Symmetries in Fluctuations Far from Equilibrium

# Symmetries in Fluctuations Far from Equilibrium

P.I. Hurtado , C. Pérez-Espigares  , J.J. del Pozo  ,    P.L. Garrido
###### Abstract

Fluctuations arise universally in Nature as a reflection of the discrete microscopic world at the macroscopic level. Despite their apparent noisy origin, fluctuations encode fundamental aspects of the physics of the system at hand, crucial to understand irreversibility and nonequilibrium behavior. In order to sustain a given fluctuation, a system traverses a precise optimal path in phase space. Here we show that by demanding invariance of optimal paths under symmetry transformations, new and general fluctuation relations valid arbitrarily far from equilibrium are unveiled. This opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations. We illustrate this concept studying symmetries of the current distribution out of equilibrium. In particular we derive an isometric fluctuation relation which links in a strikingly simple manner the probabilities of any pair of isometric current fluctuations. This relation, which results from the time-reversibility of the dynamics, includes as a particular instance the Gallavotti-Cohen fluctuation theorem in this context but adds a completely new perspective on the high level of symmetry imposed by time-reversibility on the statistics of nonequilibrium fluctuations. The new symmetry implies remarkable hierarchies of equations for the current cumulants and the nonlinear response coefficients, going far beyond Onsager’s reciprocity relations and Green-Kubo formulae. We confirm the validity of the new symmetry relation in extensive numerical simulations, and suggest that the idea of symmetry in fluctuations as invariance of optimal paths has far-reaching consequences in diverse fields.

footnotetext: Reserved for Publication Footnotes

large deviations      rare events      hydrodynamics      transport      entropy production

L arge fluctuations, though rare, play an important role in many fields of science as they crucially determine the fate of a system [1]. Examples range from chemical reaction kinetics or the escape of metastable electrons in nanoelectronic devices to conformational changes in proteins, mutations in DNA, and nucleation events in the primordial universe. Remarkably, the statistics of these large fluctuations contains deep information on the physics of the system of interest [2, 3]. This is particularly important for systems far from equilibrium, where no general theory exists up to date capable of predicting macroscopic and fluctuating behavior in terms of microscopic physics, in a way similar to equilibrium statistical physics. The consensus is that the study of fluctuations out of equilibrium may open the door to such general theory. As most nonequilibrium systems are characterized by currents of locally conserved observables, understanding current statistics in terms of microscopic dynamics has become one of the main objectives of nonequilibrium statistical physics [2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18]. Pursuing this line of research is both of fundamental as well as practical importance. At the theoretical level, the function controlling current fluctuations can be identified as the nonequilibrium analog of the free energy functional in equilibrium systems [2, 3, 4, 5], from which macroscopic properties of a nonequilibrium system can be obtained (including its most prominent features, as for instance the ubiquitous long range correlations [19, 20], etc.) On the other hand, the physics of most modern mesoscopic devices is characterized by large fluctuations which determine their behavior and function. In this way understanding current statistics in these systems is of great practical significance.

Despite the considerable interest and efforts on these issues, exact and general results valid arbitrarily far from equilibrium are still very scarce. The reason is that, while in equilibrium phenomena dynamics is irrelevant and the Gibbs distribution provides all the necessary information, in nonequilibrium physics dynamics plays a dominant role, even in the simplest situation of a nonequilibrium steady state [2, 3, 4, 5]. However, there is a remarkable exception to this absence of general results which has triggered an important surge in activity since its formulation in the mid nineties. This is the fluctuation theorem, first discussed in the context of simulations of sheared fluids [15], and formulated rigorously by Gallavotti and Cohen under very general assumptions [16]. This theorem, which implies a relation between the probabilities of a given current fluctuation and the inverse event, is a deep statement on the subtle consequences of time-reversal symmetry of microscopic dynamics at the macroscopic, irreversible level. Particularly important here is the observation that symmetries are reflected at the fluctuating macroscopic level arbitrarily far from equilibrium. Inspired by this illuminating result, we explore in this paper the behavior of the current distribution under symmetry transformations [21]. Key to our analysis is the observation that, in order to facilitate a given current fluctuation, the system traverses a well-defined optimal path in phase space [2, 3, 4, 5, 7, 8, 22]. This path is, under very general conditions, invariant under certain symmetry transformations on the current. Using this invariance we show that for -dimensional, time-reversible systems described by a locally-conserved field and possibly subject to a boundary-induced gradient and an external field , the probability of observing a current averaged over a long time obeys an isometric fluctuation relation (IFR)

 limτ→∞1τln[Pτ(J)Pτ(J′)]=\boldmathϵ⋅(J−J′), (0)

for any pair of isometric current vectors, . Here is a constant vector directly related to the rate of entropy production in the system, which depends on the boundary baths via (see below).

The above equation, which includes as a particular case the Gallavotti-Cohen (GC) result for , relates in a strikingly simple manner the probability of a given fluctuation with the likelihood of any other current fluctuation on the -dimensional hypersphere of radius , see Fig. 1, projecting a complex -dimensional problem onto a much simpler one-dimensional theory. Unlike the GC relation which is a non-differentiable symmetry involving the inversion of the current sign, , eq. (Symmetries in Fluctuations Far from Equilibrium) is valid for arbitrary changes in orientation of the current vector. This makes the experimental test of the above relation a feasible problem, as data for current fluctuations involving different orientations around the average can be gathered with enough statistics to ensure experimental accuracy. It is also important to notice that the isometric fluctuation relation is valid for arbitrarily large fluctuations, i.e. even for the non-Gaussian far tails of current distribution. We confirm here the validity of the new symmetry in extensive numerical simulations of two different nonequilibrium systems: (i) A simple and very general lattice model of energy diffusion [7, 8, 23], and (ii) a hard-disk fluid in a temperature gradient [24].

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Our starting point is a continuity equation which describes the macroscopic evolution of a wide class of systems characterized by a locally-conserved magnitude (e.g. energy, particle density, momentum, etc.)

 ∂tρ(r,t)=−\boldmath∇⋅(QE[ρ(r,t)]+\boldmathξ(r,t)). (0)

Here is the density field, is the fluctuating current, with local average , and is a Gaussian white noise characterized by a variance (or mobility) . This (conserved) noise term accounts for microscopic random fluctuations at the macroscopic level. Notice that the current functional includes in general the effect of a conservative external field, . Examples of systems described by eq. (Symmetries in Fluctuations Far from Equilibrium) range from diffusive systems [2, 3, 4, 5, 6, 7, 8, 9], where is given by Fourier’s (or equivalently Fick’s) law, , to most interacting-particle fluids [25, 26], characterized by a Ginzburg-Landau-type theory for the locally-conserved particle density. To completely define the problem, the above evolution equation must be supplemented with appropriate boundary conditions, which may include an external gradient.

We are interested in the probability of observing a space- and time-averaged empirical current , defined as

 J=1τ∫τ0dt∫drj(r,t). (0)

This probability obeys a large deviation principle for long times [27, 28], , where is the system linear size and is the current large-deviation function (LDF), meaning that current fluctuations away from the average are exponentially unlikely in time. According to hydrodynamic fluctuation theory [2, 4, 5, 6],

 G(J)=−minρ(r)∫(J−QE[ρ(r)])22σ[ρ(r)]dr, (0)

which expresses the locally-Gaussian nature of fluctuations [6, 7, 8]. The optimal profile solution of the above variational problem can be interpreted as the density profile the system adopts to facilitate a current fluctuation [7, 8, 22]. To derive eq. (Symmetries in Fluctuations Far from Equilibrium) we assumed that (i) the optimal profiles associated to a given current fluctuation are time-independent [2, 3, 4, 5, 6, 7, 8, 9, 22, 10], and (ii) the optimal current field has no spatial structure, see Supporting Information (SI). This last hypothesis, which greatly simplifies the calculation of current statistics, can be however relaxed for our purposes (as shown below). The probability is thus simply the Gaussian weight associated to the optimal profile. Note however that the minimization procedure gives rise to a nonlinear problem which results in general in a current distribution with non-Gaussian tails [2, 3, 4, 5, 6, 7, 8].

The optimal profile is solution of the following equation

 δω2[ρ(r)]δρ(r′)−2J⋅δ\boldmathω1[ρ(r)]δρ(r′)+J2δ%$ω$0[ρ(r)]δρ(r′)=0, (0)

where stands for functional derivative, and

 \boldmathωn[ρ(r)]≡∫drQnE[ρ(r)]σ[ρ(r)]. (0)

Remarkably, the optimal profile solution of eq. (Symmetries in Fluctuations Far from Equilibrium) depends exclusively on and . Such a simple quadratic dependence, inherited from the locally-Gaussian nature of fluctuations, has important consequences at the level of symmetries of the current distribution. In fact, it is clear from eq. (Symmetries in Fluctuations Far from Equilibrium) that the condition

 (0)

implies that will depend exclusively on the magnitude of the current vector, via , not on its orientation. In this way, all isometric current fluctuations characterized by a constant will have the same associated optimal profile, , independently of whether the current vector points along the gradient direction, against it, or along any arbitrary direction. In other words, the optimal profile is invariant under current rotations if eq. (Symmetries in Fluctuations Far from Equilibrium) holds. It turns out that condition (Symmetries in Fluctuations Far from Equilibrium) follows from the time-reversibility of the dynamics, in the sense that the evolution operator in the Fokker-Planck formulation of eq. (Symmetries in Fluctuations Far from Equilibrium) obeys a local detailed balance condition [17, 18]. In this case , with the system Hamiltonian, and condition (Symmetries in Fluctuations Far from Equilibrium) holds. The invariance of the optimal profile can be now used in eq. (Symmetries in Fluctuations Far from Equilibrium) to relate in a simple way the current LDF of any pair of isometric current fluctuations and , with ,

 G(J)−G(J′)=|\boldmathϵ||J|(cosθ−cosθ′), (0)

where and are the angles formed by vectors and , respectively, with a constant vector , see below. Eq. (Symmetries in Fluctuations Far from Equilibrium) is just an alternative formulation of the isometric fluctuation relation (Symmetries in Fluctuations Far from Equilibrium). By letting and differ by an infinitesimal angle, the IFR can be cast in a simple differential form, , which reflects the high level of symmetry imposed by time-reversibility on the current distribution.

The condition can be seen as a conservation law. It implies that the observable is in fact a constant of motion, , independent of the profile , which can be related with the rate of entropy production via the Gallavotti-Cohen theorem [16, 17, 18]. In a way similar to Noether’s theorem, the conservation law for implies a symmetry for the optimal profiles under rotations of the current and a fluctuation relation for the current LDF. This constant can be easily computed under very general assumptions (see SI).

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The isometric fluctuation relation, eq. (Symmetries in Fluctuations Far from Equilibrium), has far-reaching and nontrivial consequences. First, the IFR implies a remarkable hierarchy of equations for the cumulants of the current distribution, see eq. (0) in Methods. This hierarchy can be derived starting from the Legendre transform of the current LDF, , from which all cumulants can be obtained [3], and writing the IFR for in the limit of infinitesimal rotations. As an example, the cumulant hierarchy in two dimensions implies the following relations

 ⟨Jx⟩ϵ = τL2[ϵx⟨ΔJ2y⟩ϵ−ϵy⟨ΔJxΔJy⟩ϵ] (0) ⟨Jy⟩ϵ = τL2[ϵy⟨ΔJ2x⟩ϵ−ϵx⟨ΔJxΔJy⟩ϵ] 2⟨ΔJxΔJy⟩ϵ = τL2[ϵy⟨ΔJ3x⟩ϵ−ϵx⟨ΔJ2xΔJy⟩ϵ] = τL2[ϵx⟨ΔJ3y⟩ϵ−ϵy⟨ΔJxΔJ2y⟩ϵ] ⟨ΔJ2x⟩ϵ−⟨ΔJ2y⟩ϵ = τL2[ϵx⟨ΔJxΔJ2y⟩ϵ−ϵy⟨ΔJ2xΔJy⟩ϵ],

for the first cumulants, with . It is worth stressing that the cumulant hierarchy is valid arbitrarily far from equilibrium. In a similar way, the IFR implies a set of hierarchies for the nonlinear response coefficients, see eqs. (0)-(0) in Methods. In our two-dimensional example, let be the response coefficient of the cumulant to order , with and . To the lowest order these hierarchies imply Onsager’s reciprocity symmetries and Green-Kubo relations for the linear response coefficients of the current. They further predict that in fact the linear response matrix is proportional to the identity, so while . The first nonlinear coefficients of the current can be simply written in terms of the linear coefficients of the second cumulants as and , while the cross-coefficient reads (symmetric results hold for , ). Linear response coefficients for the second-order cumulants also obey simple relations, e.g. and , and the set of relations continues to arbitrary high orders. In this way hierarchies (0)-(0), which derive from microreversibility as reflected in the IFR, provide deep insights into nonlinear response theory for nonequilibrium systems [29].

The IFR and the above hierarchies all follow from the invariance of optimal profiles under certain transformations. This idea can be further exploited in more general settings. In fact, by writing explicitly the dependence on the external field in eq. (Symmetries in Fluctuations Far from Equilibrium) for the optimal profile, one realizes that if , together with the time-reversibility condition, eq. (Symmetries in Fluctuations Far from Equilibrium), the resulting optimal profiles are invariant under independent rotations of the current and the external field. It thus follows that the current LDFs for pairs and , with , independent rotations, obey a generalized isometric fluctuation relation

 GE(J)−GE∗(J′)=% \boldmathε⋅(J−J′)−\boldmathν⋅(E−E∗)+J⋅E−J′⋅E∗, (0)

where we write explicitly the dependence of the current LDF on the external field. The vector is now another constant of motion, independent of , which can be easily computed (see SI). For a fixed boundary gradient, the above equation relates any current fluctuation in the presence of an external field with any other isometric current fluctuation in the presence of an arbitrarily-rotated external field , and reduces to the standard IFR for . Condition is rather general, as most time-reversible systems with a local mobility do fulfill this condition (e.g., diffusive systems).

The IFR can be further generalized to cases where the current profile is not constant, relaxing hypothesis (ii) above. Let be the probability of observing a time-averaged current field . This vector field must have zero divergence because it is coupled via the continuity equation to an optimal density profile which is assumed to be time-independent, see SI and hypothesis (i) above. Because of time-reversibility, and it is easy to show in the equation for the optimal density profile that the term linear in vanishes, so remains invariant under (local or global) rotations of , see SI. In this way, for any divergence-free current field locally-isometric to , so , we can write a generalized isometric fluctuation relation

 limτ→∞1τln[Pτ[\boldmath\@fontswitchJ(r)]Pτ[\boldmath\@fontswitchJ′(r)]]=∫∂ΛdΓδ\@fontswitchH[ρ]δρ^n⋅[\boldmath\@fontswitchJ′(r)−\boldmath\@fontswitchJ(r)], (0)

where the integral (whose result is independent of ) is taken over the boundary of the domain where the system is defined, and is the unit vector normal to the boundary at each point. Eq. (Symmetries in Fluctuations Far from Equilibrium) generalizes the IFR to situations where hypothesis (ii) is violated, opening the door to isometries based on local (in addition to global) rotations. As a corollary, we show in the SI appendix that a similar generalization of the isometric fluctuation symmetry does not exist whenever optimal profiles become time-dependent, so the IFR breaks down in the regime where hypothesis (i) is violated. In this way, we may use violations of the IFR and its generalizations to detect the instabilities which characterize the fluctuating behavior of the system at hand [2, 9, 10].

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We have tested the validity of the IFR in extensive numerical simulations of two different nonequilibrium systems. The first one is a simple and very general model of energy diffusion [7, 8, 23] defined on a two-dimensional (2D) square lattice with sites. Each site is characterized by an energy , , and models a harmonic oscillator which is mechanically uncoupled from its nearest neighbors but interact with them via a stochastic energy-redistribution process. Dynamics thus proceeds through random energy exchanges between randomly-chosen nearest neighbors. In addition, left and right boundary sites may interchange energy with boundary baths at temperatures and , respectively, while periodic boundary conditions hold in the vertical direction. For the systems reaches a nonequilibrium steady state characterized, in the absence of external field (the case studied here), by a linear energy profile and a nonzero average current given by Fourier’s law. This model plays a fundamental role in nonequilibrium statistical physics as a testbed to assess new theoretical advances, and represents at a coarse-grained level a large class of diffusive systems of technological and theoretical interest [7, 8]. The model is described at the macroscopic level by eq. (Symmetries in Fluctuations Far from Equilibrium) with a diffusive current term with and , and it turns out to be an optimal candidate to test the IFR because: (1) the associated hydrodynamic fluctuation theory can be solved analytically [30], and (2) its dynamics is simple enough to allow for a detailed numerical study of current fluctuations.

In order to test the IFR in this model we performed a large number of steady-state simulations of long duration (the unit of time is the Monte Carlo step) for , and , accumulating statistics for the space- and time-averaged current vector . The measured current distribution is shown in the bottom inset to Fig. 2, together with a fine polar binning which allows us to compare the probabilities of isometric current fluctuations along each polar corona, see eq. (Symmetries in Fluctuations Far from Equilibrium). Taking , Fig. 2 confirms the IFR prediction that , once scaled by , collapses onto a linear function of for all values of , see eq. (Symmetries in Fluctuations Far from Equilibrium). Here , are the angles formed by the isometric current vectors , with the -axis ( in our case). We also measured the average energy profile associated to each current fluctuation, , see top inset to Fig. 2. As predicted above, profiles for different but isometric current fluctuations all collapse onto a single curve, confirming the invariance of optimal profiles under current rotations.

Standard simulations allow us to explore moderate fluctuations of the current around the average. In order to test the IFR in the far tails of the current distribution, corresponding to exponentially unlikely rare events, we implemented an elegant method recently introduced to measure large deviation functions in many-particle systems [32]. The method, which yields the Legendre transform of the current LDF, , is based on a modification of the dynamics so that the rare events responsible of the large deviation are no longer rare [32], and has been recently used with success to confirm an additivity conjecture regarding large fluctuations in nonequilibrium systems [7, 8]. Using this method we measured in increasing manifolds of constant , see Fig. 3. The IFR implies that is constant along each of these manifolds, or equivalently , , with a rotation in 2D of angle . Fig. 3 shows the measured for different values of corresponding to very large current fluctuations, different rotation angles and increasing system sizes, together with the theoretical predictions [30]. As a result of the finite, discrete character of the lattice system studied here, we observe weak violations of IFR in the far tails of the current distribution, specially for currents orthogonal to . These weak violations are expected since a prerequisite for the IFR to hold is the existence of a macroscopic limit, i.e. eq. (Symmetries in Fluctuations Far from Equilibrium) should hold strictly, which is not the case for the relatively small values of studied here. However, as increases, a clear convergence toward the IFR prediction is observed as the effects associated to the underlying lattice fade away, strongly supporting the validity of IFR in the macroscopic limit.

We also measured current fluctuations in a Hamiltonian hard-disk fluid subject to a temperature gradient [24]. This model is a paradigm in liquid state theory, condensed matter and statistical physics, and has been widely studied during last decades. The model consists in hard disks of unit diameter interacting via instantaneous collisions and confined to a box of linear size such that the particle density is fixed to . Here we choose . The box is divided in three parts: a central, bulk region of width with periodic boundary conditions in the vertical direction, and two lateral stripes of width which act as deterministic heat baths, see bottom inset to Fig. 4. This is achieved by keeping constant the total kinetic energy within each lateral band via a global, instantaneous rescaling of the velocity of bath particles after bath-bulk particle collisions. This heat bath mechanism has been shown to efficiently thermostat the fluid [24], and has the important advantage of being deterministic. As for the previous diffusive model, we performed a large number of steady state simulations of long duration ( collisions per particle) for and , accumulating statistics for the current and measuring the average temperature profile associated to each . Fig. 4 shows the linear collapse of as a function of for different values of , confirming the validity of the IFR for this hard-disk fluid in the moderate range of current fluctuations that we could access. Moreover, the measured optimal profiles for different isometric current fluctuations all nicely collapse onto single curves, see top inset to Fig. 4, confirming their rotational invariance.

It is interesting to notice that the hard-disk fluid is a fully hydrodynamic system, with 4 different locally-conserved coupled fields possibly subject to memory effects, defining a far more complex situation than the one studied here, see eq. (Symmetries in Fluctuations Far from Equilibrium). Therefore the validity of IFR in this context suggests that this fluctuation relation, based on the invariance of optimal profiles under symmetry transformations, is in fact a rather general result valid for arbitrary fluctuating hydrodynamic systems.

A few remarks are now in order. First, as a corollary to the IFR, it should be noted that for time-reversible systems with additive fluctuations, i.e. with a constant, profile-independent mobility , the optimal profile associated to a given current fluctuation is in fact independent of , see eq. (Symmetries in Fluctuations Far from Equilibrium), and hence equal to the stationary profile. In this case it is easy to show that current fluctuations are Gaussian, with . This is the case, for instance, of model B in the Hohenberg-Halperin classification [26] 222Notice that -dependent corrections to a constant mobility , which are typically irrelevant from a renormalization-group point of view [26], turn out to be essential for current fluctuations as they give rise to non-Gaussian tails in the current distribution.. On the other hand, it should be noticed that the time-reversibility condition for the IFR to hold, eq. (Symmetries in Fluctuations Far from Equilibrium) , is just a sufficient but not necessary condition. In fact, we cannot discard the possibility of time-irreversible systems such that, only for the optimal profiles, .

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The IFR is a consequence of time-reversibility for systems in the hydrodynamic scaling limit, and reveals an unexpected high level of symmetry in the statistics of nonequilibrium fluctuations. It generalizes and comprises the Gallavotti-Cohen fluctuation theorem for currents, relating the probabilities of an event not only with its time-reversal but with any other isometric fluctuation. This has important consequences in the form of hierarchies for the current cumulants and the linear and nonlinear response coefficients, which hold arbitrarily far from equilibrium and can be readily tested in experiments. A natural question thus concerns the level of generality of the isometric fluctuation relation. In this paper we have demonstrated the IFR for a broad class of systems characterized at the macroscale by a single conserved field, using the tools of hydrodynamic fluctuation theory (HFT). This theoretical framework, summarized in the path large deviation functional, eq. (3) in the SI appendix, has been rigorously proven for a number of interacting particle systems [2, 3, 4, 5], but it is believed to remain valid for a much larger class of systems. The key is that the Gaussian nature of local fluctuations, which lies at the heart of the approach, is expected to emerge for most situations in the appropriate macroscopic limit as a result of a central limit theorem: although microscopic interactions can be extremely complicated, the ensuing fluctuations of the slow hydrodynamic fields result from the sum of an enormous amount of random events at the microscale which give rise to Gaussian statistics. There exist of course anomalous systems for which local fluctuations at the macroscale can be non-Gaussian. In these cases we cannot discard that a modified version of the IFR could remain valid, though the analysis would be certainly more complicated. Furthermore, our numerical results show that the IFR remains true even in cases where it is not clear whether the HFT applies, strongly supporting the validity of this symmetry for arbitrary fluctuating hydrodynamic systems.

A related question is the demonstration of the IFR starting from microscopic dynamics. Techniques similar to those in Refs. [17, 31], which derive the Gallavotti-Cohen fluctuation theorem from the spectral properties of the microscopic stochastic evolution operator, can prove useful for this task. However, in order to prove the IFR these techniques must be supplemented with additional insights on the asymptotic properties of the microscopic transition rates as the macroscopic limit is approached. In this way we expect finite-size corrections to the IFR which decay with the system size, as it is in fact observed in our simulations for the energy diffusion model, see Fig. 3. Also interesting is the possibility of an IFR for discrete isometries related with the underlying lattice in stochastic models. These open questions call for further study.

We have shown in this paper how symmetry principles come forth in fluctuations far from equilibrium. By demanding invariance of the optimal path responsible of a given fluctuation under symmetry transformations, we unveiled a novel and very general isometric fluctuation relation for time-reversible systems which relates in a simple manner the probability of any pair of isometric current fluctuations. Invariance principles of this kind can be applied with great generality in diverse fields where fluctuations play a fundamental role, opening the door to further exact and general results valid arbitrarily far from equilibrium. This is particularly relevant in mesoscopic biophysical systems, where relations similar to the isometric fluctuation relation might be used to efficiently measure free-energy differences in terms of work distributions [33]. Other interesting issues concern the study of general fluctuation relations emerging from the invariance of optimal paths in full hydrodynamical systems with several conserved fields, or the quantum analog of the isometric fluctuation relation in full counting statistics.

Appendix:    Hierarchies for the cumulants and response coefficients

The moment-generating function associated to , defined as , scales for long times as , where is the Legendre transform of the current LDF. The cumulants of the current distribution can be obtained from the derivatives of evaluated at , i.e. for , where and is the Kronecker symbol. The IFR can be stated for the Legendre transform of the current LDF as , where is any -dimensional rotation. Using this relation in the definition of the -th order cumulant in the limit of infinitesimal rotations, , it is easy to show that

 nα\@fontswitchLβαμ(n)(n1...nα−1...nβ+1...nd)+ϵν\@fontswitchLγνμ(n+1)(n1...nγ+1...nd)=0, (0)

where is any generator of -dimensional rotations, and summation over repeated Greek indices () is assumed. The above hierarchy relates in a simple way cumulants of orders and , and is valid arbitrarily far from equilibrium. As an example, eqs. (Symmetries in Fluctuations Far from Equilibrium) and (Symmetries in Fluctuations Far from Equilibrium) above show the first two sets of relations () of the above hierarchy in two dimensions. In a similar way, we can explore the consequences of the IFR on the linear and nonlinear response coefficients. For that, we now expand the cumulants of the current in powers of

 μ(n)(n1...nd)(\boldmathϵ)=∞∑k=01k!k∑k1...kd=0∑iki=k,(n)(k)χ(k1...kd)(n1...nd)ϵk11...ϵkdd (0)

Inserting expansion (0) into the cumulant hierarchy, eq. (0), and matching order by order in , we derive another interesting hierarchy for the response coefficients of the different cumulants. For this reads

 nα\@fontswitchLβα,(0)(n)χ(0...0)(n1...nα−1...nβ+1...nd)=0, (0)

which is a symmetry relation for the equilibrium () current cumulants. For we obtain

 k∑k1...kd=0∑iki=k≥1[nαk\@fontswitchLβα,(k)(n)χ(k1...kd)(n1...nα−1...nβ+1...nd) +\@fontswitchLγν(k−1)(n+1)χ(k1...kν−1...kd)(n1...nγ+1...nd)]=0, (0)

which relates -order response coefficients of -order cumulants with -order coefficients of -order cumulants. Relations (0)-(0) for the response coefficients result from the IFR in the limit of infinitesimal rotations. For a finite rotation , which is equivalent to a current inversion, we have and we may use this in the definition of response coefficients, , see eq. (0), to obtain a complementary relation for the response coefficients

 (k)(n)χ(k1...kd)(n1...nd)=k!k1∑p1=0...kd∑pd=0(−1)n+p2p(k−p)!,(k−p)(n+p)χ(k1−p1...kd−pd)(n1+p1...nd+pd), (0)

where . A similar equation was derived in [29] from the standard fluctuation theorem, although the IFR adds further relations. All together, eqs. (0)-(0) imply deep relations between the response coefficients at arbitrary orders which go far beyond Onsager’s reciprocity relations and Green-Kubo formulae. As an example, we discuss in the main text some of these relations for a two-dimensional system.

Appendix:    Hydrodynamic fluctuation theory

The evolution of the system of interest is described by the following Langevin equation

 ∂tρ(r,t)=−\boldmath∇⋅(QE[ρ(r,t)]+\boldmathξ(r,t)), (0)

which expresses the local conservation of certain physical observable. Here is the density field, is the fluctuating current, with local average , and is a Gaussian white noise with zero mean and characterized by a variance (or mobility) . Notice that the current functional includes in general the effect of a conservative external field, . Using a path integral formulation [30], the probability of observing a given history of duration for the density and current fields can be written as

 P({ρ,j}τ0)∼exp(+LdIτ[ρ,j]), (0)

where is the system linear size, is the dimensionality, and the functional is

 Iτ[ρ,j]=−∫τ0dt∫dr(j(r,t)−QE[ρ(r,t)])22σ[ρ(r,t)], (0)

with and coupled via the continuity equation

 (0)

In this way the probability of each history has a Gaussian weight around the average local behavior given by . Eqs. (0) and (0) are equivalent to the hydrodynamic fluctuation theory recently proposed by Bertini and coworkers [4, 5, 2]. The probability of observing a space- and time-averaged empirical current , defined as

 J=1τ∫τ0dt∫drj(r,t), (0)

can be obtained from the path integral of restricted to histories compatible with a given ,

 Pτ(J)=∫\@fontswitchDρ\@fontswitchDjP({ρ,j}τ0)δ(J−1τ∫τ0dt∫drj(r,t)), (0)

This probability scales for long times as , and the current large deviation function (LDF) can be related to via a simple saddle-point calculation in the long-time limit,

 G(J)=limτ→∞1τmaxρ(r,t)j(r,t)Iτ[ρ,j], (0)

subject to constraints (0) and (0). The density and current fields solution of this variational problem, denoted here as and , can be interpreted as the optimal path the system follows in order to sustain a lont-time current fluctuation . It is worth emphasizing here that the existence of an optimal path rests on the presence of a selection principle at play, namely a long time, large size limit which selects, among all possible paths compatible with a given fluctuation, an optimal one via a saddle point mechanism. Eq. (0) defines a complex spatiotemporal problem whose solution remains challenging in most cases [30, 4, 5, 2, 6, 7, 8, 9, 10]. However, the following hypotheses greatly reduce its complexity:

1. We assume that the optimal profiles responsible of a given current fluctuation are time-independent [6], and . This, together with the continuity equation, implies that the optimal current vector field is divergence-free, .

2. A further simplification consists in assuming that the optimal current field is in fact constant across space, so .

Provided that these hypotheses hold, the current LDF can be written as

 G(J)=−minρ(r)∫(J−QE[ρ(r)])22σ[ρ(r)]dr, (0)

The optimal density profile is thus solution of the following differential equation

 δω2[ρ(r)]δρ(r′)−2J⋅δ\boldmathω1[ρ(r)]δρ(r′)+J2δ%$ω$0[ρ(r)]δρ(r′)=0, (0)

where stands for functional derivative, and

 \boldmathωn[ρ(r)]≡∫dr% Wn[ρ(r)]withWn[ρ(r)]≡QnE[ρ(r)]σ[ρ(r)]. (0)

For time-reversible systems, one can see that the evolution operator in the Fokker-Planck formulation of eq. (0) obeys a local detailed balance condition, and

 W1[ρ(r)]=QE[ρ(r)]σ[ρ(r)]=−\boldmath∇δ\@fontswitchH[ρ]δρ, (0)

where is the system Hamiltonian. In this case, by using vector integration by parts, it is easy to show that

 δδρ(r′)∫drW1[ρ(r)]⋅\boldmath\@fontswitchA(r)=−δδρ(r′)∫∂ΛdΓδ\@fontswitchH[ρ]δρ\boldmath\@fontswitchA(r)⋅^n=0, (0)

for any divergence-free vector field . The second integral is taken over the boundary of the domain where the system is defined, and is the unit vector normal to the boundary at each point. In particular, by taking constant, eq. (0) implies that . Hence for time-reversible systems the optimal profile remains invariant under rotations of the current , see eq. (0), and this allows us to prove the isometric fluctuation relation (IFR), eqs. (1) and (8) in the main text.

We can now relax hypothesis (ii) above and study cases where the current profile is not constant. Let be the probability of observing a time-averaged current field . Notice that this vector field must be divergence-free because of hypothesis . This probability also obeys a large deviation principle, , with a current LDF equivalent to that in eq. (0) but with a space-dependent current field . The optimal density profile is now solution of

 δδρ(r′)∫dr(W2[ρ(r)]−2\boldmath\@fontswitchJ(r)⋅W1[ρ(r)]+\boldmath\@fontswitchJ2(r)W0[ρ(r)])=0, (0)

which is the equivalent to eq. (0) in this case. For time-reversible systems condition (0) holds and remains invariant under (local or global) rotations of . In this way we can simply relate with the probability of any other divergence-free current field locally-isometric to , i.e. , via a generalized isometric fluctuation relation, see eq. (12) in the paper. Notice that in general an arbitrary local or global rotation of a divergence-free vector field does not conserve the zero-divergence property, so this constraints the current fields and/or local rotations for which this generalized IFR applies.

The large deviation function for the space- and time-averaged current, , can be related to via a contraction principle

 G(J)=max\boldmath\@fontswitchJ(r):\boldmath∇⋅\boldmath\@fontswitchJ(r)=0J=∫dr\boldmath\@fontswitchJ(r)G[\boldmath\@fontswitchJ(r)].