The hydrodynamic limit

The following theorem is our main result, detailing a hydrodynamic limit with an explicit rate of convergence in one dimension.

Theorem 2.4.6 (Hydrodynamic limit for zero range processes). Assume d = 1 and let F ∈C_b²(R), ϕ∈C³(T^d), k > (d+ 2)/2, C_H >0, and ρ >0 be given. Then there exists a rate of convergence r_HL(, N) with polynomial dependence on and N, whose exponents only depend onk (and d), such that the following hydrodynamic limit holds. For allt ≥0, N ∈ N, 1/N < <1, f₀ ∈ H, and µ^N₀ ∈ P(X_N) such that the entropy and the measure itself are bounded relative to the grand-canonical measure ν_ρ^N, i.e.

H^N µ^N₀ |ν_ρ^N

≤C_HN^d and µ^N₀ ≤ν_ρ^N,

it holds that there exists a rate of convergence rHL(, N) such that (2.31)

µ^N_t , F hα^N_η , ϕi_L²

−F hf_t, ϕi_L²

≤r_HL(, N)

+kF⁰kL^∞kϕkL^∞

µ^N₀ ,kα^N,_η −f0k_L¹ where f_t is given by the solution to the filtration equation (2.3) and µ^N_t is given by the zero range process, as detailed above. For all T >0, % >0, and 0≤2l+ 1≤N, the rate of convergence r_HL(, N) is bounded by

CkFk_C²(1 +kϕk_C³ +kϕk²_H1)

T ^{−(d+4)((1+d2)2k2+k2k+2+kd)−d(1−θ2)}N^1−θ(d+1)

+T^12−4+d2 N⁻²++e^−cTN^2+d−2d+T¹²r_RL(%, l, , N) for some finite, positive constants c and C depending only on d, k, C_H, and ρ. Here θ =θ(k) = (2k−d−2)/(2k+ 2)and the additional rate functionr_RL(%, l, , N) is bounded by

N⁻¹²l¹² +¹²l¹⁴

%l^d4 +%l^−d4 +%⁻¹⁴ + l N. It stems from the replacement lemma, Lemma 2.4.19.

Remark 2.4.7. Thus we see that the contributions to the hydrodynamic limit can be divided into three parts. The first term describes the propagation of the initial error due to the approximation of f₀ by the discrete particle configurations at time t = 0 and the other two are errors coming from stability properties of the limit equation and an error term due to the replacement lemma, respectively. The replacement lemma is an older result, appearing first in [36].

Note that the choice of the density ρ in the assumption regarding the Gibbs measure ν_ρ^N is a slightly arbitrary. Indeed, if the relative entropy

1

N^dH^N(µ^N₀ |ν_ρ^N)≤C is bounded, then so is

1

N^dH^N(µ^N₀ |ν_ρ^N_˜ )≤C( ˜ρ)

for any ˜ρ >0. Furthermore, if kf₀k_L^∞ ≤ρ, the local Gibbs measure (2.11) satisfies ν_f^N_0(·) ≤ν_ρ^N

and this choice for µ^N₀ satisfies the bound on the entropy as well. The rate here is much slower than the rate for independent random walks, given in Theorem 2.3.1. This is mainly due to the fact that we shall need to replace the rate function g with its local

average via a replacement lemma.

Remark 2.4.8. We can replace the term estimating the compatibility of the initial data by a slightly modified version. Under the assumptions of Theorem 2.4.6, its proof also yields

µ^N_t , F hα_η^N, ϕi_L²

−F hf_t, ϕi_L²

≤rHL(, N) +kF⁰kL^∞kϕk_H¹

µ^N₀ ,kα^N,_η −f0k_H⁻¹ +kF⁰k_L^∞|R

T^dϕ(u)du|

µ^N₀ ,|N^−dP

xη(x)−R

T^df₀(u)du|

.

This is a consequence of the following variant of the basic H⁻¹–stability estimate in Lemma 2.4.12:

kf˜_t−f˜∞−f_t+f∞k_H⁻¹ ≤ kf˜₀−f˜∞−f₀+f∞k_H⁻¹ +C|f˜∞−f∞|, where f∞ = R

T^df_t(u)du is the (constant) integral of the solution f_t and likewise for ˜f∞. The above estimate of the hydrodynamic limit can be useful if we only have information about the H⁻¹–convergence of the initial data and some information on the number of particles of the zero range process. For example, we might require that there exists some N₀ ∈Nsuch that

1 N^d

X

x∈T^dN

η(x) = Z

T^d

f₀(u)du

holds µ^N₀ –almost surely for all N ≥ N₀. Of course, this is only possible if the integral R

T^df₀(u)du is an integer. The convergence of α^N,_η in H⁻¹(T^d) is more favourable in terms of powers of than the convergence in L¹(T^d). On the other hand, the initial data converges much faster in general than the bounds we can give forr_HL(, N), so it will not matter much in which norm we measure the convergence of the initial data, see the proof of Corollar 2.4.9 below.

As before Theorem 2.4.6 yields convergence to the hydrodynamic limit, conditional on convergence of the initial data.

Corollary 2.4.9. Assume d= 1 and let F ∈C_b²(R), and ϕ∈C³(T^d) be given. Then for all N ∈N and f₀ ∈C¹(T^d), there exists a µ^N₀ ∈P(X_N) such that it holds

µ^N_t , F hα^N_η , ϕi_L²

−F hf_t, ϕi_L²

≤CN^−κ

for some κ > 0, e.g. any κ <1/6700 works. Here µ^N_t is given by the zero range process and f_t by the solution to the corresponding filtration equation ∂_tf_t = ∆σ(f_t).

In other words, if η is distributed according to law µ^N_t , then α^N_η *^∗ f_t in distribution as N → ∞.

Remarks 2.4.10. (1) The statements (3)-(6) of Remark 2.3.3 remain valid and relevant to this case.

(2) The condition H^N(µ^N₀ |ν_ρ^N) ≤ CN^d is quite useful in connection with the so-called entropy inequality. This inequality states that for all γ > 0, f ∈ C_b(X_N) and any two probability measures µ, ν ∈P(X_N), it holds that

(2.32) hµ, fi ≤ 1

γ

H^N(µ|ν) + log

ν,exp(γf) . For example, if we setf(η) =P

xη(x),µ=µ^N₀ , and ν=ν_ρ^N in estimate (2.32), we obtain that

hµ^N₀ , N^−dP

xη(x)i ≤ 1 γ

N^−dH^N(µ^N₀ |ν_ρ^N) + log

ν_ρ^N,exp(γη(0)) ,

since ν_ρ^N is a product measure. Thus we obtain a bound on the average particle density hµ^N₀ , N^−d X

x∈T^d_N

η(x)i ≤C.

Note here that the infinite radius of convergence ρ^∗ = +∞ of the partition function Z(·) yields finite exponential moments

hν_ρ^N,exp(γη(0))i= Z(ρe^γ)

Z(ρ) <+∞

for all γ ∈ R. Of course in our case, the bound on the average number of particles also follows from f₀^N ≤ν_ρ^N and the monotonicity ofN^−dP

xη(x) in η.

(3) While the size of κ is certainly not optimal, it is qualitatively correct since has the expected polynomial dependence onN. As far as we are aware, it also constitutes the first example of an explicit rate of convergence of the zero range process to the hydrodynamic limit and the first example of a uniform-in-time rate of convergence. It has the added advantage that the exponents do not depend on the function g. Thus, for example, it should allow for perturbative arguments to be applied to prove the hydrodynamic limit of small perturbations of the zero range process as given in the assumptions.

Proof of Corollary 2.4.9. First of all we can choose µ^N₀ := ν_f^N

0(·) which was defined in (2.11). Then in a rough first estimate it holds that

hµ^N₀ ,kα^N,_η −f₀k_L¹i ≤ hµ^N₀ ,kα^N,_η −f₀k_L²i¹² ≤ C

√N.

We can deduce this as follows. First, it holds that µ^N₀ ,kα^N,_η −f₀k²_L2

= Z

T^d

D

µ^N₀ , 1 N^2d

X

x,y

η(x)η(y)δ^()x

N(u)δ^()y

N

(u)

− 2 N^d

X

x

η(x)δ⁽⁾x

Nf₀(u) +f₀(u)²E du.

Sinceη(x) and η(y) are independently distributed under µ^N₀ as long asx6=y, this equals Z

T^d

1 N^2d

X

x6=y

f0(_N^x)f0(_N^y)δ^()x

N (u)δ^()y

N

(u)

+ 1

N^2d X

x∈T^dN

hν_f^N₀₍^x

N), η(0)²iδ⁽⁾x

N(u)²− 2 N^d

X

x∈T^dN

f₀(_N^x)δ⁽⁾x

Nf₀(u) +f₀(u)²

du.

Sincef0 ∈C¹(T^d) is bounded, it follows that the second momenthν_f^N

0(_N^x), η(0)²iis bounded uniformly in x and N. Furthermore it holds that kδ_x/N⁽⁾ k²_L2 ≤ C^−d. Thus the above is bounded from above by

1 N^d

X

x∈T^d_N

f₀(_N^x)δ⁽⁾x N

−f₀

2

L²

+O 1 (N)^d

,

Note thatR

T^dχ(u)du= 1 and hence (2.33)

1 (N)^d

X

x∈T^dN

χ x N

−1

≤ k∇χk∞

C N.

Since f₀ ∈ C¹(T^d) is uniformly Lipschitz and the support of χ in (2.30) is compact, this shows that

1 N^d

X

x∈T^d_N

f₀(_N^x)δ⁽⁾x N

(u)−f₀(u)

≤ C N. Thus we have shown that

µ^N₀ ,kα^N,_η −f₀k²_L2

≤ C N.

In order to estimate a rate of convergence for the hydrodynamic limit, we make the Ansatz that=(N), l =l(N), %=%(N), and T =T(N) are all monomials inN. If we optimize over the set of possible powers, we find that indeed

µ^N_t , F hα_η^N, ϕi_L²

−F hf_t, ϕi_L²

=O(N^−κ)

for someκ >0. Since d= 1, we obtain that, for example, we can achieve any rateκ with κ <1/6700. Note that this bound is much larger than one would expect in view of the

law of large numbers and not optimal. On the other hand, it is independent of the rate function g, as long as g satisfies Assumption 1.

As in Section 2.3 we have a collection of semigroups S_t^N and T_t^N describing the dynamics of the particle process as well asS_t^∞andT_t^∞describing the evolution of the limit equation.

The semigroup S_t^∞ is defined by

S_t^∞f₀ :=f_t,

for all f₀ ∈H, where f_t ∈ H is the solution (2.3) corresponding to the initial datum f₀. Uniqueness of the solutions shows that it is indeed a semigroup. ThenT_t^∞ is again given by

T_t^∞Ψ(f) = Ψ(S_t^∞f)

for all Ψ∈C_b(H) andf ∈H. The relationships of the semigroups can be summarized as S_t^N :P(X_N)→P(X_N) with dual T_t^N :C_b(X_N)→C_b(X_N),

S_t^∞:H →H with pullback T_t^∞:C_b(H)→C_b(H).

The semigroupT_t^N has generatorG^N given in equation (2.2), whereas the time-derivative G^∞ of T_t^∞ will be given in Lemma 2.4.18 below. Using the empirical measures, we also define an embedding π^N, :C_b(H)→C_b(X_N) via

π^N,Ψ

(η) = Ψ(α^N,_η ),

where we shall usually drop the superscript for ease of notation. This embedding will allow us to compare the semigroups T_t^N and T_t^∞ directly.

Remark 2.4.11. The heat equation admits measure-valued solutions and hence we could give sense to ∆α^N_η . Even though there exist solutions to the nonlinear heat equation

∂_tf = ∆σ(f) started from a measure, this approach is not feasible here, since we cannot give sense to G^∞Ψ(f) = DΨ(f)(∆σ(f)) if f is just a measure. This is the main reason we need to use the mollified empirical measure

α^N,_η =α^N_η ∗δ⁽⁾,

which can be seen as obtained through a convolution. The mollified empirical measure induces the following local averages: For any function h:N→R, we set

(2.34) (h◦η)⁽⁾(u) := 1

N^d X

x∈T^dN

h(η(x))δ⁽⁾x N(u)

where we recall that δ⁽⁾(u) is given in (2.30). The mollification introduces another scale, which might be called mesoscopic. It has microscopic size N and macroscopic size .

Such an intermediate scale appears in virtually all works on the hydrodynamic limit, see for example the replacement lemma. Here the intermediate scale is inherent in the (mollified) empirical measure.

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