Typical and rare fluctuations in nonlinear driven diffusive systems with dissipation
TTypical and rare fluctuations in nonlinear driven diffusive systems with dissipation
Pablo I. Hurtado ∗ Instituto Carlos I de F´ısica Te´orica y Computacional,and Departamento de Electromagnetismo y F´ısica de la Materia, Universidad de Granada, Granada 18071, Spain
A. Lasanta † and A. Prados ‡ F´ısica Te´orica, Universidad de Sevilla, Apdo. de Correos 1065, Sevilla 41080, Spain (Dated: October 31, 2018)We consider fluctuations of the dissipated energy in nonlinear driven diffusive systems subject tobulk dissipation and boundary driving. With this aim, we extend the recently-introduced macro-scopic fluctuation theory to nonlinear driven dissipative media, starting from the fluctuating hydro-dynamic equations describing the system mesoscopic evolution. Interestingly, the action associatedto a path in mesoscopic phase-space, from which large-deviation functions for macroscopic observ-ables can be derived, has the same simple form as in non-dissipative systems. This is a consequenceof the quasi-elasticity of microscopic dynamics, required in order to have a nontrivial competitionbetween diffusion and dissipation at the mesoscale. Euler-Lagrange equations for the optimal densityand current fields that sustain an arbitrary dissipation fluctuation are also derived. A perturbativesolution thereof shows that the probability distribution of small fluctuations is always gaussian, asexpected from the central limit theorem. On the other hand, strong separation from the gaussianbehavior is observed for large fluctuations, with a distribution which shows no negative branch,thus violating the Gallavotti-Cohen fluctuation theorem as expected from the irreversibility of thedynamics. The dissipation large-deviation function exhibits simple and general scaling forms forweakly and strongly dissipative systems, with large fluctuations favored in the former case butheavily supressed in the latter. We apply our results to a general class of diffusive lattice modelsfor which dissipation, nonlinear diffusion and driving are the key ingredients. The theoretical pre-dictions are compared to extensive numerical simulations of the microscopic models, and excellentagreement is found. Interestingly, the large-deviation function is in some cases non-convex beyondsome dissipation. These results show that a suitable generalization of macroscopic fluctuation theoryis capable of describing in detail the fluctuating behavior of nonlinear driven dissipative media.
I. INTRODUCTION
Fluctuations are inherent to many physical phenom-ena, reflecting the hectic microscopic dynamics at macro-scopic scales. In spite of their apparent random ori-gin, essential physical information is encoded therein [1].A classical example is the fluctuation-dissipation theo-rem, which relates the linear response of a system toan external perturbation to the fluctuation propertiesof the system in thermal equilibrium [2, 3]. More re-cently, the investigation of general properties of fluctu-ations in nonequilibrium steady states is opening newpaths for understanding physics far from equilibrium [4–14]. The study of fluctuation statistics of macroscopicobservables provides an alternative path to obtain ther-modynamic potentials, a complementary approach to theusual ensemble description. This observation, valid bothin equilibrium [1] and nonequilibrium [7, 9], is most rele-vant in the latter case because no general bottom-up ap-proach, connecting microscopic dynamics to macroscopicnonequilibrium properties, has been found yet. In thisway, the large deviation function (LDF) controlling the ∗ [email protected] † [email protected] ‡ [email protected] statistics of these fluctuations may play in nonequilib-rium statistical mechanics a role similar to the equilib-rium free energy [4, 10]. A central point for this emerg-ing paradigm is the identification of the relevant macro-scopic observables characterizing the out of equilibriumbehavior of the system at hand. The system dynamicsoften conserves locally some magnitude (a density of par-ticles, energy, momentum, charge, etc.), and the essentialnonequilibrium observable is thus the current or flux sus-tained by the system when subject to boundary-inducedgradients or external fields. Therefore, the understandingof current statistics in terms of the microscopic dynam-ics represents one of the main problems of nonequilib-rium statistical mechanics, triggering an intense researcheffort in recent years. In this context, key results arethe Gallavotti-Cohen fluctuation theorem [5], which re-lates the probability of observing a given current fluctu-ation (cid:126)J with the probability of the reversed event − (cid:126)J , orthe recently-introduced Isometric Fluctuation Relation[14], which relates the probability of any pair of isomet-ric current fluctuations ( (cid:126)J, (cid:126)J (cid:48) ), with | (cid:126)J | = | (cid:126)J (cid:48) | . Theseintriguing “symmetries” appear as a consequence of theinvariance under time reversal of the underlying micro-scopic dynamics. These and other recent results [5–14]are however restricted to nonequilibrium “conservative”systems characterized by currents.On the other hand, many nonequilibrium systems areinherently dissipative , that is, they need a continuous in- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b put of energy in order to reach a steady state. In this classof systems, the relevant macroscopic observable is notonly the current: the dissipated energy is also expected toplay a main role. These systems include granular media[16], dissipative biophysical systems [18], turbulent fluids[19], active matter [20], chemical reactions [21], popula-tion dynamics [22], etc. In general, this class comprisesall sort of reaction-diffusion systems where dissipation,diffusion and driving are the main physical mechanisms.Fluctuations in dissipative media have been much less in-vestigated, most probably because their physics is morecomplicated as a result of the irreversibility of their mi-croscopic dynamics. In principle, most of the results fornonequilibrium steady states we have referred to in theprevious paragraph are not applicable to dissipative sys-tems, since they stem from the reversibility of the un-derlying microscopic dynamics in the conservative case.Therefore, a question naturally arises as to whether it ispossible to extend some of these ideas to dissipative me-dia. One of the main goals of the present paper is to givea (partial) answer to this question.In this work, we analyze both typical and rare fluc-tuations in nonlinear driven diffusive systems with dis-sipation. This is done by combining a suitable general-ization of macroscopic fluctuation theory (MFT) [7] tothe realm of dissipative media, and extensive numericalsimulations of a particular albeit broad class of micro-scopic models. Our starting point is a general fluctu-ating balance equation for the (energy) density, with adrift term proportional to the spatial derivative of thecurrent and a sink term. This mesoscopic description isexpected to be valid for many driven dissipative mediaover a certain “hydrodynamic” time scale, much largerthan the one characteristic of the microscopic dynam-ics. Over the fast (microscopic) time scale, the systemforgets the initial conditions and relaxes to a local equi-librium state in which all the properties of the systembecome functionals of a few “hydrodynamic” fields, herethe density, the current and the dissipation. Afterwards,over the much slower hydrodynamic time scale, the sys-tem eventually approaches the steady state following themesoscopic balance equation. We focus on the fluctua-tions of the system in this nonequilibrium steady state,in which the dissipation and the injection of energy bal-ance each other. By using this fluctuating hydrodynamicpicture together with a path integral formulation, we de-rive a general form for the action associated to a history of the density, current and dissipation fields (that is, apath in mesoscopic phase space). Remarkably, this actiontakes the same form as in conservative nonequilibriumsystems [7–14], simplifying the analysis in the dissipativecase. This is both an important and a surprising result,which stems from the quasi-elastic character of the un-derlying microscopic dynamics in the large system sizelimit. This quasi-elasticity is necessary in order to havea balanced competition between diffusion and dissipationat the mesoscopic level. From the derived action func-tional, and using the recently-introduced additivity con- jecture [8, 11–13], a general form for the LDF of the dis-sipated energy is derived, with a “Lagrangian” includingsecond order derivatives. Therefrom, we derive the Euler-Lagrange equation (a fourth-order differential equation)for the optimal fields responsible of an arbitrary fluc-tuation. This Lagrangian variational problem can bemapped onto an equivalent Hamiltonian problem (fourcoupled first-order differential equations) which turns outto simplify the analysis. We use this Hamiltonian pic-ture to analyze in detail three different limits, namelysmall fluctuations around the average for arbitrary dissi-pation coefficient, and the whole spectrum of fluctuations(typical and rare) for weakly- and strongly-dissipativesystems. The statistics of typical (that is, small) fluc-tuations is gaussian as expected from the central limittheorem. However, strong separation from gaussian be-havior is observed for rare fluctuations, with a distri-bution which shows no negative branch, thus violatingthe Gallavotti-Cohen fluctuation theorem as otherwiseexpected from the irreversible character of microscopicdynamics. We study in general the weakly-dissipativesystem limit using a singular perturbation expansion.This yields a simple scaling form for the dissipation LDF,showing that large dissipation fluctuations are favored inthis weakly-dissipative limit, with a LDF which extendsover a broad regime and decays slowly in the far positivetail. On the other hand, a different perturbative analy-sis in the strongly-dissipative system limit can be carriedbased on the formation of boundary energy layers in thislimit for all fluctuations, which effectively decouples thesystem in two almost-independent parts. This analysisshows that large dissipation fluctuations are heavily sup-pressed in this limit, as opposed to the weakly-dissipativeregime result.We apply this theoretical scheme to a general classof d -dimensional dissipative lattice models with stochas-tic microscopic dynamics, for which the hydrodynamicfluctuating picture used above as starting point can bedemonstrated in the large system size limit [39] (we willfocus here in one dimension for simplicity). In these mod-els there is one particle at each lattice site, characterizedby its energy. Dynamics is stochastic and proceeds viacollisions between nearest neighbors, at a rate which de-pends on the energy of the colliding pair. In a collision,a certain fraction of the pair energy is dissipated, andthe remaining energy is randomly distributed within thepair. This mechanism gives rise to a nonlinear competi-tion between diffusion and dissipation in the macroscopiclimit, provided that the microscopic dissipation coeffi-cient scales adequately with the system size. This class ofmodels represents at a coarse-grained level the physics ofmany reaction-diffusion systems of technological as wellas theoretical interest. In particular, when the collid-ing pair is chosen completely at random, independentlyof the value of its energy, the Kipnis-Marchioro-Presutti(KMP) model [23] for heat conduction is recovered in theconservative case. The KMP model plays a main role innonequilibrium statistical physics as a touchstone to testtheoretical advances [5, 6, 8, 11–15, 23]. Our generalclass of models contains the essential ingredients char-acterizing most dissipative media, namely: (i) diffusivedynamics, (ii) bulk dissipation, and (iii) boundary injec-tion. The chances are that our results remain valid formore complex dissipative media described at the meso-scopic level by a similar evolution equation. Here wereport analytical and simulation results for the statisticsof the dissipated energy in this general class of modelsusing both standard simulations and an advanced MonteCarlo method [27]. The latter allows the sampling of thetails of the distribution, and implies simulating a largenumber of clones of the system.The plan of the paper is as follows. Section II describesa suitable generalization of macroscopic fluctuation the-ory to nonlinear driven dissipative media. The large-deviation statistics of the dissipated energy, a centralobservable in this type of systems, is investigated herewithin both the Lagrangian and Hamiltonian equivalentframeworks. Section III is devoted to the detailed studyof different asymptotic behaviors within the Hamiltonianformulation, which turns out to simplify the analysis. Insection IV we define a general class of microscopic latticemodels whose stochastic dynamics is dissipative [25, 39].The theoretical framework developed in the previous sec-tions is applied to this family of models in section V, andthe LDF for the dissipated energy is explicitly workedout. The analytical predictions are compared to exten-sive numerical simulations of the microscopic models, anda very good agreement is found. A summary of the mainresults of the paper, together with a physical discussionthereof, is given in sec. VI. Finally, the appendix dealswith some technical details that, for the sake of clarity,we have preferred to omit in the main text. II. MACROSCOPIC FLUCTUATION THEORYFOR DRIVEN DISSIPATIVE SYSTEMS
In this work, we will analyze a general class of sys-tems whose dynamics at the mesoscale is described bythe following fluctuating evolution equation ∂ t ρ ( x, t ) = − ∂ x j ( x, t ) + d ( x, t ) . (2.1)We focus here in one dimension for simplicity, but ouranalysis can be carried out in an equivalent manner in d -dimensions. In eq. (2.1), ρ ( x, t ), j ( x, t ) and d ( x, t ) are thedensity, current and dissipation fields, respectively, and t and x ∈ [ − / , /
2] are the macroscopic time and spacevariables, obtained after a diffusive scaling limit such that x = ˜ x/L and t = ˜ t/L , with ˜ x and ˜ t the microscopicspace and time variables and L the system length. Thesecoarse-grained spatial and temporal scales emerge froma suitable continuum limit of the underlying microscopicdynamics [39]. The current field is a fluctuating quantity,and can be written as j ( x, t ) = − D ( ρ ) ∂ x ρ ( x, t ) + ξ ( x, t ) . (2.2) The first term is Fourier’s law, where D ( ρ ) is the diffu-sivity (which might be a nonlinear function of the localdensity), and ξ ( x, t ) is the current noise that is gaussianand white, (cid:104) ξ ( x, t ) (cid:105) = 0 , (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = σ ( ρ ) L δ ( x − x (cid:48) ) δ ( t − t (cid:48) ) , (2.3)with σ ( ρ ) being the so-called mobility. This gaussianfluctuating field is expected to emerge for most situa-tions in the appropriate mesoscopic limit as a result of acentral limit theorem: although microscopic interactionsfor a given model can be highly complicated, the ensuingfluctuations of the slow hydrodynamic fields result fromthe sum of an enormous amount of random events atthe microscale which give rise to gaussian statistics, withan amplitude of the order of L − / , in the mesoscopicregime in which eq. (2.1) emerges. On the other hand,the dissipation field d ( x, t ) is d ( x, t ) = − νR ( ρ ( x, t )) , (2.4)where ν is the macroscopic dissipation coefficient, and R ( ρ ) is a certain function of the density ρ . For the calcu-lations which follow throughout this section, it is usefulto introduce a new variable y , such that y = R ( ρ ) , (2.5a) d ( x, t ) = − νy ( x, t ) . (2.5b)The dissipation field is present at the mesoscopic levelbecause the microscopic stochastic dynamics of the mod-els of interest dissipates some energy, that is, we havethe equivalent of a microscopic restitution coefficient α ,so that the amount of dissipated energy is proportionalto 1 − α . The macroscopic dissipation coefficient ν isthus proportional to 1 − α . Note however that there isno noise term in eq. (2.4), so the local fluctuations ofthe dissipation field are enslaved to those of the density ρ ( x, t ). The physical reason for this behavior is that themicroscopic dynamics must be quasi-elastic in order toensure that dissipation and diffusion take place over thesame time scale in the thermodynamic limit. Typically,1 − α must scale as L − that is the order of magnitudeof the diffusive term in a system of length L [25, 39].The boundary conditions for eq (2.1) depend on thephysical situation of interest. For instance, we may con-sider that the system is kept in contact with two ther-mal reservoirs at x = ± /
2, at the same temperature T ,so ρ ( ± / , t ) = T . In that case, the system eventuallyreaches a steady state in the long time limit, for whichthe injection of energy through the boundaries and thedissipation balance each other. The stationary average(macroscopic) solution of (2.1) verifies j (cid:48) av ( x ) + νR ( ρ av ( x )) = 0 , j av ( x ) = − D ( ρ av ( x )) ρ (cid:48) av ( x ) , (2.6)where the prime indicates spatial derivative. The firstequation in (2.6) follows from (2.1), and the second oneis Fourier’s law for the averages. Equivalently, a closedsecond-order equation for ρ may be written, ddx [ D ( ρ av ) ρ (cid:48) av ] = νR ( ρ av ) , (2.7)with the boundary conditions ρ av ( ± /
2) = T . Equations(2.6) and (2.7) can be also written for the variable y introduced in eq. (2.5a), j (cid:48) av ( x ) + νy av ( x ) = 0 , j av ( x ) = − ˆ D ( y av ( x )) y (cid:48) av ( x ) , (2.8)with ˆ D ( y ) = (cid:18) dydρ (cid:19) − D ( ρ ) , (2.9)since j av ( x, t ) = − ˆ D ( y av ) ∂ x y ( x, t ) . (2.10)Thus ˆ D is an “effective” diffusivity: it is the factor multi-plying the spatial gradient when writing Fourier’s equa-tion in terms of the new variable y . Equation (2.8) canalso be summarized in a second order differential equa-tion for y av , (cid:104) ˆ D ( y av ) y (cid:48) av (cid:105) (cid:48) = νy av , y av ( ± /
2) = R ( T ) . (2.11)Interestingly, it can be shown (see below) that ˆ D is con-stant, independent of y , whenever y = R ( ρ ) depends al-gebraically in ρ , a case we will study in detail in section V.This observation considerably simplifies the subsequentanalysis.The probability of observing a history { ρ ( x, t ) , j ( x, t ) } τ of duration τ for the density and current fields, startingfrom a given initial state, can be written now as a pathintegral over all the possible realizations of the currentnoise { ξ ( x, t ) } τ , weighted by its gaussian measure, andrestricted to those realizations compatible with eq. (2.1)at every point of space and time [12]. This probabilityhence obeys a large deviation principle of the form [4, 7–13, 25] P ( { ρ, j } τ ) ∼ exp (+ L I τ [ ρ, j ]) , (2.12)with a rate functional [7, 9] I τ [ ρ, j ] = − (cid:90) τ dt (cid:90) / − / dx [ j + D ( ρ ) ∂ x ρ ] σ ( ρ ) (2.13)with ρ ( x, t ) and j ( x, t ) coupled via the balance equa-tion (2.1), and the dissipation d ( x, t ) given in terms of ρ ( x, t ) by (2.4). Equation (2.13) expresses the gaussiannature of the local current fluctuations around its aver-age (Fourier’s law) behavior. The functional in (2.13)is the same as in the conservative case (that is, with nobulk dissipation), due to the quasi-elasticity of the micro-scopic dynamics, which makes the current noise be the FIG. 1. (Color online) Convergence of the space&time-integrated dissipation to its ensemble value for many differentrealizations, and sketch of the probability concentration astime increases, associated with the large deviation principle,eq. (2.15). only relevant one in the hydrodynamic description, seediscussion in section IV [39]. We focus now on the fluc-tuations of the dissipated energy, integrated over spaceand time d = − τ (cid:90) τ dt (cid:90) / − / dx d ( x, t ) (2.14)= ντ (cid:90) τ dt (cid:90) / − / dx R ( ρ ( x, t )) > , where we have introduced a minus sign for the sake ofconvenience, in order to make d positive. As discussedabove, this is a fundamental observable to understandthe statistical physics of driven dissipative media. Theprobability of such a fluctuation P τ ( d ) scales in the long-time limit as P τ ( d ) ∼ exp [+ τ L G ( d )] , G ( d ) = 1 τ max ρ,j I τ [ ρ, j ] . (2.15)This defines a new large deviation principle for d , see Fig.1, such that G ( d ) is obtained from I τ [ ρ, j ] via a saddle-point calculation for long times (that is, it follows fromthe contraction of the original rate function I τ [10]). Theoptimal fields ρ ( x, t ; d ), j ( x, t ; d ) which are the solutionof the variational problem (2.15) must be consistent withthe prescribed value of the dissipated energy d in (2.14),and are also related by the balance equation (2.1), sup-plemented with (2.4) and the appropriate boundary con-ditions. These optimal fields can be interpreted as theones adopted by the system to sustain a long-time fluc-tuation of the space&time-integrated dissipation d . Forthe sake of simplicity, we have not explicitly introducedin our notation the parametric dependence of the LDF G ( d ) and the associated optimal profiles on the bound-ary temperature T , though this should be borne in mindfor latter reference. A. The constrained variational problem
We now assume that these optimal profiles do not de-pend on time. In conservative systems, this conjecturehas been shown [7] to be equivalent to the additivity prin-ciple recently introduced to study current fluctuations indiffusive media [8]. The validity of this additivity sce-nario has been recently confirmed in extensive numericalsimulations for a broad interval of fluctuations [11, 14],though it may eventually break down for extreme fluctu-ations via a dynamic phase transition [30, 31]. As we willsee below, the applicability of this generalization of theadditivity conjecture to dissipative systems is well sup-ported by numerical evidence. Under this simplifying hy-pothesis, the fluctuating balance equation (2.1) reducesto j (cid:48) ( x ) + νy ( x ) = 0 , y ( x ) = − j (cid:48) ( x ) /ν, (2.16)making use of the variable y defined in eq. (2.5a). More-over, we can integrate over time in the definition (2.14)of the integrated dissipation d , d = ν (cid:90) / − / dx y ( x ) , (2.17a)or, equivalently, d = − (cid:90) / − / dx j (cid:48) ( x ) = j ( − / − j (1 / > . (2.17b)In this way, by using the additivity hypothesis we caneliminate ρ ( x ) and write G ( d ) in terms of only one vari-able as G ( d ) = − min j ( x ) S [ j ] , with S [ j ] = (cid:90) / − / dx L ( j, j (cid:48) j (cid:48)(cid:48) ) , (2.18a) L ( j, j (cid:48) , j (cid:48)(cid:48) ) = [ j − ˆ D ( − j (cid:48) /ν ) j (cid:48)(cid:48) ν ] σ ( − j (cid:48) /ν ) , (2.18b)where ˆ D is the effective diffusivity defined in eq. (2.9),and ˆ σ is the mobility, defined in eq. (2.3), both writtenin terms of y = − j (cid:48) /ν . The function L ( j, j (cid:48) , j (cid:48)(cid:48) ) is ageneralized Lagrangian with dependence on first and alsosecond order derivatives, see the appendix.We have to find the optimal current profile j ( x ; d ),that is, the solution of the variational problem (2.18),with the constraint that the integrated dissipation d hasa definite value, as given by (2.17). Therefore we mustuse the Lagrange multiplier procedure [32, 33], that is,look for an extremum of S λ [ j ] = S [ j ] − λ (cid:90) / − / dx ( j (cid:48) + d ) (2.19)= (cid:90) / − / dx L λ ( j, j (cid:48) , j (cid:48)(cid:48) ) (2.20) where L λ ( j, j (cid:48) , j (cid:48)(cid:48) ) = L ( j, j (cid:48) , j (cid:48)(cid:48) ) − λ ( j (cid:48) + d ) , (2.21)with λ being the Lagrange multiplier. The extremum of S λ follows from two conditions: (i) δ S λ = 0, and (ii) ∂S λ /∂λ = 0. The first condition implies d dx (cid:18) ∂ L λ ∂j (cid:48)(cid:48) (cid:19) − ddx (cid:18) ∂ L λ ∂j (cid:48) (cid:19) + ∂ L λ ∂j = 0 , (2.22)which is the Euler-Lagrange equation for a Lagrangian L λ containing second order derivatives (see the ap-pendix). Condition (ii) leads to the constraint on theintegrated dissipation, given by Eq. (2.17). The bound-ary conditions for the Euler-Lagrange equation are j (cid:48) ( ± /
2) = − νR ( T ) , p λj ( ± /
2) = 0 . (2.23)We have introduced the generalized momentum p j con-jugate of j , for the new Lagrangian L λ , as p λj = ∂ L λ ∂j (cid:48) − ddx (cid:18) ∂ L λ ∂j (cid:48)(cid:48) (cid:19) . (2.24)These boundary conditions arise from (1) the valuesof the density at the boundaries, which are prescribed, ρ ( ± /
2) = T , and (2) the condition δS λ = 0, which pro-vides the additional needed conditions when there arenot enough values of the variables fixed at the bound-aries (see the appendix, and also [33]). B. Mapping the constraint to boundary conditions
Taking into account the relation between L λ and L ,Eq. (2.21), the generalized momentum p λj verifies p λj = p j − λ, (2.25)where p j is the generalized momentum for the Lagrangian L , that is, p j = ∂ L ∂j (cid:48) − ddx (cid:18) ∂ L ∂j (cid:48)(cid:48) (cid:19) . (2.26)Moreover, the Euler-Lagrange equation (2.22) impliesthat d dx (cid:18) ∂ L ∂j (cid:48)(cid:48) (cid:19) − ddx (cid:18) ∂ L ∂j (cid:48) (cid:19) + ∂ L ∂j = 0 , (2.27)that is, we also obtain the Euler-Lagrange equation cor-responding to the original Lagrangian L . The definitionof p j in eq. (2.26) guarantees that p (cid:48) j = ∂ L /∂j , as in thecase of the usual variational problem with a Lagrangianincluding only first-order derivatives. Now, the boundaryconditions can be written as j (cid:48) ( ± /
2) = − νR ( T ) , p j ( ± /
2) = λ, (2.28)which follow from Eqs. (2.23) and (2.25). The aboveresult imply that our constrained variational problemcan be mapped onto a unconstrained variational prob-lem with the original Lagrangian L , its associated Euler-Lagrange equation (2.27) and the boundary conditions(2.28). The unknown value λ for the generalized momen-tum p j at the boundaries must be determined by impos-ing the prescribed value of the integrated dissipation, asgiven by Eq. (2.17), that is, λ = λ ( d ). In particular, λ = 0 is equivalent to imposing no restrictions on theintegrated dissipation, so that we should recover (as wewill see later) the average profiles and dissipation in thiscase. In this sense, a non-zero value of λ = p j ( ± /
2) is ameasure of the departure from the average hydrodynamicbehaviour.On physical grounds, we expect the corresponding op-timal density profile to be an even function of x , becauseof the symmetry of our system around the center x = 0.In fact, the Euler-Lagrange equation (2.27) admits so-lutions with well-defined parity. Since the Lagrangianhas the symmetry property L ( − j, j (cid:48) , − j (cid:48)(cid:48) ) = L ( j, j (cid:48) , j (cid:48)(cid:48) ),Eq. (2.27) has solutions with j being an odd functionof x , which implies that y (and therefore ρ ) is an evenfunction of x . From now on, we will restrict ourselvesto these symmetric solutions of the variational problem.Thus, Eq. (2.17) reduces to d = 2 j ( − / d ) = − j (1 / d ) , (2.29)so the boundary conditions for the Euler-Lagrange equa-tion boil down to j (cid:48) ( ± / d ) = − νR ( T ) , j ( − / d ) = − j (1 / d ) = d/ , (2.30)i.e. much simpler than Eq. (2.28). This follows from p j being an even function of x for the solutions with well-defined parity we are considering. In this way, symmetryconsiderations lead to the simpler boundary conditions(2.30), which in turn allow us to get rid of the Lagrangemultiplier λ . In summary, we have mapped our originalvariational problem with the subsidiary condition thatthe dissipation has a given value to an unconstrained vari-ational problem, with the original Lagrangian L ( j, j (cid:48) , j (cid:48)(cid:48) )and prescribed values of j and j (cid:48) at the boundaries.Once the optimal current profile is obtained, the op-timal density profile can be calculated from the balanceequation (2.16). Of course, the density profile so ob-tained obeys the boundary conditions ρ ( ± / d ) = T .It must be stressed that the Euler-Lagrange equation(2.27) with boundary conditions (2.30) gives the correctsolution to the constrained variational problem when theoptimal profiles have a well-defined parity. Nevertheless,one cannot rule out the existence of symmetry-breakingsolutions without well-defined parity, since in general avariational problem may have multiple solutions [32]. Inthat case, one must solve the more complex variationalproblem comprising the Euler-Lagrange equation (2.27)with the boundary conditions (2.28), where the Lagrangemultiplier λ = λ ( d ) is determined by imposing the con-straint (2.17). We note however that numerical evidence strongly supports the validity of symmetric solutions (seebelow).The LDF G ( d ) depends on d and T through the bound-ary conditions (recall that its T -dependence has beenomitted in notation for simplicity). The derivation ofthe Euler-Lagrange equation (2.27) (see the appendix)shows that, for the solutions with well-defined parity, δG ( d ) = p j (1 / δd + 2 p j (cid:48) (1 / ν dR ( T ) dT δT, (2.31)where p j is the generalized momentum conjugate of j , de-fined in Eq. (2.26), and p j (cid:48) is the generalized momentumconjugate of j (cid:48) , p j (cid:48) = ∂ L ∂j (cid:48)(cid:48) , (2.32)which is an odd function of x for the solutions with well-defined parity. In this way Eq. (2.31) offers a geometricinterpretation for the values of the generalized momentaat the boundaries, as they are directly related to the par-tial derivatives of the LDF, ∂G∂d = p j (1 / , ∂G∂T = 2 ν dR ( T ) dT p j (cid:48) (1 / . (2.33) C. A Hamiltonian formulation of the problem
We will not write the detailed form of the generalfourth-order differential equation (2.27) for the optimalprofile j ( x ; d ), since it is not particularly illuminating.Instead we now write a set of four equivalent first orderdifferential equations arising in the equivalent “Hamilto-nian” description. In the following, we sketch the proce-dure to introduce the Hamiltonian for a Lagrangian withhigher-order derivatives [32, 33], adapted to the presentcase. As the Euler-Lagrange equation is a fourth-orderdifferential equation, we should have two canonical coor-dinates and their two corresponding canonical momenta.The first canonical coordinate is the current j , and wechoose the second one to be y , which is proportional to j (cid:48) , as given by Eq. (2.16). This choice is suggested bythe structure of the Lagrangian in eq. (2.18). It is worthrecalling that the density profile can be directly obtainedfrom y by making use of its definition, Eq. (2.5a). Next,we introduce the canonical momenta p y and p j conjugateto y and j , respectively. The momentum p j has been de-fined in Eq. (2.26), and p y is given by p y ≡ − ν ∂ L ∂j (cid:48)(cid:48) , (2.34)which follows from the definition of p j (cid:48) , Eq. (2.32), andEq. (2.16) for y . The Hamiltonian is then introduced inthe usual way, H ≡ y (cid:48) p y + j (cid:48) p j − L ≡ y (cid:48) p y − νyp j − L . (2.35)After some algebra, we get H = 12 Q ( y ) p y − ˆ D − ( y ) jp y − νyp j , (2.36a) Q ( y ) ≡ ˆ σ ( y )ˆ D ( y ) , (2.36b)where we have defined the auxiliary function Q ( y ), with Q ( y ) > y . We have also made use of eq. (2.16),of the expression for p y which follows from its definition(2.34), p y = ˆ D ( y ) j + ˆ D ( y ) y (cid:48) ˆ σ ( y ) , (2.37)and of the Lagrangian L = ˆ σ ( y ) p y D ( y ) = 12 Q ( y ) p y , (2.38)written in terms of the canonical variables, with the aidof eq. (2.37). As usual, H is a function of ( y, j, p y , p j ),which satisfy the following set of four ”canonical” first-order differential equations, y (cid:48) = ∂ H ∂p y = Q ( y ) p y − ˆ D − ( y ) j, (2.39a) j (cid:48) = ∂ H ∂p j = − νy, (2.39b) p (cid:48) y = − ∂ H ∂y = − dQ ( y ) dy p y d ˆ D − ( y ) dy jp y + νp j , (2.39c) p (cid:48) j = − ∂ H ∂j = ˆ D − ( y ) p y (2.39d)that are equivalent to the fourth-order Euler-Lagrangeequation (2.27). Note that, as is usual in physics and inorder not to clutter our formulae, we have dropped thesubindex 0 for the optimal profiles, which are now solu-tions of the above canonical equations; the same notationis used for the canonical variables in the Hamiltonian andfor the solutions of Hamilton’s equations. On the otherhand, as the Hamiltonian does not depend explicitly on x , it is a first integral of the system (2.39): H = const.over any of its solutions. This property may be used tosimplify the integration of the system.In general, for a given value of the dissipation d , wehave to solve the system of equations (2.39) with theboundary conditions y ( ± /
2) = R ( T ) , j ( − /
2) = − j (1 /
2) = d/ , (2.40)which follow from Eqs. (2.30) and (2.16). Again, wehave to look for solutions of Eq. (2.39) with well-defined parity, that is, y and j are even and odd functions of x ,respectively (and, therefore, p y is odd and p j even). Thesolution of these canonical equations is then inserted intothe expression of the LDF G ( d ), which can be written interms of the canonical variables as G ( d ) = − (cid:90) / − / dx L = − (cid:90) / − / dx Q ( y ) p y , (2.41)by combining eqs. (2.18) and (2.38). In this way, weobtain the LDF for an arbitrary value of the integrateddissipation d within the Hamiltonian formulation of thevariational problem. Equation (2.41) shows clearly thatthe most probable (average) profiles correspond to a so-lution with p y = 0 for all x, for which G ( d ) vanishes.By substituting p y = 0 in Eqs. (2.39c)-(2.39d), we alsohave that p j = 0 for all x . Moreover, Eqs. (2.39a) and(2.39b) simplify to eq. (2.8), that is, the average profilesare reobtained. Therefore, there is always a solution ofthe canonical equations (2.39) with identically vanishingcanonical momenta, which corresponds to the averagesolution of the hydrodynamic equation (2.8) [40]. Theseaverage hydrodynamic profiles { ρ av , j av } lead to the av-erage value of the integrated dissipation d av = ν (cid:90) / − / dx R ( ρ av ) = ν (cid:90) / − / dx y av ( x )= 2 j av ( − / . (2.42)This discussion is consistent with the one below Eq.(2.28), which was done within the framework of theequivalent Lagrangian description. Fluctuations involvenon-zero values for the canonical momenta, whose magni-tude is then a measure of the departure from the averagebehaviour ( d − d av ) /d av . III. ANALYSIS OF THE LDF IN SOMELIMITING CASES
In the following subsections, we further analyze theform of the LDF in certain limits of interest, for whichsome general results can be obtained. First, we focus onthe behavior of G ( d ) for small fluctuations around theaverage, where a quadratic shape of the LDF is expected(corresponding to gaussian fluctuations). We then ana-lyze the limit of weakly-dissipative systems, ν (cid:28)
1, forwhich an adequate perturbative expansion allows us toobtain a non-trivial and interesting scaling form for theLDF. Finally, we consider the opposite limit of strongly-dissipative systems, ν (cid:29)
1, for which a different scalingfor the LDF is found.
A. Small fluctuations around the average
As the average behavior corresponds to the particularsolution of the canonical equations corresponding to van-ishing momenta p j = 0, p ρ = 0, small fluctuations can bethus analyzed by assuming that the canonical momentaare small. Let us define the dimensionles parameter (cid:15) = d − d av d av (3.1)to measure the separation from the average integrateddissipation d av . As we have just discussed, the canonicalmomenta vanish for (cid:15) = 0. We write y = y av + (cid:15) ∆ y, j = j av + (cid:15) ∆ j, p y = (cid:15) ∆ p y , p j = (cid:15) ∆ p j , (3.2)and linearize Eqs. (2.39) around the average solution,that is, we only retain terms linear in (cid:15) . Then,∆ y (cid:48) = Q ( y av )∆ p y − ˆ D − ( y av )∆ j − j av d ˆ D − ( y av ) dy av ∆ y, (3.3a)∆ j (cid:48) = − ν ∆ y, (3.3b)∆ p (cid:48) y = j av d ˆ D − ( y av ) dy av ∆ p y + ν ∆ p j , (3.3c)∆ p (cid:48) j = ˆ D − ( y av )∆ p y . (3.3d)The boundary conditions for these equations are∆ y ( ± /
2) = 0, ∆ j ( − /
2) = − ∆ j (1 /
2) = d av /
2. Thesolution of this system of equations must be inserted inthe large deviation function (2.18). Using the expression(2.41) for the LDF, it follows that G ( d ) ∼ − (cid:15) (cid:90) / − / dx Q ( y av )∆ p y , (3.4)for small fluctuations of the dissipation around the av-erage. Taking into account (3.1) and the large deviationprinciple (2.15), eq. (3.4) means that the probability ofsuch small fluctuations of the integrated dissipation d isapproximately gaussian, P τ ( d ) | (cid:15) |(cid:28) ∝ exp (cid:20) − Lτ ( d − d av ) d Λ ν (cid:21) (3.5)with Λ ν given byΛ ν = (cid:32)(cid:90) / − / dx Q ( y av )∆ p y (cid:33) − . (3.6)In this way, the gaussian estimation for the standard de-viation of the dissipation, by comparing (3.5) to (3.6), isgiven by χ ≡ d av Λ ν / √ τ L . In order to make a more de-tailed study of the LDF, concrete functional dependencesof the diffusivity D , the mobility σ and the dissipation R on the density ρ must be considered. This is done in thefollowing sections of the paper, where we will considera broad family of models for which the transport coef-ficients can be explicitly obtained. On the other hand, it is important to notice that gaussian statistics is onlyexpected for small fluctuations around the average dissi-pation. In general, the solution of the variational problemgiven by the integration of eq.(2.39), when inserted into(2.41), will give rise to non-gaussian statistics (that is, anon-quadratic dependence of the LDF) for an arbitraryfluctuation of the dissipated energy d . B. Weakly-dissipative systems, ν (cid:28) We proceed now by analysing the canonical equations(2.39) in the limit ν (cid:28)
1. Unsurprisingly, a regular per-turbation expansion in powers of ν breaks down, sinceit is not possible to impose the necessary boundary con-ditions for the current. This singularity of the elasticlimit was to be expected on a physical basis, as it is notpossible to obtain the behavior of weakly dissipative sys-tems ( ν (cid:28)
1) as a correction around the conservativecase ν = 0, for which ρ ( x ) = T and j ( x ) = 0. Therefore,a singular perturbation analysis should be done, lookingfor a suitable rescaling of the variables for ν (cid:28)
1. Equa-tion (2.8) for the averages implies that y av = R ( T ) + O ( ν ) , j av = − νR ( T ) x + O ( ν ) , ν (cid:28) . (3.7)The average current vanishes linearly in ν in the limit ν → + , as expected. Moreover, the average dissipation,obtained by combining eqs. (2.42) and (3.7), is given by d av = νR ( T ) + O ( ν ) . (3.8)Therefore, it is sensible to propose the following rescalingof variables j ( x ) = νψ ( x ) , p j ( x ) = Π ψ ( x ) ν , (3.9)which is consistent with the canonical equations (2.39),since ψ (cid:48) = 1 ν j (cid:48) = 1 ν ∂ H ∂p j = ∂ H ∂ Π ψ , (3.10a)Π (cid:48) ψ = νp (cid:48) j = − ν ∂ H ∂j = − ∂ H ∂ψ , (3.10b)with the same Hamiltonian H . In other words, eq. (3.9)defines a “canonical transformation” from the pair ofcanonical conjugate variables { j, p j } to { ψ, Π ψ } , a trans-formation that heals the singular behavior in the ν → + limit. The Hamiltonian can be now written as H = 12 Q ( y ) p y − y Π ψ − ν ˆ D ( y ) − ψp y (3.11)in the rescaled variables. Notice that the transformationintroduced is essential to obtain the correct “dominantbalance” [42] to the lowest order. In particular, beforethe rescaling, the term proportional to yp j was of theorder of ν and the term proportional to jp y was of theorder of unity; after the rescaling the orders of magnitudeare interchanged, the term proportional to y Π ψ is of theorder of unity while the term proportional to ψp y is of theorder of ν . We now start from the zero-th order rescaledHamiltonian by putting ν = 0 in eq. (3.11), H = 12 Q ( y ) p y − y Π ψ . (3.12)from which we we arrive at y (cid:48) = ∂ H ∂p y = Q ( y ) p y , (3.13a) p (cid:48) y = − ∂ H ∂y = − dQ ( y ) dy p y + Π ψ , (3.13b) ψ (cid:48) = ∂ H ∂ Π ψ = − y , (3.13c)Π (cid:48) ψ = − ∂ H ∂ψ = 0 . (3.13d)In order not to clutter our formulas, we do not introducea different notation for the canonical variables, althoughthe approximate canonical equations (with H ) are dif-ferent from the exact ones (with H ). We have only toremember that our results are valid to the lowest orderin ν . The canonical equations (3.13) have to be solvedwith the boundary conditions y ( ± /
2) = R ( T ) , ψ ( − /
2) = − ψ (1 /
2) = ∆ / , (3.14)where ∆ = dν = R ( T ) dd av (3.15)is assumed to be of the order of unity, that is, d = O ( ν )or d/d av = O (1). Thus, our rescaling allows us to obtaina solution for the optimal profiles for the density ρ , byinverting the relation y = R ( ρ ), and the current j = νψ , for integrated dissipations d very different from itsaverage value d av .It is worth noticing that ψ is a cyclic variable and itsconjugate momentum is thus constant, Π ψ ≡ Π ψ =const., see Eq. (3.14); this fact allows us to obtain aclosed first order differential equation for y ( x ) in the ν (cid:28) ψ = ∂G∂ ∆ , (3.16)which gives the physical interpretation of this first in-tegral of the approximate canonical equations: it is thepartial derivative of the LDF with respect to the rescaleddissipation. The Hamiltonian H is also constant, sinceit does not depend explicitly on x , and combining (3.12)and (3.13a), y (cid:48) = 2 Q ( y )( H + y Π ψ ) , y ( ± /
2) = R ( T ) . (3.17)Once this is solved, the rescaled current ψ can be ob-tained from (3.13c) ψ (cid:48) = − y, ψ ( − /
2) = − ψ (1 /
2) = R ( T ) d d av , (3.18) so that the two constants H and Π ψ will be given interms of the temperature T and d/d av . There are no moreconstants to be adjusted in the solution of eqs. (3.17) and(3.18) due to the parity properties of ( y, ψ ): y is and evenfunction of x and ψ is an odd function of x in the interval[ − / , / y av ( x ) and j av ( x ) are reobtained from the canonical equations byputting Π ψ = 0 and p y = 0 therein. Equation (3.12)implies that H = 0 over the average profiles.The simple form of the differential equation (3.17) al-lows us to infer some of the properties of the optimalprofile y ( x ) associated to a given dissipation fluctua-tion in the limit of weakly-dissipative systems. First,notice that in general the solution of eq. (3.17) willbe non-monotonic, exhibiting extrema in the interval x ∈ [ − , ]. Moreover, taking into account that the func-tion Q ( y ) is positive defined, it follows that the profile atthe extrema will take an unique value y ≡ − H Π ψ (3.19)Note that Π ψ (cid:54) = 0 for d (cid:54) = d av and, moreover, it musthave a different sign that H , i.e. sgn(Π ψ ) (cid:54) = sgn( H ),since y ( x ) > ∀ x . Therefore, the optimal profile y ( x )can only have a single extremum (minimum or maxi-mum) [41], that is located at x = 0 because of symmetryreasons. By rewriting eq. (3.17) as y (cid:48) = 2 Q ( y ) H (cid:18) − yy (cid:19) (3.20)we conclude that the constant H and y − y ( x ) must havethe same sign ∀ x ∈ [ − , ]. Thus, for H > y ( x ) has a single maximum, y ( x ) > y ( ± /
2) = R ( T ) ∀ x ,and thus d > d av . On the other hand, H < y ( x ) < R ( T ) ∀ x and d < d av . All theseproperties are confirmed below for particular examples,both analytically and numerically.Interestingly, the leading behavior for the LDF can bealso easily obtained in terms of the first integrals H andΠ ψ . In fact G ( d ) ∼ − (cid:90) / − / dx Q ( y ) p y = − (cid:90) / − / dx y (cid:48) Q ( y ) , (3.21)and making use of eq. (3.17), G ( d ) ∼ − (cid:32) H + Π ψ (cid:90) / − / dx y ( x ) (cid:33) = − ( H + Π ψ ∆) = −H (cid:18) − ∆ y (cid:19) . (3.22)Thus, the remaining task consists in writing the con-stants H and y (or equivalently H and Π ψ ) in termsof the integrated dissipation d and the temperature atthe boundaries T . Once this is done, the LDF followsfrom the simple expression given by eq. (3.22). Further-more, we may obtain bounds for the profile extremum y G ( d ) < d (cid:54) = d av : (i)for H > y is a maximum andthis implies that y > ∆, while (ii) for H < y corresponds to a minimum and thus y < ∆.Interestingly, we can also use Eq. (3.22) together withEq. (3.16) to obtain a simple relation between H , G and ∂G/∂d , namely − H = G + ∆ ∂G∂ ∆ = G ( d ) + d ∂G ( d ) ∂d = const. (3.23)It must be stressed that this relation only holds in theweakly dissipative system limit, in the sense that themacroscopic dissipation coefficient is small, ν (cid:28)
1, andthe considered integrated dissipation verifies that d = O ( ν ).For many systems of interest, the function Q ( y ) is typ-ically a homogeneous function of y of degree γ , Q ( cy ) = c γ Q ( y ) , (3.24)where c is an arbitrary real number, that is, Q ( y ) ∝ y γ .This type of dependence is common to many driven dis-sipative media, as for instance the general family of mod-els that we will study in section IV [25, 39] or differentreaction-diffusion systems [43]. However, it should benoted that not all systems obey this homogeneity condi-tion, e.g. symmetric simple exclusion processes with dis-sipative dynamics have a non-homogeneous Q ( y ) [9, 37].By introducing the scaling y ( x ) = R ( T ) Y ( x ) , y = R ( T ) Y , H = R ( T ) − γ (cid:101) H , (3.25)Eq. (3.17) is transformed into Y (cid:48) ( x ) = 2 (cid:101) H Q ( Y ) (cid:18) − Y ( x ) Y (cid:19) , Y ( ± /
2) = 1 . (3.26)This is quite a natural transformation: we scale the vari-able y with its value R ( T ) at the boundaries; besides, wewill be able to find a physically-relevant scaling variablefor the dissipation LDF. Accordingly, we also introduce ψ ( x ) = R ( T )Ψ( x ) , Ψ (cid:48) ( x ) = − Y ( x ) , (3.27a)Ψ( − /
2) = Ψ(1 /
2) = ∆2 R ( T ) = d d av . (3.27b)The (even) solution of Eq. (3.26) has the form Y = Y ( x, (cid:101) H , Y ) . (3.28)As said before, there are no more integration constantswhen solving Eqs. (3.26)-(3.27), since Y (resp. Ψ) is aneven (resp. odd) function of x . The boundary conditionis Y ( x = 1 / , (cid:101) H , Y ) = 1 , (3.29)which implies that Y = Y ( (cid:101) H ), the scaled height Y isonly a function of the scaled Hamiltonian (cid:101) H . Now, taking into account that the (odd) solution of Eq. (3.27) has theform Ψ = Ψ( x, (cid:101) H , Y ) , (3.30)we have thatΨ( x = − / , (cid:101) H , Y ( (cid:101) H )) = d d av . (3.31)Therefore, (cid:101) H = (cid:101) H (cid:18) dd av (cid:19) , (3.32)that is, (cid:101) H is a function only of the integrated dissipation d relative to its average value d av .This observation will be used in what follows to finda simple scaling form for the dissipation LDF. In fact,equation (3.22) can be readily rewritten as R ( T ) γ − G ( d ) ∼ − (cid:34) − d/d av Y ( (cid:101) H ) (cid:35) (cid:101) H (3.33)The equation above gives the general scaling of the LDFin the limit of weakly-dissipative systems: since both (cid:101) H and Y are only functions of d/d av , (cid:18) d av ν (cid:19) γ − G ( d ) = − (cid:34) − d/d av Y ( (cid:101) H ) (cid:35) (cid:101) H (3.34)is only a function of d/d av [we have made use of Eq. (3.8)for R ( T )]. This is quite a strong result: it means that, foreach value of γ , all the curves of ( d/d av ) γ − G ( d ) plottedas a function of d/d av fall on a certain “master” curvefor all values of the dissipation coefficient ν , providedthat ν (cid:28) Q ( y ) must be a ho-mogeneous function of y , with an arbitrary degree γ , arather general assumption satisfied in many cases of in-terest (see below). That being said, it is important tostress that the differential equation (3.26) for Y ( x ) con-tains the function Q ( y ), that has γ as a parameter. Thus,both ( Y, Ψ) and the rhs of eq. (3.34) also contain γ asa parameter; in principle, different physical models withdifferent functions Q ( y ) have different scaling functions.The simplest situation appears for γ = 2, in that caseEq. (3.34) predicts that G ( d ) is only a function of therelative dissipation d/d av , with no additional dependenceon ν .Finally, it is also interesting to note that the optimalprofiles y and ψ also have simple scaling forms. In fact,eq. (3.28), together with Eq. (3.25), implies that y ( x ) = d av ν Y ( x, (cid:101) H , Y ( (cid:101) H )) . (3.35)On the other hand, eq. (3.30), together with Eqs. (3.26),yields that ψ ( x ) = d av ν Ψ( x, (cid:101) H , Y ( (cid:101) H )) . (3.36)1Therefore, both y ( x ) and ψ ( x ) multiplied by ν/d av (thatis, divided by R ( T )) plotted as a function of x collapseonto a single curve for constant d/d av for each value of γ and all possible ν (cid:28) y = R ( ρ ) and j = νψ . On the other hand, the first integrals H and Π ψ follow from Eqs. (3.19) and (3.25), H = (cid:18) d av ν (cid:19) − γ (cid:101) H , Π ψ = − (cid:18) d av ν (cid:19) − γ (cid:101) H Y ( (cid:101) H ) . (3.37) C. Strongly-dissipative systems, ν (cid:29) We now proceed to analyze the limit ν (cid:29)
1, i.e. thelimit of strongly-dissipative dynamics. Equation (2.11)for the average profile implies that y av ( x ) develops twoboundary layers of width (cid:96) ν ∼ ν − / close to x = ± / ν (cid:29) x ∈ [ − / ,
0] and use the boundary conditions y ( − /
2) = R ( T ) , j ( − /
2) = d/ , y (cid:48) (0) = 0 , j (0) = 0 , (3.38)because of the symmetry of the solutions ( y even, j odd).Now, by introducing the following rescaling (suggested bythe typical lengthscale (cid:96) ν ∼ ν − / ), j = √ νψ, X = √ ν (cid:18) x + 12 (cid:19) , p y = √ ν Π y , (3.39)we arrive at the equivalent system of equations dydX = Q ( y )Π y − ˆ D − ( y ) ψ, (3.40a) dψdX = − y, (3.40b) d Π y dX = − dQ ( y ) dy Π y d ˆ D − ( y ) dy ψ Π y + p j , (3.40c) dp j dX = ˆ D − ( y )Π y (3.40d)with the boundary conditions y ( X = 0) = R ( T ) , dydX (cid:12)(cid:12)(cid:12)(cid:12) X = √ ν/ = 0 , (3.41a) ψ ( X = 0) = d √ ν , ψ ( X = √ ν/
2) = 0 . (3.41b) Interestingly, ν does not appear explicitly in the rescaledcanonical equations (3.40), but only in the boundary con-ditions. Therefore, in the limit ν → ∞ we have to solve(3.40) with the boundary conditions y ( X = 0) = R ( T ) , ψ ( X = 0) = ˜ d, (3.42a)lim X →∞ dydX = 0 , lim X →∞ ψ = 0 . (3.42b)where we have defined ˜ d = d √ ν , (3.43)which is assumed to be of the order of unity. In fact,from Eq. (2.17) one gets d = 2 ν (cid:90) − / dxy ( x ) = 2 √ ν (cid:90) √ ν/ dX y ( X ) , (3.44)that is, ˜ d ∼ (cid:90) ∞ dX y ( X ) , ν (cid:29) . (3.45)In this strongly-dissipative regime, the canonical equa-tions themselves are not simplified, but a physically ap-pealing picture emerges: the system decouples in two in-dependent boundary shells of width O ( √ ν ) close to theboundaries, where the rescaled variable X = O (1) (wehave restricted ourselves to the semi-interval [-1/2,0], thesolution in [0,1/2] is found by the symmetry argumentsalready used). Moreover, a simple scaling can be derivedfor the LDF G ( d ), Eq. (2.41), G ( d ) = − (cid:90) − / dxQ ( y ) p y ∼ −√ ν (cid:90) ∞ dXQ ( y )Π y , (3.46)where y and Π y are the solutions of (3.40) with theboundary conditions (3.42). Therefore, both y and Π y depend on R ( T ) and ˜ d through the boundary conditionsand G ( d ) = √ νF ( R ( T ) , ˜ d ) where F is a certain function.A particularly simple situation appear when both themobility σ ( y ) and the diffusivity D ( y ) are proportional tosome power of y , so that (i) the function Q ( y ) is homoge-neous, Q ( y ) ∝ y γ , as in Eq. (3.24), and (ii) the effectivediffusivity ˆ D does not depend on y , ˆ D ( y ) = ˆ D = const.as discussed in Sec. II. In fact, this is the case for thegeneral class of dissipative models analyzed in Sec. IV.The average dissipation for ν (cid:29) d av ∼ (cid:112) ν ˆ DR ( T ) ⇒ ˜ d = (cid:112) ˆ DR ( T ) dd av . (3.47)The canonical equations (3.40) can be analyzed followinga line of reasoning similar to the one used in the weaklydissipative system limit. Since the details are not neces-sary for the work presented here, we only give the finalresult for the LDF, that is G ( d ) = − (cid:114) ν ˆ D F (cid:18) [ R ( T )] γ − , dd av (cid:19) , (3.48)2where F is a certain scaling function. The scaling in Eq.(3.48) is more complex than in the weakly dissipativesystem limit. For instance, in the case γ = 2 we getthat G ( d ) = (cid:113) ν/ ˆ D F (1 , d/d av ), so the LDF curves, oncerescaled by ( ˆ D/ν ) / , collapse for all ν (cid:29) d/d av . The factor √ ν in front of the scaling function accounts for the strongsupression of the fluctuations of the dissipation that takesplace in strongly-dissipative systems: for a given value ofthe relative dissipation d/d av , the probability of such afluctuation decreases exponentially with √ ν . IV. A GENERAL CLASS OF NONLINEARDRIVEN DISSIPATIVE MODELS
In order to investigate in detail the validity of the gen-eral framework presented in previous sections, we nowintroduce a broad class of dissipative lattice models withstochastic microscopic dynamics that contain the essen-tial ingredients characterizing many dissipative media,namely: (i) nonlinear diffusive dynamics, (ii) bulk dissi-pation, and (iii) boundary driving. For the sake of sim-plicity, we will present them for the one-dimensional (1D)case, but the extension to arbitrary dimension is straight-forward.We thus consider a system defined on a 1D lattice with N sites. A configuration at a given time step p is givenby ρ = { ρ l,p } , l = 1 , . . . , N , where ρ l,p ≥ en-ergy of the l -th site at time p , so the total energy of thesystem at this time is E p = (cid:80) Nl =1 ρ l,p , see Fig. 1. Thedynamics is stochastic and sequential, and proceeds viacollisions between nearest neighbors. In an elementarystep, a nearest neighbor pair of sites ( l, l + 1) interactswith probability P l,p ( ρ ) = f (Σ l,p ) (cid:80) Ll (cid:48) =1 f (Σ l (cid:48) ,p ) , Σ l,p = ρ l,p + ρ l +1 ,p , (4.1)where f is a given function of the pair energy Σ l,p , and L is the number of possible pairs. Clearly L ∼ N , but theparticular relation depends on the boundary conditionsimposed (e.g., L = N + 1 for open boundaries while L = N for the periodic case). Once a pair is chosen, a certainfraction of its energy, namely (1 − α )Σ l,p , is dissipated tothe environment, mimicking the energy drain observedin real dissipative media. The remaining energy α Σ l,p isthen randomly redistributed between both sites, ρ l,p +1 = z p α Σ l,p , ρ l +1 ,p +1 = (1 − z p ) α Σ l,p , (4.2)with z p an homogeneously distributed random numberin the interval [0 , l = 1 , . . . , N −
1. In addition, and depending on the
FIG. 2. The model is defined on lattice sites, each one charac-terized by an energy ρ l . The dynamics is stochastic and pro-ceeds via random collisions between nearest neighbors wherepart of the pair energy is dissipated to the environment andthe rest is randomly redistributed within the pair. Such dy-namics mimics at the mesoscopic level the evolution of a wideclass of systems characterized by a nontrivial competition be-tween diffusion and dissipation. boundary conditions imposed, boundary sites might in-teract with thermal baths at both ends, possibly at dif-ferent temperatures T L (left) and T R (right). In this casethe dynamics is ρ ,p +1 = z p α ( e ,p + (cid:101) e L ) , ρ N,p +1 = z p α ( e N,p + (cid:101) e R ) , (4.3)when the first (last) site interacts with its neighboringthermal reservoir. Here (cid:101) e k , k = L, R , is an energyrandomly drawn at each step from the canonical dis-tribution at temperature T k , that is, with probabilityprob( (cid:101) e k ) = T − k exp( − (cid:101) e k /T k ) (our unit of temperatureis fixed by making k B = 1), see Fig. 1. We may considerinstead an isolated system with periodic boundary condi-tions, such that L = N and Eqs. (4.1) and (4.2) remainvalid for l = 0 ( l = N ) with the substitution ρ ,p = ρ N,p ( ρ N +1 ,p = ρ ,p ).The simplest dynamics corresponds to f (Σ l,p ) = 1 inEq. (4.1). In this case all (nearest neighbor) pairs collidewith equal probability P l,p = L − , independently of theirenergy. This choice (together with α = 1 above) corre-sponds to the Kipnis-Marchioro-Presutti (KMP) modelof energy transport [23], which can be considered as acoarse-grained description of the physics of a large classof quasi-1D real diffusive systems. For instance, it isone of the very few instances where Fourier’s law canbe rigorously proved [23]. In addition, the KMP modelhas been used to investigate the validity of the additivityprinciple for current fluctuations [8] and the Gallavotti-Cohen fluctuation theorem [5] and its generalization inrefs. [11, 14]. Another simple, but physically relevant,choice is f (Σ l,p ) = Σ l,p , so that P l,p ∼ Σ l,p / (2 E p ) for alarge system. A variant of this model has been recentlyused to study compact wave propagation in microscopicnonlinear diffusion [38]. In general, the models here in-troduced can be regarded as a toy description of densegranular gases: particles cannot freely move but may col-lide with their nearest neighbors, losing a fraction of thepair energy and exchanging the rest thereof randomly.The inelasticity parameter can be thus considered as theanalogue to the restitution coefficient in granular systems[26]: energy is conserved in the dynamics only for α = 1,while it is continuously dissipated for any 0 ≤ α < ρ ( x, t ), current j ( x, t ) and dissipation d ( x, t ). This mesoscopic descrip-tion is expected to be valid not only for this particularclass of models but for many driven dissipative mediaover a certain hydrodynamic time scale, much larger thanthe one characteristic of the microscopic dynamics. Overthe fast (microscopic) time scale, the system forgets theinitial conditions and relaxes to a local equilibrium statein which all the properties of the system become func-tionals of the hydrodynamic fields. Afterwards, over themuch slower macroscopic time scale, the system even-tually approaches the steady state following certain hy-drodynamic law. For the family of models introduced inthis paper, the time evolution of the energy density canbe shown to obey a fluctuating balance equation of thegeneral form [39] ∂ t ρ ( x, t ) = − ∂ x j ( x, t ) + d ( x, t ) , which is just the starting point for the generalization ofMacroscopic Fluctuation Theory to dissipative systemsdeveloped in previous sections. The first term in therhs of this equation accounts for the diffusive spreadingof the energy, and it is also present in the conservativecase, while the second one gives the rate of energy dissi-pation in the bulk. It is important to note here that themodels microscopic dynamics must be quasi-elastic (with(1 − α ) ∼ L − , see eq. (4.8) below) in order to ensurethat both diffusion and dissipation take place over thesame time scale in the continuum limit. Using a localequilibrium approximation, the current and dissipationfields can be expressed as functions of the local energydensity [39]. In particular, the fluctuating current can bewritten as j ( x, t ) = − D ( ρ ) ∂ x ρ + ξ , (4.4)where the first term is nothing but Fourier’s (equivalentlyFick’s) law with a diffusivity D ( ρ ). This transport coeffi-cient can be explicitely calculated for the general familyof models here presented [39], obtaining D ( ρ ) = 16 (cid:90) ∞ dr r f ( ρr ) e − r , (4.5) where f (Σ) is the function defining the microscopic col-lision rate, see eq. (4.1). On the other hand, the secondterm ξ in eq. (4.4) is a noise perturbation, white andgaussian. These gaussian fluctuations are expected toemerge for most situations in the appropriate mesoscopiclimit as a result of a central limit theorem. Microscopicinteractions can be highly complicated, but the ensuingfluctuations of the slow hydrodynamic fields result fromthe sum of an enormous amount of random events at themicroscale which give rise to Gaussian statistics at themesoscale. In the present case, a proof of the gaussiancharacter of the noise can be given, due to the simplicityof the class of models considered [39]. The current noiseamplitude is σ ( ρ ) /L (that is, the noise strength scales as L − / ), where σ ( ρ ) is often referred to as the mobility inthe literature. This coefficient can be again explicitelycomputed within the local equilibrium approximation, σ ( ρ ) = ρ (cid:90) ∞ dr r f ( ρr ) e − r . (4.6)Remarkably, a direct inspection of eqs. (4.5) and (4.6)reveals a simple relation between mobility and diffusivity σ ( ρ ) = 2 ρ D ( ρ ) , (4.7)which is nothing but a general fluctuation-dissipation re-lation for the dissipative case. In fact, it is the same oneas in the conservative case, because of the quasi-elasticityof the underlying microscopic (stochastic) dynamics, seeeq. (4.8). On the other hand, the dissipation field canbe written as d ( x, t ) = − νR ( ρ ), where ν is a macroscopicdissipation coefficient which can be related to the inelas-ticity of the underlying microscopic dynamics1 − α ≡ ν L , (4.8)and R ( ρ ) is a new transport coefficient, absent in conser-vative systems R ( ρ ) = ρ (cid:90) ∞ drr f ( ρr ) e − r . (4.9)Interestingly, R ( ρ ) can be related to the diffusivity. Bydifferentiating eq. (4.9) with respect to ρ after a changeof variables z = r √ ρ , it is found that D ( ρ ) = 16 dR ( ρ ) dρ + R ( ρ )3 ρ . (4.10)In fact, given the diffusivity, this equation can be consid-ered as a first order differential equation for R ( ρ ). Thesolution thereof, with an appropriate boundary condition(normally R ( ρ = 0) = 0), is equivalent to calculate theintegral in eq. (4.9).It should be stressed that the dissipation field d ( x, t )has no intrinsic noise, so its observed local fluctuationsare enslaved to those of the density. This stems from thesubdominant role of the noise affecting the dissipativeterm: its strength scales as L − / as a consequence ofthe quasi-elasticity of the microscopic dynamics, see eq.(4.8), so it is negligible against the current noise in themesoscopic limit [39].4 V. LARGE DEVIATIONS OF THE DISSIPATEDENERGY
In what follows we will apply the theory developed insections II-III to the general class of dissipative modelsintroduced in section IV. More concretely, we will re-strict ourselves to the following choice for the collisionrate function f ( ρ ) = 2Γ( β + 3) ρ β , (5.1)that is, f ( ρ ) ∝ ρ β , with β > − / Γ( β + 3) [44] for thesake of convenience, as it simplifies the expressions of thetransport coefficients, see below. For β = 0, f ( ρ ) = 1 andall the pairs collide with equal probability, independentlyof their energy value. Thus, the dissipative generalizationof the KMP model introduced in ref. [25] is recovered.For β = 1, f ( ρ ) = ρ/ R ( ρ ) = 2Γ( β + 3) ρ β +1 (cid:90) ∞ dr r β e − r = ρ β +1 , (5.2)that gives the rationale behind the choice of the propor-tionality constant in Eq. (5.1). The diffusivity is calcu-lated by substituting Eq. (5.2) into (4.10), D ( ρ ) = β + 36 ρ β . (5.3)Of course, the same result is obtained with eq. (4.5).Finally, the mobility σ ( ρ ) follows from the fluctuation-dissipation relation (4.7), which gives it in terms of thediffusivity, σ ( ρ ) = 2 ρ D ( ρ ) = β + 33 ρ β +2 . (5.4)Of course, for β = 0 the values of the transport coeffi-cients of the dissipative version of the KMP model arerecovered, D ( ρ ) = 1 / σ ( ρ ) = ρ , and R ( ρ ) = ρ [25].Interestingly, the algebraic dependence of transport co-efficients with the energy density ρ appears ubiquitouslyin real systems. One example is granular materials [26],where the density field ρ may be assimilated to the lo-cal granular temperature. For the hard sphere model,the average collision rate is proportional to the squareroot of the granular temperature. Thus, this granulargas case should correspond to β = 1 /
2, and in fact it isfound that D ( ρ ) ∝ ρ / while the dissipative term goesas R ( ρ ) ∝ ρ / . The latter is responsible for the algebraicdecay with time of the granular temperature (Haff’s law; ∝ t − for large times) observed in the homogeneous casewhen the system is isolated [26]. Before going into the details, it is convenient to writethe explicit form of the auxiliary variable y , related to thedensity ρ , defined in eq.(2.5a). For our family of models, y = ρ β , ρ = y β , (5.5)where we have made use of eq. (4.9). Following the nota-tion introduced in section II, we also need the “effective”diffusivity ˆ D ( y ), given by eq. (2.9), and the mobilityˆ σ = σ ( y ), both written in terms of y ,ˆ D ( y ) = D ( ρ ( y )) dρdy = 3 + β β ) , (5.6a)ˆ σ ( y ) = σ ( ρ ( y )) = 3 + β y β β . (5.6b)Hence, as anticipated in section II, while the “true” mo-bility D ( ρ ) depends on ρ , see eq. (4.5), the “effective”diffusivity is constant, ˆ D ( y ) = ˆ D . This allows us to cal-culate explicitly the average profiles for the density andthe current, using the linearity on y of eq. (2.11) above.The steady average solution is thus y av ( x ) = T β cosh (cid:16) x (cid:113) ν ˆ D (cid:17) cosh (cid:113) ν D , (5.7) j av ( x ) = − ˆ Dy (cid:48) av = − T β (cid:112) ν ˆ D sinh (cid:16) x (cid:113) ν ˆ D (cid:17) cosh (cid:113) ν D . (5.8)Moreover, the average density is readily obtained by com-bining (5.5) and (5.7), ρ av ( x ) = T cosh (cid:16) x (cid:113) ν ˆ D (cid:17) cosh (cid:113) ν D β . (5.9)The validity of these hydrodynamic predictions has beentested via extensive numerical experiments in ref. [39].Notice that one can define an unique natural lengthscaleassociated to a given ν from the hydrodynamic profilesabove, namely (cid:96) ν = (cid:113) ˆ D/ν . This is the lengthscaleover which profiles vary appreciably, and it decreaseslike ∼ ν − / as ν grows. In fact this observation sug-gests that, in the limit of strongly-dissipative systems ν (cid:29)
1, boundary energy layers develop localized aroundthe thermal baths, effectively decoupling the system intwo almost-independent parts, an observation that wehave already used in Section III C to obtain a simplescaling relation for the dissipation LDF for strongly dis-sipative systems ν (cid:29) d , integrated over space andtime, as defined in eq. (2.14). As described in section II,this probability obeys a large deviation principle P τ ( d ) ∼ exp [+ τ L G ( d )] , (5.10)5where the large deviation function G ( d ) is obtained bysolving a variational problem for the so-called “optimal”profiles { ρ ( x ; d ) , j ( x ; d ) } which sustain the consideredfluctuation, after an (additivity) conjecture on the time-independence of optimal paths. In fact, making use ofeq. (2.41), G ( d ) = − (cid:90) / − / dx Q ( y ) p y , (5.11)with { y ( x ) , p y ( x ) } being the solutions of the canonicalequations (2.39) with the appropriate boundary condi-tions. For the family of models introduced in eq. (5.1)for the collision probability, the Hamiltonian defined ineq. (2.36a) is H = 12 Q ( y ) p y − D jp y − νyp j . (5.12)The auxiliary function Q ( y ) in the equations above hasbeen defined in eq. (2.36b), for the case we are analyzing Q ( y ) = ˆ σ ( y )ˆ D = 12(1 + β ) β y β β , (5.13)that is, Q ( y ) is basically the mobility, since ˆ D is constant,independent of y . Thus, Q ( y ) is an homogeneous functionof degree γ , Q ( y ) = c y γ , γ = 2 + β β , (5.14)with c = 12(1+ β ) / (3+ β ). The parameter γ varies fromthe value γ = 2 for the case β = 0, which correspondsto the dissipative KMP model, to γ = 1 for the limit β → ∞ . The canonical equations determine the optimalprofiles, y (cid:48) = ∂ H ∂p y = Q ( y ) p y − j ˆ D , (5.15a) j (cid:48) = ∂ H ∂p j = − νy, (5.15b) p (cid:48) y = − ∂ H ∂y = − dQ ( y ) dy p y νp j , (5.15c) p (cid:48) j = − ∂ H ∂j = p y ˆ D . (5.15d)The boundary conditions for this system of differentialequations are y ( ± /
2) = T β , j ( − /
2) = − j (1 /
2) = d . (5.16)The main difference with the general system of canonicalequations (2.39) comes from ˆ D being constant and not -4 -2 0 2 4 6 8 (d-d av )/ c P t ( d ) n =10 -3 n =10 -2 n =10 -1 n =1 n =10 n =10 n =10 -5 0 5 (d-d av )/ c -8 -6 -4 -2 P t ( d ) FIG. 3. (Color online) Probability distribution for the dissi-pated energy, integrated over space and a long time τ , plottedversus the reduced variable ( d − d av ) /χ (of unit variance) formany different values of ν ∈ [10 − , ], for the case β = 0.Inset: Semilog plot of the same data. In both cases the lineis the normal distribution. Gaussian statistics is observed fortypical fluctuations, but the tails already show signs of asym-metry. a function of y . Of course, just as in the general case,the canonical equations (5.15) have a particular solutionwith vanishing canonical momenta, p y = 0, p j = 0, y (cid:48) = − j ˆ D , j (cid:48) = − νy, (5.17)that is the particularization of eq. (2.8) for our familyof models. The solution thereof are the average profiles(5.7)-(5.8). A. Typical fluctuations and gaussian behavior
As described in subsection III A, gaussian fluctuationsare expected for small deviations from the average dissi-pation behavior, P τ ( d ) ≈ exp[ − Lτ ( d − d av ) / (2 d Λ ν )],see eq. (3.5). Following the general theory, small fluc-tuations correspond to small canonical momenta, sincethe average behavior is obtained for p y = 0, p j = 0. Welinearize the canonical equations by introducing the pa-rameter (cid:15) = ( d − d av ) /d av (cid:28)
1. Recalling eq. (2.42), andmaking use of eq. (5.7), the average dissipation is d av = ν (cid:90) / − / dx y av ( x ) = T β (cid:112) ν ˆ D tanh (cid:114) ν D . (5.18)Writing the canonical variables as their averages plus alinear correction in (cid:15) , see eq. (3.2), we find at first orderin (cid:15) the following set of equations∆ y (cid:48) = Q ( y av )∆ p y − D ∆ j, (5.19a)6∆ j (cid:48) = − ν ∆ y, (5.19b)∆ p (cid:48) y = ν ∆ p j , (5.19c)∆ p (cid:48) j = 1ˆ D ∆ p y , (5.19d)which particularizes eq. (3.3) for our family of mod-els. The boundary conditions are ∆ y ( ± /
2) = 0,∆ j ( − /
2) = − ∆ j (1 /
2) = d av /
2. The solution of thissystem must be inserted into eq. (3.6), which gives thevariance of the gaussian distribution, χ ≡ d Λ ν /Lτ .The canonical momentum ∆ p y is directly obtained byintegrating eqs. (5.19c) and (5.19d),∆ p y = K sinh (cid:18) x (cid:114) ν ˆ D (cid:19) , (5.20)since ∆ p y must be an odd function of x as a consequenceof y being even. The constant K is to be determined withthe aid of the boundary conditions, but this can onlybe done after solving eqs. (5.19a) and (5.19b), havingpreviously inserted (5.20) into them. Substitution of eq.(5.20) into eq. (3.6) givesΛ ν = K (cid:32)(cid:90) / − / dx Q ( y av ) sinh (cid:18) x (cid:114) ν ˆ D (cid:19)(cid:33) − . (5.21)In order to evaluate the integral, eq. (5.14) for Q ( y )must be used, Q ( y ) ∝ y γ , with the parameter γ being afunction of β , 1 < γ ≤
2. We now analyze the simplestchoice β = 0, that is, γ = 2, that corresponds to thedissipative version of the KMP model introduced in [25].In this case, the calculation is straightforward and yieldsΛ ν = sinh(2 √ ν ) − √ ν √ ν sinh ( √ ν ) . (5.22)Interestingly, Λ ν ∼ / ν in the limit ofweakly-dissipative systems ν (cid:28)
1. This can be under-stood as a reminiscence of the scaling of G ( d ) derived insection III. In fact, eq. (3.34) tells us that, for γ = 2, G ( d ) is just a function of d/d av . In the gaussian ap-proximation, this implies the convergence of Λ ν to a con-stant value in the quasi-elastic limit as ν → + . On theother hand, Λ ν ∼ (2 √ ν ) − for ν (cid:29)
1, which is con-sistent with the suppression of dissipation fluctuationspreviously found in the strongly inelastic regime, as ex-pressed by the general scaling of the LDF given by Eq.(3.48). The same qualitative observations apply to othervalues of β , though the calculation is more convoluted.We have tested the above predictions in standardMonte Carlo simulations of the dissipative KMP modeldescribed in this section, for the particular case β = 0.Figure 3 shows the probability density function (pdf) forthe dissipated energy, integrated over the whole systemand over a long time τ for many different values of the -3 -2 -1 n -3 -2 -1 d a v , Ln ~ n ~ n -1/2 L n ~1/3d av ~ n FIG. 4. (Color online) Measured average dissipation and itsvariance as a function of ν for β = 0. The solid line corre-sponds to the theoretical prediction for d av , eq. (5.18), whilethe dashed line is the gaussian estimation of the dissipationvariance parameter, Λ ν , see eq. (5.22). The agreement isexcellent in all cases. Notice in particular the scaling with ν of both observables in the weakly- and strongly-dissipativesystem limits. macroscopic dissipation coefficient ν ∈ [10 − , ]. In or-der to minimize finite-size effects in the measurements,we performed simulations for systems with increasing sizeas ν grows, L ∝ (cid:96) − ν , in such a way that the number oflattice sites per unit typical length is constant and largeenough so we are within the hydrodynamic regime. Fur-thermore, the integration time τ = O (1) for the contin-uous, diffusive, timescale over which the hydrodynamicpredictions should hold [45]. Standard Monte Carlo sim-ulations do not allow us to sample the tails of the distri-bution, but they are useful to study the typical fluctua-tions around the average we are interested in here (e.g.,a regime of 5 standard deviations around the average).Figure 3 shows that, when plotted against the reducedvariable z ≡ ( d − d av ) /χ , the distribution P τ ( z ) followsapproximately a normal distribution for typical fluctu-ations. Moreover, all curves for different ν collapse inthis regime. However, even at this standard simulationlevel, it becomes apparent that the tails of the distribu-tion (corresponding to moderate dissipation fluctuations)deviate from gaussian behavior, see inset in Fig. 3, show-ing asymmetric tails and breaking the collapse to gaus-sian behavior observed for small fluctuations. As we willshow below, the analysis of the dissipation LDF showsthat the large fluctuations statistics is far from gaussian.In order to further check our theory, we have also com-pared the measured average dissipation and its variancewith the analytical results above, as a function of themacroscopic dissipation coefficient ν , varying in a rangewhich covers 6 orders of magnitude. Again, we see inFig. 4 that the agreement is excellent in all cases. In par-7ticular, the average dissipation grows as ν (resp. ν / )in the weakly (resp. strongly) dissipative system limit,while the variance remains constant for ν (cid:28) ν − / for ν (cid:29)
1. Remarkably, the gaussian ap-proximation for the variance turns out to be an excel-lent estimator of the empirical dissipation variance. For Lτ (cid:29)
1, large fluctuations of the dissipation are very rareand most of the probability concentrates in a region ofwidth proportional to ( Lτ ) − / around the average value,a regime described by the gaussian approximation. B. Complete fluctuation spectrum for theintegrated dissipation
We now investigate the whole spectrum of fluctua-tions (both typical and rare) of the integrated dissi-pation. Thus, we need to evaluate the LDF G ( d ) forarbitrary values of d , in general not close to its aver-age value d av , both analytically and numerically. Ex-ploring in standard simulations the tails of the dissipa-tion distribution associated to the nontrivial structure of G ( d ) is an daunting task, since LDFs involve by defini-tion exponentially-unlikely rare events, see eq. (2.12).This has been corroborated in Fig. 3, where the dissi-pation distribution has been measured directly but weare unable to gather enough statistics in the tails ofthe pdf to obtain clear-cut results in the non-gaussianregime. A recent series of works have addressed this is-sue, developing an efficient method to measure directlyLDFs in many particle systems [27–29]. The method isbased on a modification of the dynamics so that the rareevents responsible of the large deviation are no longerrare [27], and it has been developed for discrete- [27]and continuous-time Markov dynamics [28]. For a re-cent review, which also discusses Hamiltonian systems,see ref. [29]. The method yields the Legendre-Fencheltransform of the dissipation LDF, which is usually de-fined as µ ( s ) = max d [ G ( d ) + sd ] [10, 46]. In particular,if U C (cid:48) C is the transition rate from configuration C to C (cid:48) of the associated stochastic process, the modified dy-namics is defined as ˜ U C (cid:48) C ( s ) = U C (cid:48) C exp( s d C (cid:48) C ), where d C (cid:48) C is the energy dissipated in the elementary transition C → C (cid:48) . It can be then shown [11, 27–29] that the nat-ural logarithm of the largest eigenvalue of matrix ˜ U ( s )gives µ ( s ), which in turn can be Legendre-transformedback to obtain a Monte Carlo estimate of G ( d ). Themethod of refs. [27–29] thus provides a way to measure µ ( s ) by evolving a large number M of copies or clonesof the system using the modified dynamics ˜ U ( s ). Thismethod is exact in the limit M → ∞ , but in practicewe are able to simulate a large but finite population ofclones, typically M ∈ [10 , ]. This introduces addi-tional finite-size effects related to the population of cloneswhich must be considered with care, see [13] for furtherdiscussion along this line. The numerical results for theLDF in the following sections have been obtained usingthese advanced Monte Carlo techniques.
1. Weakly-dissipative systems, ν (cid:28) We now focus our attention on the analysis of LDF ofthe integrated dissipation for weakly dissipative systems,in which ν (cid:28)
1. In the general framework developed insection II, we found a scaling property for G ( d ), as givenby eq. (3.34), (cid:18) d av ν (cid:19) γ − G ( d ) = − (cid:34) − d/d av Y ( (cid:101) H ) (cid:35) (cid:101) H , (5.23)where γ = (2 + β ) / (1 + β ), Y ( (cid:101) H ) is determined by Eq.(3.29), and the constant (cid:101) H depends only on the ratio d/d av , as given by eq. (3.31).For the sake of concreteness, let us consider now thesimplest case β = 0, corresponding to the dissipativeKMP model introduced in [25]. Equation (3.26) for therescaled density profile now reads Y (cid:48) ( x ) = 8 (cid:101) H Y (cid:18) − YY (cid:19) , Y ( ± /
2) = 1 , (5.24)which can be explicitly integrated, with the solution Y ( x, (cid:101) H ) = Y sech ( x (cid:112) (cid:101) H ) , Y = cosh (cid:115) (cid:101) H , (5.25)where we have already used that Y ( x ) must be an evenfunction of x . The rescaled current profile Ψ( x ) intro-duced in (3.27) isΨ( x, (cid:101) H ) = − cosh (cid:113) (cid:101) H (cid:112) (cid:101) H tanh( x (cid:112) (cid:101) H ) . (5.26)The optimal profiles for the density and the current canbe now readily written by combining the previous twoequations with Eqs. (3.35)-(3.36), yielding ρ ( x ) = T Y ( x ) = T cosh (cid:115) (cid:101) H ( x (cid:112) (cid:101) H ) , (5.27a) j ( x ) = d av Ψ( x ) = − d av cosh (cid:113) (cid:101) H (cid:112) (cid:101) H tanh( x (cid:112) (cid:101) H )(5.27b)in terms of (cid:101) H = (cid:101) H ( d ). We have taken into account that y ≡ ρ for β = 0. Note that the curves ρ ( x ) /T = Y ( x, (cid:101) H )for different values of ν plotted as a function of x onlydepend on the relative dissipation d/d av . Now, eq. (3.31)implies that dd av = 2Ψ( − /
2) = sinh (cid:112) (cid:101) H (cid:112) (cid:101) H , ( d av ∼ νT ) , (5.28)which gives the constant (cid:101) H implicitly in terms of d/d av .Finally, particularizing eq. (5.23) for the case γ = 2 we8 d/d av -8-6-4-20 G ( d ) ν =0.01 ν =0.10 1 2 3 4 5 6d/d av -4-3-2-10 G ( d ) β =1 β =0.5 β =0 β =1 β =0.5 β =0 FIG. 5. (Color online) Scaling of the dissipation LDF in thequasi-elastic limit ( ν (cid:28)
1) for N = 50, T = 1 and varying β for two different values of ν , namely ν = 0 .
01 (filled symbols)and ν = 0 . G ( d av ) = 0, ∀ ν, β . For the case β = 0, thesimulation curves are plotted for d < d I , with d I being theinflection point at which G ( d ) changes convexity in the limit ν (cid:28) G ( d ) for different β , where it is clear thatincreasing β favors larger dissipation fluctuations. are analyzing (that is, β = 0), we obtain G ( d ) = (cid:112) (cid:101) H tanh (cid:115) (cid:101) H − (cid:101) H . (5.29)Note that eq. (5.28) for (cid:101) H ( d ) requires some careful anal-ysis. From the general discussion in Section III B, wehave that (cid:101) H > d > d av , and thus (cid:112) (cid:101) H is a realnumber, while (cid:101) H < d < d av and (cid:112) (cid:101) H is imaginary.The latter case poses no problem for G ( d ), which is al-ways real-valued. In fact, if we write (cid:112) (cid:101) H = i (cid:113) | (cid:101) H| wearrive at G ( d ) = − (cid:113) | (cid:101) H| tan (cid:113) | (cid:101) H| + | (cid:101) H| for (cid:101) H <
0. Inthe limit as d →
0, we have that (cid:101)
H → − π /
2, and thus G ( d ) → −∞ as expected on a physical basis.Equation (5.29) gives a simple scaling form for G ( d ),independent of ν , for the linear ( β = 0) dissipative KMPmodel in the low-dissipation limit ν (cid:28) G ( d ) vs the relative dissipa-tion d/d av is independent of ν in this quasi-elastic regime.This scaling is fully confirmed in Fig. 5, in which we plot G ( d ) for different, small values of ν ∈ [10 − , − ] mea-sured in simulations of the dissipative KMP model usingthe advanced Monte Carlo technique described at the be-ginning of this subsection. In particular, the agreementbetween theory and simulations is excellent in the broadfluctuation regime that we could measure (see below).The dissipation LDF is highly skewed with a fast decreasefor fluctuations d < d av and no negative branch, so fluctu- -100 -80 -60 -40 -20 0 s d av -25-20-15-10-505 m ( s ) n =0.01 n =0.1 s d av m ( s ) d/d av -1-0.50 G ( n ) ( d ) GG'G''
FIG. 6. (Color online) Scaling plot of the Legendre transformof the dissipation LDF, µ ( s ) = max d [ G ( d ) + sd ], in the quasi-elastic limit ν (cid:28) N = 50, T = 1, β = 0 and two differentvalues of ν , namely ν = 0 .
01 (circles) and ν = 0 . µ ( s ) is defined up to a threshold value s I = 0 . /d av ,beyond which the Legendre-Fenchel transform diverges. Thetop-right inset shows a zoom around the threshold s I . Thisis related to the existence of an inflection point in G ( d ) for d I = 2 . d av , i.e. a point at which G (cid:48)(cid:48) ( d I ) = 0, see middle-left inset, beyond which the dissipation LDF is non-convex,see discussion in main text. ation theorem-type relations linking the probabilities of agiven integrated dissipation d and the inverse event − d donot hold [5, 6]. This was of course expected from the lackof microreversibility, a basic tenet for the fluctuation the-orem to apply [36]. The limit (cid:101) H (cid:29) G ( d ) ≈ − [ln( d/d av )] ,that is, a very slow decay which shows that such largefluctuations are far more probable than expected withingaussian statistics ( ∼ − ( d/d av ) ). In fact, such slowdecay implies the presence of an inflection point in G ( d ):there is a value d I such that G (cid:48)(cid:48) ( d I ) = 0. The convex-ity of G ( d ) changes at d = d I , G (cid:48)(cid:48) ( d ) < d < d I while G (cid:48)(cid:48) ( d ) > d > d I . Specifically, Eqs. (5.28)and (5.29) imply that d I /d av = 2 . µ ( s ) = max d [ G ( d ) + sd ] = G [ d ∗ ( s )] + s d ∗ ( s ) , (5.30)9where d ∗ ( s ) is solution of the equation ∂G ( d ) ∂d = − s . (5.31)Note that, mathematically, µ ( s ) is the Legendre-Fencheltransform of − G ( d ), because the Legendre-Fenchel trans-form is defined for convex functions [10]. The partialderivative of G with respect to d is related to the first in-tegral of Hamilton equations Π ψ , see Eq. (3.16), whichin turn can be obtained from eqs. (3.37) and (5.25),yielding Π ψ = ν ∂G∂d = − (cid:101) H T sech (cid:115) (cid:101) H . (5.32)Equivalently, s = − ∂G∂d = (cid:101) H νT sech (cid:115) (cid:101) H . (5.33)In this way, making use of Eqs. (5.28), (5.29) and (5.33),the Legendre transform of the dissipation LDF can bewritten as µ ( s ) = 2 (cid:112) (cid:101) H tanh (cid:115) (cid:101) H − (cid:101) H , (5.34)in terms of (cid:101) H , which is obtained implicitly as a functionof s from Eq. (5.33). Note that the scaling of G ( d )with d/d av , see Eq. (5.23), implies a similar collapse for µ ( s ) when plotted as a function of s d av . Eq. (5.31)has a single solution d ∗ ( s ) for s < G (cid:48) ( d ) exhibits a minimum at d I ,increasingly smoothly afterward to reach asymptoticalyzero in the limit d → ∞ , see middle-left inset in Fig.6. Therefore, for s > d ∗ ( s ) ≤ d I ≤ d ∗ ( s ) for eq. (5.31), but only the first one maximizeseq. (5.30). This means that we cannot obtain G ( d ) byinverse Legendre-transforming µ ( s ) for dissipations abovethe inflection point d I = 2 . d av . In fact, µ ( s ) isdefined up to a critical s I , such that s I = 0 . /d av (the slope of − G ( d ) at the inflection point), beyond which µ ( s ) diverges. This can be seen by noticing the mainproperties of µ ( s ), namely ∂µ∂s = d , ∂ µ∂s = − (cid:20) ∂ G∂d (cid:21) − . (5.35)Therefore, µ has a singularity at the value of the slope s I corresponding to the inflection point d I , where ∂ µ/∂s diverges. The transition to non-convex behavior thusimplies that we can only measure the statistics of raredissipation fluctuations up to d I using the cloning algo-rithm [27–29]. Fig. 6 shows a comparison between themeasured µ ( s ) for two different values of ν (cid:28) s I . The agree-ment is excellent in all cases, and the collapse of µ ( s ) -0.4 -0.2 0 0.2 0.4 x r ( x ; d ) d/d av =0.30d/d av =0.37d/d av =0.46d/d av =0.54d/d av =0.78d/d av =1.01d/d av =1.20d/d av =1.46d/d av =1.73d/d av =1.97d/d av =2.22 FIG. 7. (Color online) Top: Optimal energy profiles for vary-ing d/d av and β = 0, measured for ν = 10 − (symbols) and ν = 10 − (dashed lines), and MFT predictions (solid lines).Agreement is very good in all cases. Bottom: MFT predictionfor the optimal density profiles for varying d/d av . when plotted against s d av is confirmed. The challengeremains to devise computational techniques capable ofexploring rare-event statistics even in regimes where theassociated LDF is non-convex.We may solve in a similar way the MFT for the in-tegrated dissipation for arbitrary values of the exponent β , though mathematical expressions are far more convo-luted that in the illustrative case β = 0 described above.Fig. 5 also shows the dissipation LDF for other exponents β >
0, as well as the results of numerical experiments inthese cases. Qualitatively, the results are equivalent tothose discussed above, with a ν -independent scaling formof the LDF in the ν (cid:28) d → d (cid:29) d av . Thistail changes convexity (based on a numerical analysis)at a large dissipation d I which increases with (a) ν forfixed β (b) β for fixed ν . For β =0, we have d I /d av (cid:39) . ν = 1, while d I /d av > ν = 10. On the otherhand, for β = 1, G (cid:48)(cid:48) ( d ) < β = 0 .
5, the positive values of G (cid:48)(cid:48) ( d ) are so small that0 FIG. 8. (Color online) Collapse of the optimal energy profilesmeasured for N = 50 and T = 1 as a function of the relativedissipation d/d av for different values of ν (cid:28)
1, namely ν =10 − (dotted lines) and ν = 10 − (dashed lines), for β = 0(top, green), β = 0 . β = 1 (bottom, blue).The larger β , the less pronounced the central overshoot is for d > d av . Solid lines correspond to MFT predictions. we have chosen not to eliminate the points behind thenumerical inflection point, d I /d av (cid:39) . ν (cid:28) d I /d av (cid:39) . ν = 1, although they roughly coincidewith the values at which the theoretical and the simula-tion curves begin to separate. Furthermore, comparisonwith numerical results is excellent in all cases. Interest-ingly, see inset in Fig. 5, increasing β results in a broaderdissipation LDF, meaning that large dissipation fluctua-tions are enhanced as β grows away from the linear case β = 0.We have also measured the typical energy profile as-sociated to a given dissipation fluctuation for the case β = 0, see top panel in Fig. 7, finding also very goodagreement with the macroscopic fluctuating theory de-veloped in this paper. Remarkably, optimal profiles forvarying ν (cid:28) d/d av (all thesimulations have been done with the same value of theenergy density at the boundaries T = 1), as predicted byeq. (5.27a). Furthermore, profiles exhibit the x ↔ − x symmetry conjectured in section II B in all cases, with asingle extremum which can be minimum or maximum de-pending on the value of the relative dissipation d/d av , aproperty which was deduced from the general formalismin section III B. Interestingly, profiles associated to dissi-pation fluctuations above the average exhibit an energyovershoot in the bulk. This observation suggests thatthe mechanism responsible for large dissipation fluctua-tions consists in a continued over-injection of energy fromthe boundary bath, which is transported to and storedin the bulk before being dissipated. The same quali-tative observations and good agreement between theoryand simulations is observed for other values of the ex-ponent β >
0, see Fig. 8. Notice in particular the nicecollapse of optimal profiles for different values of ν (cid:28) d/d av -12-10-8-6-4-20 G ( d ) ν =10 -2 ν =10 -1 ν =1 ν =10 β =1.0 β =0.5 β =0.0 FIG. 9. (Color online) Dissipation LDF for N = 50, T = 1and varying β = 0 , . , . ν ∈ [10 − , β =0 . G ( d av ) = 0 ∀ ν, β ), so that β = 1, 0 . β = 1, dashed for β = 0 .
5, and dotted for β = 0. As in Fig. 5, for a fixed ν increasing β results in larger dissipation fluctuations. but equal relative dissipation. An interesting observationis that optimal density profiles are less pronounced thelarger de nonlinearity exponent β is, see Fig. 8. Thisgives a plausible explanation of the widening of G ( d ) as β increases: for the same value of d/d av and increasing β , the associated optimal profile is closer to the hydrody-namic solution the larger β is, and hence this fluctuationcost decreases, having a larger associated probability.
2. Arbitrary dissipation coefficient ν For arbitrary values of ν (cid:38) G ( d ). For each par-ticular case, the whole variational problem, eqs. (5.11)-(5.16), must be solved, which is often analytically in-tractable. In order to further advance, we resort nowto a numerical evaluation of the optimal profiles, whichare used in turn to compute the dissipation LDF. Fig. 9shows the theoretical predictions for G ( d ) for increasing,non-perturbative values of ν , together with numerical re-sults from simulations, for different values of β . As forthe weakly-dissipative system limit previously discussed,the agreement between theory and measurements in Fig.9 is quite good. We attribute the observed differencesbetween theory and simulation to finite size effects in thelatter, which are more apparent for large ν as comparedto the weakly-dissipative system limit ν (cid:28)
1, comparewith Fig. 5, see also [39]. Such strong finite-size effectsare expected since the natural length scale associated toa given ν is (cid:96) ν = (cid:113) ˆ D/ν . As follows from Eq. (5.7)and the associated discussion, (cid:96) ν decreases as ν grows1 FIG. 10. (Color online) Top: Optimal energy profiles as afunction of the relative dissipation measured for ν = 10, N = 50 and T = 1 for the particular case β = 0. Thick (green)lines correspond to measurements while thin (pink) lines areMFT predictions. Bottom: Measured optimal energy profilesfor ν = 10, N = 50 and T = 1, and varying values of the non-linearity exponent β . For a given relative dissipation, energylocalization around thermal baths decreases as β increases. so larger system sizes are needed to observe convergenceto the macroscopic limit. In addition, finite-size effectsrelated to the number of clones M used for the samplingbecome an issue in this limit [13, 31].In any case, the sharpening of G ( d ) as ν increasesfor any β shows that large dissipation fluctuations arestrongly suppressed in this regime, as was argued for ν (cid:29) ν (cid:29) (cid:96) ν → d mea-sured for ν = 10, see Fig. 10, in contrast to the behaviorobserved for ν (cid:28)
1, see Figs. 7-8. The agreement of theobserved profiles with MFT predictions is rather good,taking into account the non-negligible finite-size effectsaffecting these measurements. Bottom panel in Fig. 10shows the measured energy profiles as a function of therelative dissipation and for different values of the nonlin-earity exponent β . From this figure, it is clear that fora given relative dissipation, energy localization aroundthermal baths decreases as β increases. This suggestsagain that, as in the ν (cid:28) d/d av , increases as β grows, giving rise to a broadening of G ( d ) with β . VI. SUMMARY AND CONCLUSIONS
In this paper we have developed a general theoreti-cal framework for calculating the probability of large de-viations for the dissipated energy in a general class ofnonlinear driven diffusive systems with dissipation. Ourstarting point is a mesoscopic fluctuating hydrodynamictheory for the energy density in terms if a few slow hy-drodynamic fields, that is, a fluctuating reaction-diffusionequation with a drift term compatible with Fourier’s lawand a sink term which can be written in terms of thelocal energy density. The validity of this hydrodynamicdescription can be demonstrated for a large family ofstochastic microscopic models [39], but it is expected todescribe the coarse-grained physics of many real systemssharing the same main ingredients, namely: (i) nolineardiffusive dynamics, (ii) bulk dissipation, and (iii) bound-ary driving. From this fluctuating hydrodynamic descrip-tion, and using a standard path integral formulation ofthe problem, we can write the probability of a path inmesoscopic phase space, that is, the space spanned bythe slow hydrodynamic fields. Interestingly, the actionassociated to this path, from which large-deviation func-tions for macroscopic observables can be derived, has thesame simple form as in non-dissipative systems. This isa consequence of the quasi-elasticity of microscopic dy-namics, required in order to have a nontrivial competitionbetween diffusion and dissipation at the mesoscale [39].We use the derived action functional to investigate thelarge deviation function of the dissipated energy. Theenergy dissipated in a non-conserving diffusive systemis, together with the energy current, the relevant macro-scopic observable characterizing nonequilibrium behav-ior. A simple and powerful additivity conjecture simpli-fies the resulting variational problem for the dissipationLDF, from which we arrive at Euler-Lagrange equationsfor the optimal density and current fields that sustainan arbitrary dissipation fluctuation. A Hamiltonian re-formulation of this variational problem greatly simplifiesthe calculations, allowing us to analyze the general the-ory in certain interesting limits. A perturbative solutionthereof shows that the probability distribution of small(that is, typical) fluctuations of the dissipated energy isalways gaussian, as expected from the central limit the-orem. Moreover, a general expression for the variance ofthe distribution in the gaussian approximation has beenderived which compares nicely with numerical results.On the other hand, strong separation from the gaussianbehavior is expected for large dissipation fluctuations,with a distribution which shows no negative branch, thusviolating the Gallavotti-Cohen fluctuation theorem as ex-pected from the irreversibility of the dynamics. Further-more, the dissipation LDF exhibits simple and generalscaling forms in the weakly- and strongly-dissipative sys-tem limits, which can be analyzed in general withoutknowing the explicit solution of the canonical equations.We apply our results to a general class of diffusivelattice models for which dissipation, nonlinear diffusion2and driving are the key ingredients. The theoretical pre-dictions, which can be explicitely worked out in certaincases, are compared to extensive numerical simulations ofthe microscopic models (which cover both typical fluctu-ations and rare events), and excellent agreement is foundin all cases. In particular, the simple scaling for the dissi-pation large-deviation function in the weakly-dissipativesystem limit is fully confirmed for different values of thenonlinearity exponent β , exhibiting non-convex behaviorfor large enough fluctuations. Interestingly, in this limit ν (cid:28) ν (cid:29) ACKNOWLEDGMENTS
We acknowledge financial support from Spanish Min-isterio de Ciencia e Innovaci´on projects FIS2011-24460and FIS2009-08451, EU-FEDER funds, and Junta de An-daluc´ıa projects P07-FQM02725 and P09-FQM4682.
Appendix: Variational problem with a Lagrangianincluding second-order derivatives
Let us analyze a variational problem in which the “ac-tion” is defined as the integral of a “Lagrangian” withsecond order derivatives, that is S [ j ] = (cid:90) x x dx L ( j, j (cid:48) , j (cid:48)(cid:48) ) . (A.1)The action S [ j ] is a functional of the profile j ( x ) in thefixed interval x ≤ x ≤ x . The variational problemarises when one looks for the “optimal” profile j ( x ) forwhich the functional S [ j ] is a extremum. For the sake ofconcreteness, let us consider a problem similar to the oneanalyzed in this paper: we are interested in calculating G defined as G = − min j ( x ) S [ j ] . (A.2)Then, we consider the variation δ S of the functional whena given profile j ( x ) is slightly changed to j ( x ) + δj ( x ), δ S = (cid:90) x x dx (cid:18) ∂ L ∂j δj + ∂ L ∂j (cid:48) δj (cid:48) + ∂ L ∂j (cid:48)(cid:48) δj (cid:48)(cid:48) (cid:19) . (A.3)Now, we take into account that δj (cid:48) = ddx δj, δj (cid:48)(cid:48) = d dx δj (A.4)in order to integrate by parts (i) once the term propor-tional to δj (cid:48) (ii) twice the term proportional to δj (cid:48)(cid:48) . Wearrive thus at δ S = (cid:26)(cid:20) ∂ L ∂j (cid:48) − ddx (cid:18) ∂ L ∂j (cid:48)(cid:48) (cid:19)(cid:21) δj + ∂ L ∂j (cid:48)(cid:48) δj (cid:48) (cid:27) x x + (cid:90) x x dx (cid:20) ∂ L ∂j − ddx (cid:18) ∂ L ∂j (cid:48) (cid:19) + d dx (cid:18) ∂ L ∂j (cid:48)(cid:48) (cid:19)(cid:21) , (A.5)3where [ f ] x x = f ( x ) − f ( x ). By analogy with the case of the usual Lagrangian with only first-order derivatives, weintroduce the generalized momenta as p j = ∂ L ∂j (cid:48) − ddx (cid:18) ∂ L ∂j (cid:48)(cid:48) (cid:19) , p j (cid:48) = ∂ L ∂j (cid:48)(cid:48) . (A.6)In this way, the boundary term has the usual form and Eq. (A.5) can be rewritten as δ S = [ p j δj + p j (cid:48) δj (cid:48) ] x x + (cid:90) x x dx (cid:20) ∂ L ∂j − ddx (cid:18) ∂ L ∂j (cid:48) (cid:19) + d dx (cid:18) ∂ L ∂j (cid:48)(cid:48) (cid:19)(cid:21) δj. (A.7)The extremum condition is δS = 0. If the values of j and j (cid:48) are prescribed at the boundaries, both δj and δj (cid:48) vanish at x , and the boundary term vanishes. Then,as δj is arbitrary for x < x < x , the “optimal” pro-file solution of the variational problem verifies the Euler-Lagrange equation d dx (cid:18) ∂ L ∂j (cid:48)(cid:48) (cid:19) − ddx (cid:18) ∂ L ∂j (cid:48) (cid:19) + ∂ L ∂j = 0 , (A.8)which is a fourth-order differential equation. Interest-ingly, Eq. (A.8) can be written as dp j /dx = ∂ L /∂j thatis formally identical to the usual Euler-Lagrange equationfor Lagrangians with only first-order derivatives. Theboundary conditions for the Euler-Lagrange equation arethe prescribed values of j and j (cid:48) at the boundaries;four conditions for the fourth-order differential equation.However, in physical problems there are sometimes lessprescribed quantities at the boundaries than necessary.In that case, as pointed out by Lanczos [33], the ex-tremum condition δ S = 0 provides the “missing” bound-ary conditions. For instance, if we only have fixed valuesof j (cid:48) at the boundaries (as in the LDF problem we havedealt with in the main text), δj (cid:48) ( x ) = δj (cid:48) ( x ) = 0 but δj ( x ) and δj ( x ) are free parameters. Equation (A.7)still implies the Euler-Lagrange equation but also that p j ( x ) = p j ( x ) = 0 . (A.9) The generalized momentum conjugate of the variablethat is not fixed at the boundary must vanish: the so-lution of the variational problem verifies then the Euler-Lagrange equation (A.8) with the prescribed values of j (cid:48) at the boundaries and the “extra” conditions providedby Eq. (A.9). In this way, we obtain the four condi-tions needed to determine completely the solution of theEuler-Lagrange equation.The function G defined in Eq. (A.2) depends on thevalues of j and j (cid:48) at the boundaries. Making use of Eq.(A.7), and taking into account that the optimal profile j ( x ) verifies the Euler-Lagrange equation, we get δG = − p j, δj − p j (cid:48) , δj (cid:48) + p j, δj + p j (cid:48) , δj (cid:48) . (A.10)We have introduced the notation p j,i = p j ( x i ), δj i = δj ( x i ), i = 1 , p j, = − ∂G∂j , p j (cid:48) , = − ∂G∂j (cid:48) , p j, = ∂G∂j , p j (cid:48) , = ∂G∂j (cid:48) . (A.11)Equation (2.33) of the main paper is the particularizationof this result for the case (i) x = − x = 1 /
2, (ii) solu-tions of the Euler-Lagrange equation with well-definedparity, in which p j, = p j, , p j (cid:48) , = − p j (cid:48) , , and (iii) theboundary conditions of Eq. (2.30). [1] L. D. Landau and E. M. Lifshitz, Statistical Physics 3rdedition, Course of Theoretical Physics Vol. 5 (PergamonPress, Oxford, 1980).[2] H. B. Callen and T. A. Welton, Phys. Rev. , 34 (1951).[3] R. Kubo, Reports on Progress in Physics , 255 (1966).[4] R.S. Ellis, Entropy, Large Deviations and Statistical Me-chanics (Springer, New York, 1985).[5] G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. ,2694 (1995).[6] J.L. Lebowitz and H. Spohn, J. Stat. Phys. , 333(1999).[7] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio andC. Landim, Phys. Rev. Lett. , 040601 (2001); Phys.Rev. Lett. , 030601 (2005); J. Stat. Mech. P07014(2007); J. Stat. Phys. , 857 (2009). [8] T. Bodineau and B. Derrida, Phys. Rev. Lett. , 180601(2004)[9] B. Derrida, J. Stat. Mech. P07023 (2007).[10] H. Touchette, Phys. Rep. ,1 (2009).[11] P.I. Hurtado and P.L. Garrido, Phys. Rev. Lett. ,250601 (2009); Phys. Rev. E , 041102 (2010).[12] C. P´erez-Espigares, J.J. del Pozo, P.L. Garrido and P.I.Hurtado, AIP Conf. Proc. , 204 (2011).[13] P.I. Hurtado and P.L. Garrido, J. Stat. Mech. P02032(2009).[14] P.I. Hurtado, C. P´erez-Espigares, J.J. del Pozo and P.L.Garrido, Proc. Natl. Acad. Sci. USA , 7704 (2011).[15] P. L. Krapivsky and B. Meerson, Phys. Rev. E , 031106(2012). [16] J. J. Brey, M. I. Garc´ıa de Soria, P. Maynar and M. J.Ruiz-Montero, Phys. Rev. Lett. , 098001 (2005).[17] J.-F. Millithaler, E. Alfinito and L. Reggiani, IEEE Con-ference Proceedings, vol. , 417 (2011), doi:10.1109/ICNF.2011.5994358[18] J. Murray, Mathematical Biology , Springer Verlag (2002).[19] S. T. Bramwell, P. C. W. Holdsworth and J.-F. Pinton,Nature , 552 (1998).[20] S Ramaswamy,
Annu. Rev. Condens. Matt. Phys. An Introduction to Nonlin-ear Chemical Dynamics , Oxford University Press, Oxford(1998); A. De Wit, Adv. Chem. Phys. , 435 (1999);P. Gaspard, J. Chem. Phys. , 8898 (2004).[22] R.S. Cantrell and C. Corner,
Spatial Ecology viaReaction-Diffusion Equations , Wiley Series (2003).[23] C. Kipnis, C. Marchioro and E. Presutti, J. Stat. Phys. , 65 (1982).[24] D. Levanony and D. Levine, Phys. Rev. E , 055102(R)(2006).[25] A. Prados, A. Lasanta, and P.I. Hurtado, Phys. Rev.Lett. , 140601 (2011).[26] T. P¨oschel and S. Luding eds., Granular Gases , LectureNotes in Physics vol.564 (Springer-Verlag, Berlin, 2001).[27] C. Giardin`a, J. Kurchan and L. Peliti, Phys. Rev. Lett. , 120603 (2006).[28] V. Lecomte and J. Tailleur, J. Stat. Mech. (2007) P03004;J. Tailleur and V. Lecomte, AIP Conf. Proc. , 212(2009).[29] C. Giardina, J. Kurchan, V. Lecomte, J. Tailleur, J. Stat.Phys. , 787 (2011).[30] T. Bodineau and B. Derrida, Phys. Rev. E , 066110(2005).[31] P.I. Hurtado and P.L. Garrido, Phys. Rev. Lett. ,180601 (2011).[32] I. M. Gelfand and S. V. Fomin, Calculus of Variations (Dover, New York, 2000).[33] C. Lanczos,
The Variational Principles of Mechanics (Dover, New York, 1986). [34] G.Eyink, J. L. Lebowitz, and H. Spohn, Commun. Math.Phys. , 253 (1990); , 119 (1991).[35] J. J. Brey, M. J. Ruiz-Montero, and F. Moreno, Phys.Rev. E , 5339 (2000).[36] A. Puglisi, P. Visco, A. Barrat, E. Trizac, and F. vanWijland, Phys. Rev. Lett. , 110202 (2005).[37] T. Bodineau and M. Lagouge, J. Stat. Phys. , 201(2010)[38] P.I. Hurtado and P.L. Krapivsky, Phys. Rev. E ,060103(R) (2012).[39] A. Prados, A. Lasanta and P. I. Hurtado, Phys. Rev. E , 031134 (2012).[40] A mathematically similar situation was found in Ref. [15]for the fluctuations of the current in conservative sys-tems, where a “standard” variational problem with a La-grangian with only first-order derivatives of the current j arose. Thus, in the canonical description, there was onlyone canonical momentum p j , and the particular solutionof the canonical equations with p j = 0 also correspondedto the average behaviour.[41] Multiple maxima (minima), if present, must be separatedby intermediate minima (maxima), but this is not possi-ble as y ( x ) takes the same value y at all these extrema.The possibility of an infinite (continuum) set of extremais ruled out because of the smoothness of Q ( y ) for thetypical systems of interest.[42] C. M. Bender and S. A. Orszag, Advanced Mathemat-ical Methods for Scientists and Engineers (New York,Springer, 1999).[43] J. P. Boon, J. F. Lutsko, and C. Lutsko,
Phys. Rev. E , 021126 (2012).[44] Γ( z ) is the gamma function, Γ( z ) = (cid:82) ∞ dt t z − e − t , withthe property Γ( z + 1) = z Γ( z )[45] In the simulations, time is measured in Monte Carlosteps. A Monte Carlo step comprises N collision eventsof the microscopic dynamics introduced in Sec. IV. Onthe other hand, the unit of our continuous timescale t comprises N Monte Carlo steps [39].[46] The parameter s appearing in the Legendre transform ofG(d) is conjugated to the dissipation via the equation s = − ∂G/∂d , and turns out to be just minus the Lagrangemultiplier introduced in Sec. II A, s = − λλ