Entropic and gradient flow formulations for nonlinear diffusion

arXiv:1508.00549v1 [math-ph] 3 Aug 2015

Entropic and gradient flow formulations for nonlinear diffusion Nicolas Dirr∗1 , Marios Stamatakis†2 , and Johannes Zimmer‡2 1

Cardiff University, School of Mathematics, Senghennydd Road, Cardiff CF24 4AG, UK 2 University of Bath, Department of Mathematical Sciences, Bath BA2 7AY, UK August 4, 2015

Abstract Nonlinear diffusion ∂t ρ = ∆(Φ(ρ)) is considered for a class of nonlinearities Φ. It is shown that for suitable choices of Φ, an associated Lyapunov functional can be interpreted as thermodynamics entropy. This information is used to derive an associated metric, here called thermodynamic metric. The analysis is confined to nonlinear diffusion obtainable as hydrodynamic limit of a zero range process. The thermodynamic setting is linked to a large deviation principle for the underlying zero range process and the corresponding equation of fluctuating hydrodynamics. For the latter connections, the thermodynamic metric plays a central role.

1

Introduction

There is a wealth of results on nonlinear diffusion equations of the kind ∂t ρ = ∆(Φ(ρ));

(1)

as the monumental monograph by Vazquez [20] for the porous medium equation, Φ(ρ) = ρm , demonstrates. Recently, Bodineau, Leibowitz, Mouhot and Villani [3] studied drift-driven nonlinear diffusion equations with inhomogeneous Dirichlet data and established the existence of Lyapunov functionals for such equations. ∗

[email protected] [email protected] ‡ [email protected] †

1

In this short note, we ask a simple question: can we, for suitable choices of Φ, identify some Lyapunov functionals as thermodynamics entropies, and derive the associated metric, which we call thermodynamic metric? The techniques we employ are basic tools and results from statistical mechanics and optimal transport. Yet, the result seems interesting in a twofold way. Firstly, from the statistical mechanics perspective, the thermodynamic free energy is well known; see, e.g., Großkinsky [11]. Yet this free energy has not been placed in a gradient flow context, i.e., linked to the associated thermodynamic metric. Secondly, from the perspective of gradient flows in metric spaces, nonlinear diffusion has been studied in a classic Wasserstein setting, with the 2-Wasserstein metric defining the metric. Here, however, we establish that the thermodynamic metric is a different (weighted) Wasserstein metric. Fornaro, Lisini, Savar´e and Toscani [9] study nonlinear diffusion as in (1) augmented by a drift term in a Wasserstein setting, while we propose here a weighted Wasserstein metric for (1). It is well known that one equation can have different incarnations as gradient flow, that is as steepest descent of a functional in a metric space; linear diffusion is a classic example with infinitely many such formulations. Then the functional is a Lyapunov functional. Here we show that in the setting of this article, the Lyapunov function has a thermodynamic interpretation, being the thermodynamic entropy of a zero range process. We point out that the existence theory can be established without this thermodynamic interpretation. That is, the knowledge of Lyapunov functionals, which may or may not be entropies, can be key to a successful existence theory, as [9] testifies. At the same time, large deviation theory singles out one particular metric in the cases under consideration. We show in Section 4 that this gives, at least formally, immediately rise to a variational scheme of Jordan-Kinderlehrer-Otto (JKO) type [16]. We remark that different particle processes can lead to the same equation as hydrodynamic limit, but that equation will then typically have different representations as pairs of metric and entropy. For example, the symmetric exclusion process and the zero range process with linear jump rates (see Subsection 2.1) both lead to the linear diffusion equation, but with two different entropies and correspondingly two different metrics. It turns out that the same pair of a weighted Wasserstein metric and the thermodynamic entropy arises both from the large deviation theory derivation of an JKO scheme and the weighted version of the classical Benamou-Brenier formula. The weighted Wasserstein metric is (formally) conformally equivalent to the classical one, with the weight being related to the diffusion of a tagged particle. Namely, it is often convenient to write (1) as ∂t ρ = ∆(σ(ρ)ρ) = div(σ(ρ)∇ρ) + div((∇σ)ρ), (2) since σ(z) = Φ(z)/ρ(z) will be related to the diffusivity of a tagged particle in the underlying zero range process, as discussed in Subsection 2.2. Roughly speaking, the equation which is the hydrodynamic limit of the zero range process looks like the Kolmogorov equation of a single tagged particle. This diffusivity (sometimes called selfdiffusion coefficient) will turn out to be closely related to the thermodynamic metric

of the process discussed below.

2 2.1

Particle models The zero range process

In this subsection, we briefly summarise the zero range process on a flat torus; see [11, 12] for more information on zero range process and the condensation phenomenon they exhibit for particular choices of jump rates. A zero range process is an interacting particle system. Let Λ be lattice, that is, an ordered collection of boxes indexed by an (at most countable) index set. Each box can be empty or contain a finite number of particles. The particles are indistinguishable. They evolve according to an irreducible Markov kernel on Λ in the following way. A particle at a given site x ∈ Λ interacts only with the particles located at the same site (hence zero range). Specifically, the rate at which one particle leaves a site x depends only on the total number of particles at site x, described by the (local) jump rate function g : Z+ −→ R+ , which satisfies g(k) = 0 if and only if k = 0. Here g is assumed to be Lipschitz continuous. In particular, for the hydrodynamic limit in the flat torus Td we consider the diffusively rescaled zero range processes with finite lattice the discrete d-dimensional torus TdN ∼ = ( Z /N Z )d ∼ = {0, 1, . . . , N − 1}d as N → +∞. The state space of the system is the TdN set MN := Z+ of all configurations of particles η : TdN −→ Z+ ; the system is in state η if x contains η(x) particles for all x ∈ TdN . Let p denote the symmetric nearest neighbour transition probability. The symmetric (nearest neighbour) zero range process is a Markov jump process on the state space MN with formal generator X LN f (η) = N 2 {f (η x,y ) − f (η)}g(η(x))p(y − x), x,y∈TdN

where η x,y := η − 1{x} + 1{y} is the configuration resulting from moving a particle from x to y and the term N 2 comes from the diffusive rescaling of the processes. The microscopic stochastic dynamics defined by this generator can be described as follows. Starting from an initial P configuration η0 , the system waits an exponential time 2 2 τ1 with rate N λ(η0 ) := N x∈Td g(η0 (x)), at which time one particle is moved from N

0 (x)) p(y − x). Then the processes starts again with ητ1 := η0x,y x to y with probability g(η λ(η0 ) in place of η0 . The equilibrium states of zero range processes are well known. They are characterised by the equation νLN = 0 [17, Prop. A.4.1]. Assuming ν is a translation invariant and product equilibrium state with fugacity ϕ := Eν g(η(0)) = Eν g(η(x)), then νLN = 0 can be solved and

ϕk ν{η η(0) = k} = ν{η η(0) = 0} · , g!(k)

with g!(k) := g(1) · . . . · g(k) and g!(0) = 1. Since ν is a probability measure, the partition function X ϕk Z(ϕ) := g!(k) k∈Z+ 1 converges at ϕ and ν{η η(0) = 0} = Z(ϕ) . Conversely, for ϕ in the proper domain DZ := {ϕ ∈ R+ Z(ϕ) < +∞}, the translation invariant product measure ν¯ϕN in MN with one-site marginal 1 ϕk ν¯ϕN {η(0) = k} := Z(ϕ) g!(k) is an equilibrium state of the zero range process. Consequently the family of the translation invariant and product equilibrium states of the zero range process is {¯ νϕN }ϕ∈DZ . From a statistical mechanics point of view, this family is the grand canonical ensemble. Of course, in order for non-trivial equilibrium states to exist, the critical fugacity p ϕc := lim inf k g!(k) (3) k→+∞

must be strictly positive to ensure that the partition function Z has non-trivial radius of convergence. For the description of the evolution of the empirical density of the zero range process, it is useful to have a reparametrisation of the grand canonical ensemble by the density. This can be achieved by introducing the density function R : DZ −→ [0, +∞], which is given by ∞

R(ϕ) := E

N ν¯ϕ

ϕZ 0 (ϕ) ϕ X ϕk−1 k = . η(0) = Z(ϕ) k=0 g!(k) Z(ϕ)

(4)

This function is analytic (on [0, ϕc )) and strictly increasing [17, Section 2.3] with DR = DZ ∩ DZ 0 . The critical density is then ρc := sup R(DR ) ∈ (0, +∞]. In this article, we restrict our attention to zero range processes with superlinear jump rate g, i.e., for some constant a0 > 0 g(k) ≥ a0 k for all k ≥ 1, (5) which enforces ϕc = ρc = +∞ (we refer the reader to [12] for zero range processes that exhibit phase transitions). Then the inverse Φ := R−1 is well defined on all of R+ ; by N setting νρN := ν¯Φ(ρ) , the grand canonical ensemble is reparametrised by the density. We note that Φ(ρ) is stochastically represented as the mean local jump rate at a state of density ρ, since ∞ X Φ(ρ)k 1 g(k) = Φ(ρ). (6) EνρN g(η(0)) = Z(Φ(ρ)) k=1 g!(k) The thermodynamic entropy density S of the zero range process is defined as the Legendre transform of the pressure p(λ) = log(Z(exp(λ)); here it is (see [12, Eq. (3.2)]; we use the opposite sign convention) S(ρ) = ρ log Φ(ρ) − log(Z(Φ(ρ)))

(7)

and thus S 0 (ρ) = log(Φ(ρ)) + ρ

Φ0 (ρ) Z 0 (Φ(ρ)) 0 − Φ (ρ). Φ(ρ) Z(Φ(ρ))

(8)

By (4) ρ= and so

Z 0 (Φ(ρ)) Φ(ρ), Z(Φ(ρ))

Z 0 (Φ(ρ)) 0 Φ0 (ρ) Φ (ρ) = ρ , Z(Φ(ρ)) Φ(ρ)

and hence (8) simplifies to S 0 (ρ) = log(Φ(ρ)) = log(σ(ρ)ρ).

(9)

The entropy functional S corresponding to the thermodynamic entropy S is given by Z S(µ) = S(µac ) dx + S 0 (∞)µ⊥ (Td ), (10) Td

where S 0 (∞) = limρ→+∞ S(ρ)/ρ = log ϕc is the rate of linear growth at infinity of the thermodynamic entropy, and µ = µac + µ⊥ is the Radon-Nikodym decomposition of µ, µac  LTd , µ⊥ ⊥ LTd (with respect to Lebesgue measure LTd on the torus). In particular, in the case that ϕc = +∞, the thermodynamic entropy has superlinear growth S 0 (∞) = +∞, and thus S(µ) = +∞ if µ 6 LTd . The large deviations functional (see also Subsection 4) of the occupation variables {η(x)}x∈Zd with respect to the equilibrium state νρ∞∗ on the infinite lattice Zd is given by [17, Lemma 6.1.7] Iρ∗ (ρ) = ρ log

Φ(ρ) Z(Φ(ρ)) − log , Φ(ρ∗ ) Z(Φ(ρ∗ ))

and thus differs from the thermodynamic entropy by an affine function. The entropy functional Iρ∗ associated to the rate functional Iρ∗ as in (10) is (see the proof of Lemma 5.1.6 in [17]) the rate functional for the large deviations principle (with speed N1d ) satisfied by the sequence  1 X  η(x)δ νρN∗ ∈ P(M+ (Td )), N ∈ N, x/N Nd ] d x∈TN

of the empirical embeddings of the zero range process with respect to an equilibrium state of density ρ∗ . In the last display M+ (Td ) denotes the set of all non-negative Borel measures on the torus, and P the set of probability measures; the push-forward f] µ of a measure µ by a (measurable) map f is defined as the measure satisfying f] µ(A) = µ(f −1 (A)) for all (measurable) sets A. Since S and Iρ∗ differ by an affine function, the functionals S and Iρ∗ differ by a constant and consequently S and Iρ∗ , ρ∗ ∈ R+ define the same gradient flow (with respect to any metric).

It is well known (see [17, Theorem 5.1.1] for a precise formulation) that the hydrodynamic limit of the zero range process is given by (1), with Φ being the mean jump rate function. This means that starting the process from a weak local equilibrium of profile ρ0 ∈ L1 (Td ), i.e., from an initial law µN 0 , N ≥ 1, such that for all δ > 0 and d G ∈ C(T )   Z   1 X  x  G(x)ρ (x) dx > δ = 0, G η(x) − lim µN 0 0 N →+∞   N d N Td d x∈T N

and (ρt )t∈R+ is the unique weak solution of (1) starting from ρ0 , then the law µN t of the process at each time t > 0 is a weak local equilibrium of profile ρt . In other words, N asymptotically as N → +∞ the law µN t looks like the product measure νρt (·) with slowly varying profile ρt characterised by the marginals νρNt (·) {η(x) = k} =

2.2

1 Φ(ρt (x/N ))k . Z(Φ(ρt (x/N ))) g!(k)

Asymptotic behaviour of tagged particles

Given a symmetric zero-range process, we can “tag” a particle that is originally at the origin, say, and follow its movements, denoting its position at time t by X(t). It is natural to ask whether there is a central limit theorem for this tagged particle, that is, whether the law of N −1 X(N 2 t) converges to a diffusion dX = σ(X)dW , where W is a standard Brownian motion, and if so, what the diffusivity σ is. This central limit theorem has been shown to hold in many situations, both in equilibrium (i.e., with the initial distribution of the particles given by an invariant measure for the process) and in non-equilibrium, see, e.g., [17] and the references therein. It turns out [15] that for ρ > 0, Φ(ρ) , σ(ρ) = ρ where Φ and ρ(t, x) as in (1). In the next section, this intrinsic diffusivity will be related to what we will call below the thermodynamic metric of the process.

3

Thermodynamic formulation of nonlinear diffusion

We now phrase (2) with nonlinear or inhomogeneous σ in a Wasserstein setting. We extend the classic Benamou-Brenier formulation [2] of optimal transport by a weighted version. We will assume that the diffusivity σ(x) is positive definite, and it may be of the form σ(x) = σ(ρ(x, t)), although we will not write it explicitly in this form in order to simplify notation.

We introduce a metric tensor, following the work of Otto [19] in the presentation of Villani [21, Section 8]. What follows can thus be seen as the Benamou-Brenier formulation of optimal transport with a nonconstant metric. For vector fields v and w, define the metric Z g(v, w) := ρσ −1 hv, wi dx (11) Ω −1

(one can think of σ hv, vi as a Riemannian metric on a manifold which is topologically Rn .) This penalises transport in regions with slow diffusion, where σ is small. (In the classic setting, the division by σ is absent). As in the standard case, the connection between a tangent vector ∂t ρ (i.e., a derivative of a path in the space of densities/measures) and a vector field is made by the transport equation ∂t ρ + div(ρv) = 0.

(12)

In order to ensure uniqueness, we minimise now the metric functional associated with (11), under the constraint (12),  Z 2 −1 inf ρσ |v| dx ∂t ρ + div(ρv) = 0 . (13) v

Ω

In the classic case (no division by σ), this term has an interpretation as twice the kinetic energy, and we thus call (13) (twice) the weighted kinetic energy. Note that the correct variation of a vector field (i.e. a section of the tangent bundle) is additive. In order to keep the constraint (12), we need to keep div(ρv) constant under the variation, which means taking variations of the form v + ρ−1 w with divergence-free w. The extremality condition for (11) leads to the condition Z σ −1 hv, wi dx = 0 Ω

for all divergence-free w, so by the Helmholtz-Hodge decomposition of vector fields into divergence-free and gradient fields σ −1 v = ∇ξ

(14)

for some real function ξ. Note that thus ∂t ρ = div(ρv) = div(σρ∇ξ). As (for given suitable f , and with appropriate boundary conditions) the equation div(ρσ∇ξ) = f has a unique solution, there is (up to additive constants) only one such ξ. Let Z S(ρ) := S(ρ) dx (15) Ω 0

be the entropy. Multiplying (12) by S , using (14) and integrating by parts, we obtain Z Z 0 ∂t ρ · S dx = − ρσ∇ξ · ∇[S 0 (ρ)] dx = −g(v, z), with z = σ∇[S 0 (ρ)]. Ω

Ω

In words, this formula says that the product of tangent vectors representing ∂t ρ and dS = S 0 is expressed by the (negative) metric tensor evaluated at the corresponding vector fields (dS is a covector). In particular, dS corresponds to the tangent vector represented by the vector field σ∇[S 0 (ρ)]. Thus, the continuity equation (12) becomes ∂t ρ = div(ρσ∇[S 0 (ρ)]).

(16)

If we now choose the thermodynamic entropy (7), with its derivative given by (8), then we recover (2). In summary if we choose the thermodynamic entropy, then precisely the choice (11) for the metric recovers the nonlinear diffusion equation (1); we thus call (11) the thermodynamic metric. Weighted kinetic energies as (13) have been introduced and studied before, in particular by Dolbeault, Nazaret and Savar´e [6]. We sketch how their results can be used to obtain existence for solutions to (1). We consider the minimisation of twice the kinetic energy (13), Z Z 1

ρσ −1 |v|2 dx dt

inf 0

Ω

under the usual continuity equation ∂t ρ + div(ρv) = 0. Equivalently, one can consider Z 1Z Φ(ρ) |v|2 dx dt inf 0

Ω

with the constraint ∂t ρ + div(Φ(ρ)v) = 0; the latter variational formulation is [6, Eq. (1.6)] in our notation. Then, as proved in [6], concavity of Φ is a sufficient condition to establish central properties of the theory of gradient flows in metric flows [1]. Let us note here that not all mean jump rate functions of zero range processes are concave. Two simple examples where it is concave are Φ(ρ) = ρ and Φ(ρ) = ρ/(ρ + 1), which correspond to g(k) = k and g(k) ≡ 1, respectively. Although the latter jump rate does not satisfy the superlinearity assumption (5), one can consider the jump rate g (k) = g(k) + k for some  > 0, which according to (6) gives the mean jump rate Φ (ρ) = Φ(ρ)+ρ. More generally, as one can see by plots the mean jump rate function corresponding to the Evans model  b 1{k≥1} gb (k) = 1 + k is concave for all parameters b ≥ 0. The density function Rb = Φ−1 b in the Evans model is given by [12, Eq. (4.5)] Rb (ϕ) = where 2 F1 (a, b, c; z) :=

P∞

k=0

ϕ2 F1 (2, 2, 2 + b, ϕ) , (1 + b)2 F1 (1, 1, 1 + b, ϕ)

(a)k (b)k z k (c)k k!

0 ≤ ϕ,

is the hypergeometric function and

( a(a + 1) · . . . · (a + k − 1) (a)k := 1

if k ≥ 1 , if k = 0

Figure 1: Graph of the density function Rb = Φ−1 b in the Evans model for b = 1, b = 2.5 and b = 3.5

Figure 2: Graph of the density function Rb = Φ−1 in the Landim model for b = 0.5, b b = 3 and b = 5 is the rising Pochhammer symbol. Fig. 1 gives plots for Rb for some values of b. An example of a zero-range process with non-convex density function is given by the Landim jump rate function given by ( k if k = 0, 1 . gb (k) = k b if k ≥ 2 ( k−1 ) P ϕk In this case, gb !(k) = k b and the partition function is Zb (ϕ) = 1+ ∞ k=1 kb =: 1+Lib (ϕ), where Lib is the polylogarithmic function. This results in the density function Rb (ϕ) =

Lib−1 (ϕ) . 1 + Lib (ϕ)

Fig. 2 indicates that Rb is convex for small b > 0, concave for large b > 0 and neither of the two for intermediate values. We close this section with the rather obvious remark that equations of porous medium type, Φ(ρ) = ρm , are not covered by the setting described here as for the zero-range process, Φ0 (0) is finite.

4 4.1

Connection to large deviation theory and fluctuations Large deviation theory

We first explain this link using a toy model of a stochastically perturbed one-dimensional gradient flow, √ dX  = −V 0 (X  )dt + dW, where dW denotes a Wiener process. Heuristically, one expects that for small noise,  small, trajectories will tend to be close to the deterministic trajectory solving x˙ = −V 0 (x). Large deviation theory describes the probability of observing a realisation, here a trajectory, by means of a rate functional I; the probability of finding a given trajectory is essentially exp(−1/ I) as sketched now. For the toy model, the large deviations rate functional depends on the final time T and is denoted IT ; it is Z T 1 (x˙ + V 0 (x))2 ds (17) IT (x, ˙ x) = 2 0 Z Z t 1 T 0 1 2 |x| ˙ ds + (V (x))2 ds + V (x(T )) − V (x(0)). (18) = 2 2 0 0 In essence, this means that   1 P({x (0) ∼ x, x (T ) ∼ y}) ∼ exp − inf IT (x, ˙ x) ,  



where the infimum is taken over paths x = x(s) with x(0) = x and x(T ) = y. In order to find the most likely position at t = T, we have to minimise first over all paths such that x(0) = x and x(T ) = y and then over all terminal values x(T ). Note that the first integral in (18) is bounded from below by the distance d(x(0), x(T )) between the initial and the final point. Choosing T = h  1 and re-scaling, we see that we have to minimise Z h 1 0 1 2 d (x(0), x(h)) + V (x(h)) + (V (x(sh)))2 ds. 2h 2 0 If d(x(0), x(h))  h, then the last term is clearly of lower order, while if d(x(0), x(h)) = O(h) and V is smooth enough, then the last integral equals Z h 1 0 h (V (x(sh))2 ds = V 0 (x(0)) + O(h2 ), 2 0 2 so again the term relevant for the minimisation is of lower order. Formally this means that we can find the most likely position x(h) by minimizing the functional [16] 1 2 d (x(0), x(h)) + V (x(h)). 2h

(19)

The analogous computation starting with the large deviation rate functional for the empirical measure of n Brownian particles (with n playing the role of the large parameter 1/ in the toy model; here the rate functional is defined on function spaces) [4], then one obtains an equation with the same structure as (19), with d being the Wasserstein norm, and the entropy S in place of V (see [7] for the proof). Thus, one obtains the celebrated Jordan-Kinderlehrer-Otto functional [16] as expansion of the associated large deviation rate functional. We now apply this reasoning in the context of nonlinear diffusion and more specifically the zero range process. The large deviation rate function for the zero range process is of the form Z T k∂t ρ(s) + ∆Φ(ρ(s))k2H −1 ds (20) Φ(ρ(s))

0

(see [18] and references therein); here the weighted negative Sobolev norm is defined as follows. Let for some positive bounded function w Z hu, viHw1 := w∇u · ∇v dx, Ω

and denote by

∆−1 w u

the solution of div(w∇(∆−1 w u)) = u.

Then hu, viHw−1 =

−1 h∆−1 1 w u, ∆w viHw

(21)

Z =−

u∆−1 w v dx,

Ω

where the last identity used integration by parts and (21). As seen from the reasoning above, by expanding the square in (20) we get a metric term Z 2 dx = g(v, v), Φ ∇(∆−1 ∂ ρ) t Φ Ω

where g is as in (11), v as in (12). This follows as we get from (14) v = σ∇ξ = (Φ/ρ)∇ξ and ξ = ∆−1 Φ (∂t ρ). The null Lagrangian (the mixed term) equals Z TZ ∂t ρ∆−1 Φ (∆Φ(ρ)) dx dt, 0

Ω

so it remains to compute ∆−1 Φ (∆Φ(ρ)). As log Φ(ρ) solves div(Φ(ρ)∇ log Φ(ρ)) = ∆Φ(ρ), 0 we have ∆−1 Φ (∆Φ(ρ)) = log(Φ(ρ)), so if S (ρ) = log(Φ(ρ)) as in (8), we obtain Z TZ Z Z −1 ∂t ρ∆Φ (∆Φ(ρ)) dx dt = S(ρ(T )) dx − S(ρ(0)) dx, 0

Ω

Ω

Ω

and recover the thermodynamic entropy S of (15). The remaining quadratic term is for small time intervals formally of lower order. Thus in summary, an expansion of the large deviation rate functional (20) for nonlinear diffusion associated with the zero range process yields a variational formulation Jordan-Otto-Kinderlehrer type, with the metric being the weighted Wasserstein metric considered in this article, and the functional being the thermodynamic entropy.

4.2

Fluctuating hydrodynamics

The previous Subsection 4.1 showed that the thermodynamic metric is related not only to the limit equation, but also to the probability of deviations from it. At least at a formal level, this connection is even stronger, as we now explain. Formally, the metric is given by a symmetric linear operator, which has an inverse that can be interpreted as covariance matrix. If we add a suitable space-time noise to the limit partial differential equation, then it formally has not only the same large deviations rate functionals, but finite moments of the fluctuations (that is, the difference between stochastic process and deterministic limit) converge to the same process as for the original particle model. This is now explained in more detail. Consider the equations p ˙) (22) dρ = ∆(Φ(ρ ))dt + div( σ(ρ )ρ W ∂t ρ = ∆(Φ(ρ)), (23) ˙ is space-time white noise. We point out that the stochastic PDE (SPDE) (22) where W is not expected to have strong (i.e., pathwise) solutions. To our knowledge, the suitable notion of generalised solutions is an open problem. Consider, however, the fluctuations, i.e., the differences between solutions of (22) and the deterministic PDE (23), re-scaled with the scaling of the central limit theorem. As Z  := ρ − ρ is small, we can linearise around ρ and obtain formally that a limit (in law) Z 0 of −1/2 Z  should solve the linear SPDE p ˙ ), (24) dZ 0 = ∆(Φ0 (ρ))Zdt + div( σ(ρ )ρ W i.e., it is an Ornstein-Uhlenbeck process. Here Φ0 is the self-diffusion coefficient of the zero range process, while σ is the bulk-diffusion coefficient. This convergence statement for the fluctuations can be stated in the following weaker form. Consider for smooth g : Rd → R the process Y  := −1/2 hg, ρ − ρ i, then by (24), an application of Ito’s formula and the (formal) limit  → 0 we get for

any smooth F : R → R that 0

0

Z

F (Y (t)) − F (Y (0)) − 0

t

F 0 (Y 0 (s))hY 0 (s), Φ0 (ρ(s))∆gids Z 1 t 00 0 F (Y (s))hσ(ρ(s))ρ(s)∇g, ∇gids (25) − 2 0

is a martingale. Note that in the last term, the expression hσ(ρ(s))ρ(s)∇g, ∇gi can be written as kgk2H 1 (ρ) , which implies that the large deviations should be related to the metric with Φ

the inverse operator, i.e., kgk2H −1 (ρ) , as found in the previous subsection. Φ We now relate the martingale (25) to the zero range process. We consider the empirical measure of the zero range process, i.e., for smooth g 1 X x N η(x) hρ , gi := d g N N d x∈TN

and the fluctuations Y N := N −d/2 hg, ρ − ρN i. It can be shown [8, 10, 14] for a large class of zero range processes that for any smooth test function, F (Y N ) converge to a solution of the martingale problem (25). In other words, the limit of the fluctuations of the zero range process is in law equal to those of the SPDE (22). This can be seen as justification of the complicated noise term in (22); precisely this noise term is related to fluctuations of the zero range process via the martingale expression (25). We remark that (22) is of fluctuating hydrodynamics type. In the linear case Φ(ρ) = ρ, the multiplicative conservative noise in (22) has been derived by Dean [5], and using quite different lines of arguments by Sturm and von Renesse [22]; the link between the noise and the geometry has been explained by a formal large deviation argument [13]. Acknowledgements We thank Mark A. Peletier for helpful and stimulating discussions. All authors gratefully acknowledge funding from the Leverhulme Trust, RPG2013-261, and JZ was partially funded by the EPSRC (EP/K027743/1). The authors thank Marcus Kaiser for valuable comments.

References [1] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´e. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨auser Verlag, Basel, 2005.

[2] Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math., 84(3):375–393, 2000. [3] Thierry Bodineau, Joel Lebowitz, Cl´ement Mouhot, and C´edric Villani. Lyapunov functionals for boundary-driven nonlinear drift-diffusion equations. Nonlinearity, 27(9):2111–2132, 2014. [4] Donald A. Dawson and J¨ urgen G¨artner. Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics, 20(4):247–308, 1987. [5] David S. Dean. Langevin equation for the density of a system of interacting Langevin processes. Journal of Physics A: Mathematical and General, 29(24):L613, 1996. [6] Jean Dolbeault, Bruno Nazaret, and Giuseppe Savar´e. A new class of transport distances between measures. Calc. Var. Partial Differential Equations, 34(2):193– 231, 2009. [7] Manh Hong Duong, Vaios Laschos, and Michiel Renger. Wasserstein gradient flows from large deviations of many-particle limits. ESAIM Control Optim. Calc. Var., 19(4):1166–1188, 2013. [8] P. A. Ferrari, E. Presutti, and M. E. Vares. Nonequilibrium fluctuations for a zero range process. Ann. Inst. H. Poincar´e Probab. Statist., 24(2):237–268, 1988. [9] Simona Fornaro, Stefano Lisini, Giuseppe Savar´e, and Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete Contin. Dyn. Syst., 32(5):1675–1707, 2012. [10] G. Gielis, A. Koukkous, and C. Landim. Equilibrium fluctuations for zero range processes in random environment. Stochastic Process. Appl., 77(2):187–205, 1998. [11] Stefan Großkinsky. Phase transitions in nonequilibrium stochastic particle systems with local conservation laws. PhD thesis, Technische Universit¨at M¨ unchen, 2004. [12] Stefan Großkinsky, Gunter M. Sch¨ utz, and Herbert Spohn. Condensation in the zero range process: stationary and dynamical properties. J. Statist. Phys., 113(34):389–410, 2003. [13] Rob Jack and Johannes Zimmer. Geometrical interpretation of fluctuating hydrodynamics in diffusive systems. J. Phys. A: Math. Theor, 47:485001, 2014. [14] Hanna Jankowski. density fluctuations for the zero range process with colour. http://arxiv.org/math/0607505v1, 2006.

[15] Milton Jara, Claudio Landim, and Sunder Sethuraman. Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes. Ann. Inst. Henri Poincar´e Probab. Stat., 49(3):611–637, 2013. [16] Richard Jordan, David Kinderlehrer, and Felix Otto. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal., 29(1):1–17 (electronic), 1998. [17] Claude Kipnis and Claudio Landim. Scaling limits of interacting particle systems, volume 320 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. [18] A. Koukkous and H Guiol. Large deviations for a zero mean asymmetric zero range process in random media. Preprint arXiv:math/0009110v1, 2003. [19] Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations, 26(1-2):101–174, 2001. [20] Juan Luis V´azquez. The porous medium equation. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory. [21] C´edric Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. [22] Max-K. von Renesse and Karl-Theodor Sturm. Entropic measure and Wasserstein diffusion. Ann. Probab., 37(3):1114–1191, 2009.