Symmetries in Fluctuations Far from Equilibrium

Symmetries in fluctuations far from equilibrium

Pablo I. Hurtado1, Carlos Pérez-Espigares, Jesús J. del Pozo, and Pedro L. Garrido

Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada 18071, Spain

Edited by Joel L. Lebowitz, Center for Mathematical Sciences Research, Piscataway, NJ, and approved February 22, 2011 (received for review September 6, 2010)

Fluctuations arise universally in nature as a reflection of the termine their behavior and function. In this way understanding discrete microscopic world at the macroscopic level. Despite their current statistics in these systems is of great practical significance. apparent noisy origin, fluctuations encode fundamental aspects of Despite the considerable interest and efforts on these issues, the physics of the system at hand, crucial to understand irreversi- exact and general results valid arbitrarily far from equilibrium bility and nonequilibrium behavior. To sustain a given fluctuation, a are still very scarce. The reason is that, whereas in equilibrium system traverses a precise optimal path in phase space. Here we phenomena dynamics is irrelevant and the Gibbs distribution show that by demanding invariance of optimal paths under sym- provides all the necessary information, in nonequilibrium physics metry transformations, new and general fluctuation relations valid dynamics plays a dominant role, even in the simplest situation of a arbitrarily far from equilibrium are unveiled. This opens an unex- nonequilibrium steady state (2–5). However, there is a remark- plored route toward a deeper understanding of nonequilibrium able exception to this absence of general results that has triggered physics by bringing symmetry principles to the realm of fluctua- an important surge in activity since its formulation in the mid tions. We illustrate this concept studying symmetries of the current nineties. This is the fluctuation theorem, first discussed in the distribution out of equilibrium. In particular we derive an isometric context of simulations of sheared fluids (14), and formulated fluctuation relation that links in a strikingly simple manner the rigorously by Gallavotti and Cohen under very general assump- probabilities of any pair of isometric current fluctuations. This re- tions (15). This theorem, which implies a relation between the lation, which results from the time-reversibility of the dynamics, probabilities of a given current fluctuation and the inverse event, includes as a particular instance the Gallavotti–Cohen fluctuation is a deep statement on the subtle consequences of time-reversal symmetry of microscopic dynamics at the macroscopic, irreversi- theorem in this context but adds a completely new perspective ble level. Particularly important here is the observation that on the high level of symmetry imposed by time-reversibility on symmetries are reflected at the fluctuating macroscopic level ar- the statistics of nonequilibrium fluctuations. The new symmetry bitrarily far from equilibrium. Inspired by this illuminating result, implies remarkable hierarchies of equations for the current cumu- we explore in this paper the behavior of the current distribution lants and the nonlinear response coefficients, going far beyond ’ – under symmetry transformations (20). Key to our analysis is the Onsager s reciprocity relations and Green Kubo formulas. We observation that, to facilitate a given current fluctuation, the sys- confirm the validity of the new symmetry relation in extensive tem traverses a well-defined optimal path in phase space (2–8, numerical simulations, and suggest that the idea of symmetry in 21). This path is, under very general conditions, invariant under fluctuations as invariance of optimal paths has far-reaching conse- certain symmetry transformations on the current. Using this in- quences in diverse fields. variance we show that for d-dimensional, time-reversible systems described by a locally conserved field and possibly subject to a ∣ ∣ ∣ ∣ large deviations rare events hydrodynamics transport boundary-induced gradient and an external field E, the probabil- entropy production ity PτðJÞ of observing a current J averaged over a long time τ obeys an isometric fluctuation relation (IFR) arge fluctuations, though rare, play an important role in many fields of science as they crucially determine the fate of a sys- 1 ðJÞ L Pτ ¼ ϵ ðJ − J0Þ; [1] tem (1). Examples range from chemical reaction kinetics or the lim ln 0 · τ→∞ τ PτðJ Þ escape of metastable electrons in nanoelectronic devices to con- formational changes in proteins, mutations in DNA, and nuclea- for any pair of isometric current vectors, jJj¼jJ0j. Here ϵ ¼ ε þ tion events in the primordial universe. Remarkably, the statistics E is a constant vector directly related to the rate of entropy of these large fluctuations contains deep information on the phy- production in the system, which depends on the boundary baths sics of the system of interest (2, 3). This is particularly important via ε (see below). for systems far from equilibrium, where no general theory exists The above equation, which includes as a particular case the up to date capable of predicting macroscopic and fluctuating Gallavotti–Cohen (GC) result for J0 ¼ −J, relates in a strikingly behavior in terms of microscopic physics, in a way similar to equi- simple manner the probability of a given fluctuation J with the librium statistical physics. The consensus is that the study of fluc- likelihood of any other current fluctuation on the d-dimensional tuations out of equilibrium may open the door to such general hypersphere of radius jJj, see Fig. 1, projecting a complex d-di- theory. As most nonequilibrium systems are characterized by mensional problem onto a much simpler one-dimensional theory. currents of locally conserved observables, understanding current Unlike the GC relation, which is a nondifferentiable symmetry statistics in terms of microscopic dynamics has become one of involving the inversion of the current sign, J → −J, Eq. 1 is valid the main objectives of nonequilibrium statistical physics (2–17). for arbitrary changes in orientation of the current vector. This Pursuing this line of research is both of fundamental as well as practical importance. At the theoretical level, the function con- Author contributions: P.I.H. and P.L.G. designed research; P.I.H., C.P.-E., J.J.d.P., and P.L.G. trolling current fluctuations can be identified as the nonequili- performed research; P.I.H., C.P.-E., J.J.d.P., and P.L.G. analyzed data; and P.I.H. wrote brium analog of the free-energy functional in equilibrium the paper. systems (2–5), from which macroscopic properties of a nonequi- The authors declare no conflict of interest. librium system can be obtained (including its most prominent This article is a PNAS Direct Submission. features, as for instance the ubiquitous long range correlations 1To whom correspondence should be addressed. E-mail: [email protected]. (18, 19), etc.) On the other hand, the physics of most modern This article contains supporting information online at www.pnas.org/lookup/suppl/ mesoscopic devices is characterized by large fluctuations that de- doi:10.1073/pnas.1013209108/-/DCSupplemental.

7704–7709 ∣ PNAS ∣ May 10, 2011 ∣ vol. 108 ∣ no. 19 www.pnas.org/cgi/doi/10.1073/pnas.1013209108 Downloaded by guest on September 24, 2021 exponentially unlikely in time. According to hydrodynamic fluctuation theory (2, 4–6),

Z 2 ðJ − QE½ρðrÞÞ GðJÞ¼−min dr; [4] ρðrÞ 2σ½ρðrÞ

which expresses the locally-Gaussian nature of fluctuations (6–8). The optimal profile ρ0ðr; JÞ solution of the above variational problem can be interpreted as the density profile the system adopts to facilitate a current fluctuation J (7, 8, 21). To derive Eq. 4 we assumed that (i) the optimal profiles associated to a given current fluctuation are time-independent (2–9, 21), and (ii) the optimal current field has no spatial structure (SI Text). This last hypothesis, which greatly simplifies the calculation of current statistics, can be however relaxed for our purposes (as shown below). The probability PτðJÞ is thus simply the Gaussian weight associated to the optimal profile. Note however that the minimization proce- Fig. 1. The isometric fluctuation relation at a glance. Sketch of the current dure gives rise to a nonlinear problem that results in general in a h i distribution in two dimensions, peaked around its average J ϵ, and isometric current distribution with non-Gaussian tails (2–8). j j contour lines for different J s. The isometric fluctuation relation, Eq. 1, estab- The optimal profile is solution of the following equation lishes a simple relation for the probability of current fluctuations along each of these contour lines. δω2½ρðrÞ δω1½ρðrÞ 2 δω0½ρðrÞ 0 − 2J · 0 þ J 0 ¼ 0; [5] makes the experimental test of the above relation a feasible pro- δρðr Þ δρðr Þ δρðr Þ

blem, as data for current fluctuations involving different orienta- δ tions around the average can be gathered with enough statistics where δρðr0Þ stands for functional derivative, and to ensure experimental accuracy. It is also important to notice Z Qn ½ρðrÞ that the isometric fluctuation relation is valid for arbitrarily large E ωn½ρðrÞ ≡ dr : [6] fluctuations; i.e., even for the non-Gaussian far tails of current σ½ρðrÞ distribution. We confirm here the validity of this symmetry in extensive numerical simulations of two different nonequilibrium Remarkably, the optimal profile ρ0ðr; JÞ solution of Eq. 5 systems: (i) A simple and very general lattice model of energy depends exclusively on J and J2. Such a simple quadratic depen- diffusion (7, 8, 22), and (ii) a hard-disk fluid in a temperature dence, inherited from the locally Gaussian nature of fluctuations, gradient (23). has important consequences at the level of symmetries of the current distribution. In fact, it is clear from Eq. 5 that the condition Results δω ½ρðrÞ The Isometric Fluctuation Relation Our starting point is a continuity 1 ¼ 0; [7] equation that describes the macroscopic evolution of a wide class δρðr0Þ of systems characterized by a locally conserved magnitude (e.g., energy, particle density, momentum, etc.) implies that ρ0ðr; JÞ will depend exclusively on the magnitude of the current vector, via J2, not on its orientation. In this way, all ∂tρðr;tÞ¼−∇ · ðQE½ρðr;tÞ þ ξðr;tÞÞ: [2] isometric current fluctuations characterized by a constant jJj will have the same associated optimal profile, ρ0ðr; JÞ¼ρ0ðr; jJjÞ, Here ρðr;tÞ is the density field, jðr;tÞ ≡ QE½ρðr;tÞ þ ξðr;tÞ is the independently of whether the current vector J points along the fluctuating current, with local average QE½ρðr;tÞ, and ξðr;tÞ is a gradient direction, against it, or along any arbitrary direction. Gaussian white noise characterized by a variance (or mobility) In other words, the optimal profile is invariant under current σ½ρðr;tÞ. This (conserved) noise term accounts for microscopic rotations if Eq. 7 holds. It turns out that condition [7] follows random fluctuations at the macroscopic level. Notice that the from the time-reversibility of the dynamics, in the sense that current functional includes in general the effect of a conservative the evolution operator in the Fokker–Planck formulation of external field, QE½ρðr;tÞ ¼ Q½ρðr;tÞ þ σ½ρðr;tÞE. Examples of Eq. 2 obeys a local detailed balance condition (16, 17). In this 2 – systems described by Eq. range from diffusive systems (2 9), case QE½ρðrÞ∕σ½ρðrÞ ¼ −∇ δH½ρ∕δρ, with H½ρðrÞ the system where Q½ρðr;tÞ is given by Fourier’s (or equivalently Fick’s) Hamiltonian, and condition [7] holds. The invariance of the op- law, Q½ρðr;tÞ ¼ −D½ρ∇ ρðr;tÞ, to most interacting-particle fluids timal profile can be now used in Eq. 4 to relate in a simple way (24, 25), characterized by a Ginzburg–Landau-type theory for the current LDF of any pair of isometric current fluctuations J the locally conserved particle density. To completely define the and J0, with jJj¼jJ0j, problem, the above evolution equation must be supplemented with appropriate boundary conditions, which may include an ex- GðJÞ − GðJ0Þ¼jϵjjJjðcos θ − cos θ0Þ; [8]

ternal gradient. PHYSICS 0 0 We are interested in the probability PτðJÞ of observing a space- where θ and θ are the angles formed by vectors J and J , respec- and time-averaged empirical current J, defined as tively, with a constant vector ϵ ¼ ε þ E; see below. Eq. 8 is just an Z Z alternative formulation of the isometric fluctuation relation [1]. 1 τ 0 J ¼ dt drjðr;tÞ: [3] By letting J and J differ by an infinitesimal angle, the IFR can τ 0 be cast in a simple differential form, ∂θGðJÞ¼jϵjjJj sin θ, which reflects the high level of symmetry imposed by time-reversibility This probability obeys a large-deviation principle for long times on the current distribution. d 0 (26, 27), PτðJÞ ∼ exp½þτL GðJÞ, where L is the system linear size The condition δω1½ρðrÞ∕δρðr Þ¼0 can be seen as a conserva- and GðJÞ ≤ 0 is the current large-deviation function (LDF), tion law. It implies that the observable ω1½ρðrÞ is in fact a meaning that current fluctuations away from the average are constant of motion, ϵ ≡ ω1½ρðrÞ, independent of the profile

Hurtado et al. PNAS ∣ May 10, 2011 ∣ vol. 108 ∣ no. 19 ∣ 7705 Downloaded by guest on September 24, 2021 0 0 0 ρðrÞ, which can be related with the rate of entropy production via GEðJÞ − GE ðJ Þ¼ε · ðJ − J Þ − ν · ðE − E ÞþJ · E − J · E ; the Gallavotti–Cohen theorem (15–17). In a way similar to [11] Noether’s theorem, the conservation law for ϵ implies a symmetry where we write explicitly the dependence of the current LDF for the optimal profiles under rotations of the current and a fluc- on the external field. The vector ν ≡ ∫ Q½ρðrÞdr is now another tuation relation for the current LDF. This constant can be easily ρðrÞ computed under very general assumptions (SI Text). constant of motion, independent of , which can be easily computed (SI Text). For a fixed boundary gradient, the above equation relates any current fluctuation J in the presence of an external Implications and Generalizations The isometric fluctuation relation, 0 1 field E with any other isometric current fluctuation J in the pre- Eq. , has far-reaching and nontrivial consequences. First, the E IFR implies a remarkable hierarchy of equations for the cumu- sence of an arbitrarily rotated external field , and reduces to E ¼ E δ ∫ Q½ρðrÞdr ¼ 0 lants of the current distribution, see Eq. 13 in Materials and Meth- the standard IFR for . Condition δρðr0Þ is ods. This hierarchy can be derived starting from the Legendre rather general, as most time-reversible systems with a local mo- σ½ρ transform of the current LDF, μðλÞ¼maxJ½GðJÞþλ · J, from bility do fulfill this condition (e.g., diffusive systems). which all cumulants can be obtained (3), and writing the IFR The IFR can be further generalized to cases where the current ii ½JðrÞ for μðλÞ in the limit of infinitesimal rotations. As an example, profile is not constant, relaxing hypothesis ( ) above. Let Pτ be the probability of observing a time-averaged current field the cumulant hierarchy in two dimensions implies the following JðrÞ¼τ−1∫ τ dtjðr;tÞ relations 0 . This vector field must have zero divergence because it is coupled via the continuity equation to an optimal 2 2 density profile that is assumed to be time-independent; see SI hJxi ¼ τL ½ϵxhΔJy i − ϵyhΔJxΔJyi ϵ ϵ ϵ [9] Text and hypothesis (i) above. Because of time-reversibility, 2 2 hJyiϵ ¼ τL ½ϵyhΔJx iϵ − ϵxhΔJxΔJyiϵ QE½ρðrÞ∕σ½ρðrÞ ¼ −∇ δH½ρ∕δρ and it is easy to show in the equation for the optimal density profile that the term linear in JðrÞ vanishes, so ρ0½r; JðrÞ remains invariant under (local or 2hΔJ ΔJ i ¼ τL2½ϵ hΔJ3i − ϵ hΔJ2ΔJ i x y ϵ y x ϵ x x y ϵ global) rotations of JðrÞ; see SI Text. In this way, for any diver- 2 3 2 gence-free current field J0ðrÞ locally isometric to JðrÞ,so ¼ τL ½ϵxhΔJy iϵ − ϵyhΔJxΔJy iϵ [10] J0ðrÞ2 ¼ JðrÞ2 ∀r, we can write a generalized isometric fluctua- 2 2 2 2 2 hΔJx iϵ − hΔJy iϵ ¼ τL ½ϵxhΔJxΔJy iϵ − ϵyhΔJx ΔJyiϵ; tion relation Z ΔJ ≡ J − hJ i 1 ½JðrÞ δH½ρ for the first cumulants, with α α α ϵ. It is worth stressing Pτ ^ 0 lim ln 0 ¼ dΓ n · ½J ðrÞ − JðrÞ; [12] that the cumulant hierarchy is valid arbitrarily far from equili- τ→∞ τ Pτ½J ðrÞ ∂Λ δρ brium. In a similar way, the IFR implies a set of hierarchies for the nonlinear response coefficients, see Eqs. 15–17 in Materi- where the integral (whose result is independent of ρðrÞ) is taken ðkÞ ðkx;kyÞ over the boundary ∂Λ of the domain Λ where the system is als and Methods. In our two-dimensional example, let ðnÞχðn ;n Þ be x y defined, and n^ is the unit vector normal to the boundary at each nx ny the response coefficient of the cumulant hΔJx ΔJy iϵ to order point. Eq. 12 generalizes the IFR to situations where hypothesis k kx y ii ϵx ϵy , with n ¼ nx þ ny and k ¼ kx þ ky. To the lowest order these ( ) is violated, opening the door to isometries based on local (in SI Text hierarchies imply Onsager’s reciprocity symmetries and Green– addition to global) rotations. As a corollary, we show in Kubo relations for the linear response coefficients of the current. that a similar generalization of the isometric fluctuation symme- They further predict that in fact the linear response matrix is pro- try does not exist whenever optimal profiles become time-dependent, so the IFR breaks down in the regime where hypothesis (i) ð1Þχð1;0Þ ¼ ð1Þχð0;1Þ ¼ ð0Þχð0;0Þ ¼ ð0Þχð0;0Þ portional to the identity, so ð1Þ ð1;0Þ ð1Þ ð0;1Þ ð2Þ ð2;0Þ ð2Þ ð0;2Þ is violated. In this way, we may use violations of the IFR and its ð1Þ ð0;1Þ ð1Þ ð1;0Þ generalizations to detect the instabilities that characterize the whereas ð1Þχð1;0Þ ¼ 0 ¼ ð1Þχð0;1Þ. The first nonlinear coefficients of fluctuating behavior of the system at hand (2, 9). the current can be simply written in terms of the linear coeffi- ð2Þχð2;0Þ ¼ 2ð1Þχð1;0Þ cients of the second cumulants as ð1Þ ð1;0Þ ð2Þ ð2;0Þ and Checking the Isometric Fluctuation Relation We have tested the ð2Þ ð0;2Þ ð1Þ ð1;0Þ validity of the IFR in extensive numerical simulations of two dif- ð1Þχð1;0Þ ¼ −2ð2Þχð1;1Þ, whereas the cross-coefficient reads ð2Þ ð1;1Þ ð1Þ ð0;1Þ ð1Þ ð0;1Þ ferent nonequilibrium systems. The first one is a simple and very ð1Þχð1;0Þ ¼ 2½ð2Þχð2;0Þ þ ð2Þχð1;1Þ (symmetric results hold for general model of energy diffusion (7, 8, 22) defined on a two- L2 nx ¼ 0, ny ¼ 1). Linear response coefficients for the second-order dimensional (2D) square lattice with sites. Each site is char- e i ∈ ½1;L2 ð1Þχð1;0Þ ¼ −ð1Þχð0;1Þ acterized by an energy i, , and models a harmonic os- cumulants also obey simple relations (e.g., ð2Þ ð1;1Þ ð2Þ ð1;1Þ cillator that is mechanically uncoupled from its nearest neighbors ð1Þ ð1;0Þ ð1Þ ð0;1Þ ð1Þ ð1;0Þ ð1Þ ð0;1Þ and ð2Þχð2;0Þ þð2Þ χð2;0Þ ¼ ð2Þχð0;2Þ þð2Þ χð0;2Þ) and the set of relations but interact with them via a stochastic energy-redistribution pro- continues to arbitrary high orders. In this way hierarchies [15–17], cess. Dynamics thus proceeds through random energy exchanges between randomly chosen nearest neighbors. In addition, left and which derive from microreversibility as reflected in the IFR, pro- right boundary sites may interchange energy with boundary baths vide deep insights into nonlinear response theory for nonequili- at temperatures TL and TR, respectively, while periodic boundary brium systems (28). conditions hold in the vertical direction. For TL ≠ TR the systems The IFR and the above hierarchies all follow from the invar- reaches a nonequilibrium steady-state characterized, in the ab- iance of optimal profiles under certain transformations. This idea sence of external field (the case studied here), by a linear energy can be further exploited in more general settings. In fact, by writ- profile ρ ðrÞ¼TL þ xðTR − TLÞ and a nonzero average current E 5 st ing explicitly the dependence on the external field in Eq. for given by Fourier’s law. This model plays a fundamental role in δ ∫ Q½ρðrÞdr ¼ 0 the optimal profile, one realizes that if δρðr0Þ , to- nonequilibrium statistical physics as a testbed to assess new the- gether with the time-reversibility condition, Eq. 7, the resulting oretical advances, and represents at a coarse-grained level a large optimal profiles are invariant under independent rotations of class of diffusive systems of technological and theoretical interest the current and the external field. It thus follows that the current (7, 8). The model is described at the macroscopic level by Eq. 2 0 1 LDFs for pairs ðJ;EÞ and ðJ ¼ RJ;E ¼ SEÞ, with R, S indepen- with a diffusive current term Q½ρðr;tÞ ¼ −D½ρ∇ ρ with D½ρ¼2 dent rotations, obey a generalized isometric fluctuation relation and σ½ρ¼ρ2, and it turns out to be an optimal candidate to test

7706 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1013209108 Hurtado et al. Downloaded by guest on September 24, 2021 the IFR because: (i) the associated hydrodynamic fluctuation theory can be solved analytically (29), and (ii) its dynamics is simple enough to allow for a detailed numerical study of current fluctuations. To test the IFR in this model we performed a large number of steady-state simulations of long duration τ >L2 (the unit of time is the Monte Carlo step) for L ¼ 20, TL ¼ 2, and TR ¼ 1, accumulating statistics for the space- and time-averaged current vector J. The measured current distribution is shown in Fig. 2, Bottom Inset, together with a fine polar binning that allows us to compare the probabilities of isometric current fluctuations along each polar corona, see Eq. 1. Taking GðJÞ¼ d −1 ðτL Þ ln PτðJÞ, Fig. 2 confirms the IFR prediction that GðJÞ − GðJ0Þ, once scaled by jJj−1, collapses onto a linear function of cos θ − cos θ0 for all values of jJj, see Eq. 8. Here θ, θ0 are the angles formed by the isometric current vectors J, J0 with the x-axis (E ¼ 0 in our case). We also measured the average energy ρ ðr; JÞ profile associated to each current fluctuation, 0 , see Fig. 2, Fig. 3. IFR for large current fluctuations. Legendre transform of the current Top Inset. As predicted above, profiles for different but isometric LDF for the energy diffusion model, for different values of jλ þ ϵj correspond- current fluctuations all collapse onto a single curve, confirming ing to very large current fluctuations, different rotation angles ϕ such 0 the invariance of optimal profiles under current rotations. that λ ¼ Rϕðλ þ ϵÞ − ϵ, and increasing system sizes. Lines are theoretical μðλÞ¼μ½R ðλ þ ϵÞ − ϵ ∀ϕ ∈ ½0;2π Standard simulations allow us to explore moderate fluctua- predictions. The IFR predicts that ϕ . The iso- tions of the current around the average. To test the IFR in the far metric fluctuation symmetry emerges in the macroscopic limit as the effects tails of the current distribution, corresponding to exponentially associated to the underlying lattice fade away. unlikely rare events, we implemented an elegant method recently strictly), which is not the case for the relatively small values of introduced to measure large-deviation functions in many-particle L L systems (30). The method, which yields the Legendre transform studied here. However, as increases, a clear convergence to- of the current LDF, μðλÞ, is based on a modification of the dy- ward the IFR prediction is observed as the effects associated to namics so that the rare events responsible of the large-deviation the underlying lattice fade away, strongly supporting the validity are no longer rare (30), and has been recently used with success to of IFR in the macroscopic limit. confirm an additivity conjecture regarding large fluctuations in We also measured current fluctuations in a Hamiltonian hard- nonequilibrium systems (7, 8). Using this method we measured disk fluid subject to a temperature gradient (23). This model is a μðλÞ in increasing manifolds of constant jλ þ ϵj, see Fig. 3. The paradigm in liquid state theory, condensed matter and statistical IFR implies that μðλÞ is constant along each of these manifolds, physics, and has been widely studied during last decades. The or equivalently μðλÞ¼μ½Rϕðλ þ ϵÞ − ϵ, ∀ϕ ∈ ½0;2π, with Rϕ a model consists in N hard disks of unit diameter interacting via rotation in 2D of angle ϕ. Fig. 3 shows the measured μðλÞ for instantaneous collisions and confined to a box of linear size L different values of jλ þ ϵj corresponding to very large current such that the particle density is fixed to Φ ¼ N∕L2 ¼ 0.58. Here fluctuations, different rotation angles ϕ and increasing system we choose N ¼ 320. The box is divided in three parts: a central, sizes, together with the theoretical predictions (29). As a result bulk region of width L − 2α with periodic boundary conditions in of the finite, discrete character of the lattice system studied here, the vertical direction, and two lateral stripes of width α ¼ L∕4 we observe weak violations of IFR in the far tails of the current distribution, specially for currents orthogonal to ϵ. These weak violations are expected because a prerequisite for the IFR to hold is the existence of a macroscopic limit (i.e., Eq. 2 should hold PHYSICS Fig. 4. IFR in a hydrodynamic hard-disk fluid. Confirmation of IFR in a two-dimensional hard-disk fluid under a temperature gradient after a polar binning of the measured current distribution. As predicted by IFR, the differ- Fig. 2. Confirmation of IFR in a diffusive system. The IFR predicts that ence of current LDFs for different isometric current fluctuations, once scaled −1 0 0 jJj ½GðJÞ − GðJ Þ collapses onto a linear function of cos θ − cos θ for all values by the current norm, collapses in a line when plotted against cos θ − cos θ0. of jJj. This collapse is confirmed here in the energy diffusion model for a wide (Top Inset) Optimal temperature profiles associated to different current range of values for jJj.(Bottom Inset) Measured current distribution together fluctuations. Profiles for a given jJj and different angles θ ∈ ½−7.5°; þ 7.5° with the polar binning used to test the IFR. (Top Inset) Average profiles for all collapse onto a single curve, thus confirming the invariance of optimal different but isometric current fluctuations all collapse onto single curves, profiles under current rotations. Notice that the profiles smoothly penetrate confirming the invariance of optimal profiles under current rotations. Angle into the heat baths. (Bottom Inset) Snapshot of the 2D hard-disk fluid with range is jθj ≤ 16.6°, see marked region in the histogram. Gaussian heat baths.

Hurtado et al. PNAS ∣ May 10, 2011 ∣ vol. 108 ∣ no. 19 ∣ 7707 Downloaded by guest on September 24, 2021 that act as deterministic heat baths, see Fig. 4, Bottom Inset. This the ensuing fluctuations of the slow hydrodynamic fields result is achieved by keeping constant the total kinetic energy within from the sum of an enormous amount of random events at each lateral band via a global, instantaneous rescaling of the ve- the microscale that give rise to Gaussian statistics. There exist locity of bath particles after bath-bulk particle collisions. This of course anomalous systems for which local fluctuations at the heat bath mechanism has been shown to efficiently thermostat macroscale can be non-Gaussian. In these cases we cannot dis- the fluid (23), and has the important advantage of being deter- card that a modified version of the IFR could remain valid, ministic. As for the previous diffusive model, we performed a though the analysis would be certainly more complicated. large number of steady-state simulations of long duration Furthermore, our numerical results show that the IFR remains (τ > 2N collisions per particle) for TL ¼ 4 and TR ¼ 1, accumu- true even in cases where it is not clear whether the HFT applies, lating statistics for the current J and measuring the average tem- strongly supporting the validity of this symmetry for arbitrary fluc- perature profile associated to each J. Fig. 4 shows the linear tuating hydrodynamic systems. −1 0 0 collapse of jJj ½GðJÞ − GðJ Þ as a function of cos θ − cos θ A related question is the demonstration of the IFR starting for different values of jJj, confirming the validity of the IFR from microscopic dynamics. Techniquessimilar to those in refs. 16 for this hard-disk fluid in the moderate range of current fluctua- and 31, which derive the Gallavotti–Cohen fluctuation theorem tions that we could access. Moreover, the measured optimal pro- from the spectral properties of the microscopic stochastic evolu- files for different isometric current fluctuations all nicely collapse tion operator, can prove useful for this task. However, to prove onto single curves, see Fig. 4, Top Inset, confirming their rota- the IFR these techniques must be supplemented with additional tional invariance. insights on the asymptotic properties of the microscopic transi- It is interesting to notice that the hard-disk fluid is a fully hy- tion rates as the macroscopic limit is approached. In this way drodynamic system, with 4 different locally conserved coupled we expect finite-size corrections to the IFR that decay with fields possibly subject to memory effects, defining a far more the system size, as it is in fact observed in our simulations for complex situation than the one studied here, see Eq. 2. Therefore the energy diffusion model, see Fig. 3. Also interesting is the pos- the validity of IFR in this context suggests that this fluctuation sibility of an IFR for discrete isometries related with the under- relation, based on the invariance of optimal profiles under sym- lying lattice in stochastic models. These open questions call for metry transformations, is in fact a rather general result valid for further study. arbitrary fluctuating hydrodynamic systems. We have shown in this paper how symmetry principles come A few remarks are now in order. First, as a corollary to the forth in fluctuations far from equilibrium. By demanding invar- IFR, it should be noted that for time-reversible systems with iance of the optimal path responsible of a given fluctuation under additive fluctuations (i.e., with a constant, profile-independent symmetry transformations, we unveiled a remarkable and very σ mobility ) the optimal profile associated to a given current general isometric fluctuation relation for time-reversible systems J 5 fluctuation is in fact independent of , see Eq. , and hence that relates in a simple manner the probability of any pair of equal to the stationary profile. In this case it is easy to show that isometric current fluctuations. Invariance principles of this kind GðJÞ¼ϵ ðJ − hJi Þþ current fluctuations are Gaussian, with · ϵ can be applied with great generality in diverse fields where fluc- σ−1ðJ2 − hJi2Þ ϵ . This is the case, for instance, of model B in the tuations play a fundamental role, opening the door to further – Hohenberg Halperin classification (25)*. On the other hand, exact and general results valid arbitrarily far from equilibrium. it should be noticed that the time-reversibility condition for 7 This is particularly relevant in mesoscopic biophysical systems, the IFR to hold, Eq. , is just a sufficient but not necessary where relations similar to the isometric fluctuation relation might condition. In fact, we cannot discard the possibility of time- be used to efficiently measure free-energy differences in terms of irreversible systems such that, only for the optimal profiles, 0 work distributions (32). Other interesting issues concern the δω1½ρðrÞ∕δρðr Þj ¼ 0. ρ0 study of general fluctuation relations emerging from the invar- Discussion iance of optimal paths in full hydrodynamical systems with several The IFR is a consequence of time-reversibility for systems in the conserved fields, or the quantum analog of the isometric fluctua- hydrodynamic scaling limit, and reveals an unexpected high level tion relation in full counting statistics. of symmetry in the statistics of nonequilibrium fluctuations. It – Materials and Methods generalizes and comprises the Gallavotti Cohen fluctuation the- We now exploit the IFR (1) to derive a set of hierarchies for the current orem for currents, relating the probabilities of an event not only cumulants and the linear and nonlinear response coefficients. The mo- with its time-reversal but with any other isometric fluctuation. ment-generating function associated to PτðJÞ, defined as ΠτðλÞ¼ d d This has important consequences in the form of hierarchies ∫ PτðJÞ expðτL λ · JÞdJ, scales for long times as ΠτðλÞ ∼ exp½þτL μðλÞ, where μðλÞ¼ ½Gð Þþλ for the current cumulants and the linear and nonlinear response maxJ J · J is the Legendre transform of the current LDF. coefficients, which hold arbitrarily far from equilibrium and can The cumulants of the current distribution can be obtained from the deriva- ðnÞ n n1 nd tives of μðλÞ evaluated at λ ¼ 0 (i.e., μ ≡ ½∂ μðλÞ∕∂λ :::λ λ¼0 ¼ be readily tested in experiments. A natural question thus con- ðn1 :::nd Þ 1 d ðτLd Þn−1hΔJn1 :::ΔJnd i n ¼ ∑ n ≥ 1 ΔJ ≡ J − ð1 − δ ÞhJ i cerns the level of generality of the isometric fluctuation relation. 1 d ϵ for i i ) where α α n;1 α ϵ and δn;m is the Kronecker symbol. The IFR can be stated for the Legendre trans- In this paper we have demonstrated the IFR for a broad class of μðλÞ¼μ½Rðλ þ ϵÞ − ϵ R d systems characterized at the macroscale by a single conserved form of the current LDF as , where is any -dimensional rotation. Using this relation in the definition of the n-th order field, using the tools of hydrodynamic fluctuation theory (HFT). cumulant in the limit of infinitesimal rotations, R ¼ I þ ΔθL, it is easy to This theoretical framework, summarized in the path large-devia- show that tion functional, Eq. S3, has been rigorously proven for a number of interacting-particle systems (2–5), but it is believed to remain ðnÞ ðnþ1Þ nαLβαμ þ ϵνLγνμ ¼ 0; [13] valid for a much larger class of systems. The key is that the Gaus- ðn1:::nα−1.::nβ þ1.::ndÞ ðn1:::nγ þ1.::ndÞ sian nature of local fluctuations, which lies at the heart of the approach, is expected to emerge for most situations in the appro- where L is any generator of d-dimensional rotations, and summation over ∈ ½1;d priate macroscopic limit as a result of a central limit theorem: repeated Greek indices ( ) is assumed. The above hierarchy relates in a n n þ 1 ∀n ≥ 1 Although microscopic interactions can be extremely complicated, simple way cumulants of orders and , and is valid arbitrarily far from equilibrium. As an example, Eqs. 9 and 10 above show the first two sets of relations (n ¼ 1;2) of the above hierarchy in two dimensions. In a similar *Notice that ρ-dependent corrections to a constant mobility σ, which are typically way, we can explore the consequences of the IFR on the linear and nonlinear irrelevant from a renormalization-group point of view (25), turn out to be essential response coefficients. For that, we now expand the cumulants of the current for current fluctuations as they give rise to non-Gaussian tails in the current distribution. in powers of ϵ

7708 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1013209108 Hurtado et al. Downloaded by guest on September 24, 2021 ∞ 1 k the response coefficients result from the IFR in the limit of infinitesimal ðnÞ ðkÞ ðk1:::kdÞ k1 kd μ ðϵÞ¼ χ ϵ1 :::ϵd [14] rotations. For a finite rotation R ¼ −I, which is equivalent to a current inver- ðn1:::ndÞ ∑ k! ∑ ðnÞ ðn1:::ndÞ k¼0 k1:::kd¼0 sion, we have μðλÞ¼μð−λ − 2ϵÞ and we may use this in the definition of k ¼k ðnÞ ðk :::k Þ n n k k ∑ i response coefficients, χ 1 d ≡ k!½∂nþkμðλÞ∕∂λ 1 :::λ d ∂ϵ 1 :::ϵ d , see i ðkÞ ðn1 :::nd Þ 1 d 1 d λ¼0¼ϵ Eq. 14, to obtain a complementary relation for the response coefficients Inserting expansion [14] into the cumulant hierarchy, Eq. 13, and matching order by order in k, we derive another interesting hierarchy for the response k ¼ 0 k1 kd nþp p coefficients of the different cumulants. For this reads ðkÞ ðk :::k Þ ð−1Þ 2 ðk−pÞ ðk −p :::k −p Þ χ 1 d ¼ k! ::: χ 1 1 d d ; [17] ðnÞ ðn1:::ndÞ ∑ ∑ ðk − pÞ! ðnþpÞ ðn1þp1:::ndþpdÞ ð0Þ ð0.::0Þ p ¼0 p ¼0 nαLβα χ ¼ 0; [15] 1 d ðnÞ ðn1:::nα−1.::nβ þ1.::ndÞ

which is a symmetry relation for the equilibrium (ϵ ¼ 0) current cumulants. where p ¼ ∑i pi . A similar equation was derived in (28) from the standard For k ≥ 1 we obtain fluctuation theorem, although the IFR adds further relations. All together, Eqs. 15–17 imply deep relations between the response coefficients at k n ’ α ðkÞ ðk1:::kdÞ ðk−1Þ ðk1:::kν−1.::kdÞ arbitrary orders that go far beyond Onsager s reciprocity relations and Lβα χ þ Lγν χ ¼ 0; ∑ k ðnÞ ðn1:::nα−1.::nβ þ1.::ndÞ ðnþ1Þ ðn1:::nγ þ1.::ndÞ Green–Kubo formulae. As an example, we discuss in the main text some k :::k ¼0 1 d of these relations for a two-dimensional system. k ¼k≥1 ∑ i i [16] ACKNOWLEDGMENTS. We acknowledge financial support from Spanish Ministerio de Ciencia e Innovación Project FIS2009-08451, the University of which relates k-order response coefficients of n-order cumulants with ðk − 1Þ- Granada, and Junta de Andalucía (Projects P07-FQM02725 and P09- order coefficients of ðn þ 1Þ-order cumulants. Relations [15] and [16]for FQM4682).

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