∞ and from the critical configurations at Ti = Tc and spanning the range of final temperatures T f ∈ [0, Tc[ we elucidate the role played by Ti and T f on the aging properties and, in particular, on the behavior of the autocorrelation C and of the integrated response function χ. Our results show that for any choice of T f , while the autocorrelation function exponent λC takes a markedly different value for Ti = ∞ [λC (Ti = ∞) 5/4] or Ti = Tc [λC (Ti = Tc) 1/8] the response function exponents are unchanged. Supported by the outcome of the analytical solution of the solvable spherical model we interpret this fact as due to the different contributions provided to autocorrelation and response by the large-scale properties of the system. As changing T f is considered, although this is expected to play no role in the large-scale/long-time properties of the system, we show important effects on the quantitative behavior of χ. In particular, data for quenches to T f = 0 are consistent with a value of the response function exponent λχ = 1 2 λC (Ti = ∞) = 5/8 different from the one [λχ ∈ (0.5 − 0.56)] found in a wealth of previous numerical determinations in quenches to finite final temperatures. This is interpreted as due to important pre-asymptotic corrections associated to T f > 0.
In order to illuminate the properties of current fluctuations in more than one dimension, we use a lattice-based Markov process driven into a non-equilibrium steady state. Specifically, we perform a detailed study of the particle current fluctuations in a two-dimensional zero-range process with open boundary conditions and probe the influence of the underlying geometry by comparing results from a square and a triangular lattice. Moreover, we examine the structure of local currents corresponding to a given global current fluctuation and comment on the role of spatial inhomogeneities for the discrepancies observed in testing some recent fluctuation symmetries.
We study the zero-range process on a simple diamond lattice with open boundary conditions and determine the conditions for the existence of loops in the mean current. We also perform a large deviation analysis for fluctuations of partial and total currents and check the validity of the Gallavotti–Cohen fluctuation relation for these quantities. In this context, we show that the fluctuation relation is not satisfied for partial currents between sites even if it is satisfied for the total current flowing between the boundaries. Finally, we extend our methods to study a chain of coupled diamonds and demonstrate co-existence of mean current regimes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.