The macroscopic dynamics of diffusive systems are described by hydrodynamic equations often provided by conservation laws and constitutive equations; that is equations expressing the current in terms of the thermodynamic variables. More precisely on the basis of a local equilibrium assumption, at the macroscopic level, the system is completely described by a local density ρ(t,x), which may have several components and the local current j(t,x). The evolution equations are of the form 5∂tρ(t)+∇⋅j(t)=0,j(t)=J(t,ρ(t)), where we omit the space variable x∈Λ⊂Rd.

For diffusive systems, the constitutive equations take the following form: 6J(ρ)=-D(ρ)∇ρ+χ(ρ)E, where the diffusion coefficient D(ρ) and the mobility χ(ρ) are d×d symmetric and positive definite matrices, and E is an external field acting on the bulk.

These equations have to be supplemented by appropriate boundary conditions on ∂Λ, the boundary of Λ. If λ(t,x), x∈∂Λ is the chemical potential of the external reservoirs, the boundary conditions read 7f′(ρ(t,x))=λ(t,x),x∈∂Λ, where f(ρ) is the equilibrium free energy density. Non-equilibrium depends on the boundary conditions and the external field. It may happen that the two equilibrate each other, in which case we have a non-homogeneous equilibrium state. Much of what we shall say applies to equations which are not in divergence form.

We assume that the microscopic evolution is given by a Markov process Xt which represents the configuration of the system at time t. The stationary non-equilibrium state (SNS) is described by a stationary, i.e., invariant with respect to time shifts, probability distribution Pst over the trajectories of Xt.

The macroscopic equations are supposed to derive from an underlying microscopic dynamics through an appropriate scaling limit where the microscopic time is divided by a factor ≈N2 and space is divided by ≈N for N tending to ∞ where N is the number of degrees of freedom of the system.

The hydrodynamic equations represent laws of large numbers with respect to the probability measure Pst conditioned on an initial state X0. The initial conditions are determined by X0. Of course many microscopic configurations give rise to the same value of ρ0(x). In general ρt(x) is an appropriate limit as the number of degrees of freedom diverges.

Classically we should start from molecules interacting with realistic forces and evolving with Newtonian dynamics. This is beyond the reach of present-day mathematical theory, and much simpler models have to be adopted in the reasonable hope that some essential features are adequately captured.

We further assume that the stationary measure Pst admits a principle of large deviations describing the fluctuations from the typical hydrodynamic behavior. This means that the probability that the macroscopic variable ρt deviates from the solutions of the hydrodynamic equations and is close to some trajectory ρ^t is exponentially small and of the following form: 8Pstρ(Xt)≈ρ^t(u),t∈[t1,t2]≈e-1ϵd[V(ρ^t1)+J(ρ^)]=e-1ϵdI(ρ^), where J(ρ^) is a positive functional which vanishes if ρ^t is a solution of Eq. (), and V(ρ^t1) is the cost to produce the initial value ρ^t1. The parameter ϵ is a scaling factor of the order of the ratio between the microscopic length scale (typical intermolecular distance) and the macroscopic one. The factor ϵ-d is of the order of the number of particles in a macroscopic volume. The role of Avogadro's number in Eq. () is played here by ϵd. We normalize V(ρ^) so that V(ρ^)=0 when ρ^=ρ¯ is the stationary solution. Therefore J(ρ^) represents the extra cost necessary in order that the system follows the trajectory ρ^t. Finally ρ(Xt)≈ρ^t means closeness in some metric. V is the generalization to the infinite dimensional case of the Freidlin–Wentzell quasi-potential and is the large deviation rate of the stationary probability measure. The exponent in Eq. () due to the factor ϵ-d is extensive in space. When we consider large deviations of a time-averaged quantity like the current , we get an extra time factor.

Let us denote by θ the time inversion operator defined by θXt=X-t=Xt*. The probability measure Pst* describing the evolution of the time-reversed process Xt* is given by the composition of Pst and θ-1 that is 9Pst*(Xt*=ϕt,t∈[t1,t2])=Pst(Xt=ϕ-t,t∈[-t2,-t1]). We assume that the time-reversed process also admits a hydrodynamic description. This hypothesis is physically very reasonable: by acting on a system from the outside, we can invert the currents flowing through the system and for example have heat passing from lower temperature to higher temperature.

At the level of large deviations, Eq. () implies 10V(ρ^t1)+J[t1,t2]*(ρ^)=V(ρ^t2)+J[-t2,-t1](θρ^), where ρ^t1, ρ^t2 represents the initial and final points of the trajectory and V(ρ^ti) represents the costs associated with the creation of the fluctuations ρ^ti starting from the stationary non-equilibrium state (SNS). The functional J* vanishes on the solutions of the hydrodynamics associated to the time-reversed process.

The physical situation we are considering is the following. The system is in the stationary state ρ¯ at t=-∞, but at t=0 we find it in the state ρ^0. We want to determine the most probable trajectory followed in the spontaneous creation of this fluctuation. According to Eq. (), this trajectory is the one that minimizes J among all trajectories connecting ρ¯ to ρ^0 in the time interval [-∞,0]. From Eq. () we have 11J[-∞,0](ρ^)=J[0,∞]*(θρ^)+V(ρ^0). The right-hand side is minimal if J[0,∞]*(θρ^)=0; that is if θρ^ is a solution of the time-reversed hydrodynamics. The existence of such a relaxation solution is due to the fact that the stationary solution ρ¯ is attractive also for the time-reversed hydrodynamics. We have therefore the following generalization of Onsager–Machlup to non-equilibrium stationary states:

In a SNS the spontaneous emergence of a macroscopic fluctuation takes place most likely following a trajectory which is the time reversal of the relaxation path according to the time-reversed hydrodynamics.

The functional J for diffusive systems has the Freidlin–Wentzell form generalized to the infinite dimensional situation. 13J=12∫dt∫dx(∂tρ+∇J(ρ))K(ρ)-1(∂tρ+∇J(ρ)), where the kernel K(ρ) is the elliptic operator defined on functions π:Λ→R vanishing at the boundary ∂Λ by 14K(ρ)π=-∇⋅(χ(ρ)∇π). This operator is the generalization of the Onsager matrix. Interpreting J(ρ,∂tρ) as a Lagrangian, there corresponds by Legendre duality the Hamiltonian 15H=12〈∇H,χ(ρ)∇H〉+12〈H,∇⋅(D(ρ)∇ρ)〉, where H is the momentum conjugate to ρ; that is H=δLδ(∂tρ). The scalar product 〈,〉 means integration over x.

The associated Hamilton equations are 16∂tρ=δHδH=12∇⋅D(ρ)∇ρ-∇⋅(χ(ρ)∇H)∂tH=-δHδρ=-12∑1≤i,j≤dχi,j′(ρ)∂xiH∂xjH+Di,j(ρ)∂xi∂xjH. These equations with appropriate boundary conditions are the variational equations to calculate the optimal trajectory creating ρ. They characterize the MFT. They are difficult to solve for generic dependence on ρ of the transport coefficients D(ρ),χ(ρ). The variational problem has been solved for constant D and quadratic χ; see for example . For the special model zero-range where V turns out to be local, see e.g., .

The quasi-potential or non-equilibrium free energy satisfies the associated Hamilton–Jacobi equation: 17〈∇δVδρ⋅χ(ρ)∇δVδρ〉-〈δVδρ∇⋅J(ρ)〉=0. As we shall see, the expression of V(ρ) for SSEP can be obtained by solving Eq. ().

For lattice gases, the following expression has been derived for the joint fluctuations of density and current: 18P(ρ,j)≍exp⁡{-1ϵd14∫dt∫dx(j-J(ρ))⋅χ(ρ)-1(j-J(ρ))}=exp⁡{-1ϵdI[t0,t1](ρ,j)}, where 19I[T0,T1](ρ,j)=14∫T0T1dt∫Λdx[j-J(ρ)]⋅χ(ρ)-1[j-J(ρ)]. Here, j is the actual value of the current, which is connected to ρ by the continuity equation ∂tρ+∇⋅j=0, while J(ρ) is the hydrodynamic current for the given value of ρ. For a simple interpretation of the exponent, think of an electric circuit. In this case, χ-1 is the resistance, and the double integral is the energy dissipated by j(t)-J(ρ(t)).

By minimizing first with respect to the current j, it is possible to show that 20V(ρ)=inf⁡I[-∞,0](ρ,j),ρ(t),j(t):∇⋅j=-∂tρρ(-∞)=ρ¯,ρ(0)=ρ, where ρ¯ is the stationary solution.

In a stationary state, it is natural to consider the fluctuations of the local time-averaged current, 21j̃(x)=1τ∫0τdtj(x,t). For fluctuations of j̃, the following large deviation principle has been derived: 22P(j̃)≍exp⁡-ϵ-dτΦ(j̃), where 23Φ(j̃)=lim⁡τ→∞inf⁡j∈Aτ,j1τI[0,τ](ρ,j̃). The set Aτ,j is the set of all currents j such that τ-1∫0τdtj(t,x)=j̃(x).

This is a more general form of a large fluctuation principle proposed by called an additivity principle. Suppose we split a one-dimensional system in two segments of different length L1 and L2. In this case, we must specify the boundary condition in the intermediate point; that is a value ρ of the density. The additivity principle takes the following form: 24PL1+L2(j,ρa,ρb)≈max⁡ρ[PL1(j,ρa,ρ)PL2(j,ρ,ρb)], where PL is the probability corresponding to the length L, and ρa and ρb are the boundary values of the density. This principle is correct for various models and equivalent to Eq. (). However in this approach there is no time dependence, and it is assumed that in the variational calculation is enough to consider minimizers that are independent of time, while in Eq. () we admit time dependence. This is a non-trivial difference because as it has been clarified a phase transition may be involved. In such a case, the method of would underestimate the probability of a current fluctuation. Such a transition has been proved to exist in the model of Kipnis–Marchioro–Presutti in equilibrium and found numerically in .

In general, the appearance of singularities in the large deviation rates denotes the presence of a non-equilibrium phase transition. Actually the variational principle of Bodineau and Derrida may provide several time-independent solutions which in fact have been found in models discussed in representing different phases. There is another type of phase transition whose appearance is signaled by the non-differentiability of the quasi-potential V(ρ). This type of transition has been found in the weakly asymmetric exclusion process (WASEP), that is, in presence of an external field, for sufficiently high values of the field . The existence of non-equilibrium phase transitions, often impossible in equilibrium, is a generic feature which has to be taken in account when analyzing a phenomenon.

We are concerned with macroscopic correlations, which are a generic feature of non-equilibrium non-linear models. Microscopic space correlations which decay as a summable power law disappear at the macroscopic level.

We introduce the pressure functional as the Legendre transform of the quasi-potential V: G(h)=sup⁡ρ{〈hρ〉-V(ρ)}. By Legendre duality, we have the change of variable formulae h=δVδρ, ρ=δGδh, so that the Hamilton–Jacobi equation (Eq. ) can then be rewritten in terms of G as 25〈∇h⋅,χ(δGδh)∇h〉-〈∇h⋅D(δGδh)∇δGδh-χ(δGδh)E〉=0, where h vanishes at the boundary of Λ. As for equilibrium systems, G is the generating functional of the correlation functions; see e.g., .

We define 26Cn(x1,…,xn)=δnGδh(x1)⋯δh(xn)|h=0. By expanding Eq. () around the stationary state one obtains, after non-trivial manipulations and combinatorics, recursive equations for the Cn(x1,…,xn); see . We discuss the pair correlation function by splitting it into the local equilibrium part and a possibly non-local term as follows: 27C(x,y)=Ceq(x)δ(x-y)+B(x,y), where 28Ceq(x)=D-1(ρ¯(x))χ(ρ¯(x)), and ρ¯ is the stationary solution. We then obtain from the general equations the following equation for B: 29L†B(x,y)=α(x)δ(x-y), where L† is the formal adjoint of the elliptic operator L=Lx+Ly given, using the usual convention that repeated indices are summed, by the following: 30Lx=Dij(ρ¯(x))∂xi∂xj+χij′(ρ¯(x))Ej(x)∂xi, and 31α(x)=∂xi[χij′(ρ¯(x))Djk-1(ρ¯(x))J¯k(x)], where J¯=J(ρ¯)=-D(ρ¯(x))∇ρ¯(x)+χ(ρ¯(x))E(x) is the macroscopic current in the stationary state. In order that Eq. () may have a non-trivial solution, α must be different from zero, which is generically the case when χ and D have a non-linear dependence on ρ. For non-trivial α, long-range correlations appear. In particular, for the SSEP where χ(ρ)=ρ(1-ρ), 32B(x,y)=-12(ρ1-ρ2)2Δ-1(x,y). Here, Δ-1 is the Green function of the Dirichlet Laplacian.

In , it was shown that long-range correlations may appear also in equilibrium in reaction–diffusion dynamics when microscopic time-reversal invariance is strongly violated.

We consider the variational problem defining V(ρ) for the one-dimensional simple exclusion process characterized by D=1 and χ=ρ(1-ρ). By performing the change of variable and inserting it in the Hamilton–Jacobi equation (Eq. ), 33δVδρ(x)=log⁡ρ(x)1-ρ(x)-ϕ(x;ρ), for some functional ϕ(x;ρ) to be determined satisfying the boundary conditions ϕ(±1)=log⁡ρ(±1)/[1-ρ(±1)]. We obtain a solution of the variational problem if we solve the following ordinary differential equation which relates the functional ϕ(x)=ϕ(x;ρ) to ρ: 34Δϕ(x)[∇ϕ(x)]2+11+eϕ(x)=ρ(x)x∈(-1,1),ϕ(±1)=log⁡ρ(±1)/[1-ρ(±1)]. This equation admits a unique monotone solution, which is the relevant one for the quasi-potential. A computation shows that 35V(ρ)=F(ρ)+∫Λdx(1-ρ)ϕ+log⁡∇ϕ∇ρ¯1+eϕ, where F(ρ) is the equilibrium free energy 36F(ρ)=∫dxf(ρ(x))=∫dx{ρlog⁡ρ+(1-ρ)log⁡(1-ρ)}. This expression was first obtained by Derrida, Lebowitz and Speer solving the microscopic model . They also proved that if one splits the system in two parts, the rates obey an additivity rule more complicated than a simple sum and similar to the additivity principle of . The above macroscopic calculation via the Hamilton–Jacobi equation was done in .

If an external field is present, the large deviation rate has been calculated by .

The time evolution depends on the initial condition. Basic work on non-stationary problems for SSEP is due to with interesting developments in recent work by Mallick–Moriya–Sasamoto . They studied the evolution of a step initial condition like in Fig. 1. By considering the time integral of the local current through the origin Qt=∫0tj(0,t′)dt′ where t is the microscopic time, we expect in a diffusive one-dimensional regime a law of large numbers for the quantity Qt/t for large t and a large deviation principle, 37PQtt=q≍e-tϕ(q).

Define the cumulant generating function 〈eλQ〉≃e√tμ(λ) where λ is a real parameter and 〈,〉 now stands for stochastic average. The functions Φ(q) and μ(λ) are Legendre transform of each other. This problem was studied by .

The variational equations are 38∂tρ=∂x(∂xρ-χ(ρ)∂xH),∂tH=-∂x2H-12χ′(ρ)∂x2H, where χ(ρ) is the mobility. Notice that the equation for H is backward in time, the boundary conditions being H(t,x)=λθ(x) and H(0,x)=λθ(x)+f′(ρ(0,x))-f′(ρ¯)), f(ρ) is the equilibrium free energy density.

In , the authors calculated the moment generating function with quenched and fluctuating initial conditions, microscopically with the Bethe ansatz and macroscopically with the MFT. The variational equations could be solved only in special cases. Recently Mallick–Moriya–Sasamoto have discovered that the following non-local transformations, which generalize the Cole–Hopf transformation, allow to map the variational equations for the SSEP to the Ablowitz–Kaup–Newell–Segur (AKNS) equations: 39u(t,x)=1χ′(ρ)∂xχ(ρ)exp⁡-∫-∞xdyχ′(ρ)2∂yH,40v(t,x)=-2χ′(ρ)∂xexp⁡∫-∞xdyχ′(ρ)2∂yH. The AKNS equations, 41∂tu(t,x)=∂xxu(t,x)-2u2(t,x)v(x,t),42∂tv(t,x)=∂xxv(t,x)-2v2(t,x)u(x,t), were solved for the SSEP, that is χ=ρ(1-ρ), with the inverse scattering method. These transformations are valid for quadratic χ. The boundary conditions for the step initial condition are 43u(0,x)=ωδ(x)v(t,x)=δ(x). From their solution, they obtained the moment generating function 44μ(λ)=1√π∑1∞(-1)n-1ωnn3/2, where 45ω=(eλ-1)ρa(1-ρb)+(e-λ-1)ρb(1-ρa). For integrability results in the Kipnis–Marchioro–Presutti model of heat transfer, see the recent work by Bettelheim–Smith–Meerson and references therein.