Convergence of the entropy

since the average number of particles is bounded. Collecting all available bounds, this completes the proof of the hydrodynamic limit.

Remark. As in the proof of the hydrodynamic limit for independent random walks, Theorem 2.3.1, the term T1 is a measure for the difference of the particle semigroup and the limit semigroup on the level of observables and is (for fixed Ψ) bounded by a consistency result on the two generators. Furthermore we have seen that this consistency estimate is transported along the flow of the limit equation by the stability result. The second term T2 measures the propagation of the difference between the initial data of the particle system and the limit partial differential equation along the flow of the limit equation. Hence it is bounded by a stability result on the limit equation. The last term T3 did not appear in the proof of Theorem 2.3.1 and measures the error due to the mollification of the empirical measure.

lemma with a rate rRL(, N), i.e.

1 T N^d

Z T 0

X

x∈T^dN

µ^N_t ,

(g◦η)⁽⁾(x)−σ(η⁽⁾(x))

dt≤rRL(, N)

for >0, N ∈N such that N <1 and all T >0, where we suppose that lim sup

→0

lim sup

N→∞

r_RL(, N) = 0.

To keep notation simple, we allow all constants in this section (in contrast to the last) to depend on (f_t)t∈[0,∞) ⊂ C³(T^d), although it is possible in principle to keep track of this dependence as well. We set

f∞= Z

T^d

f_t(u)du,

which is independent of t since f_t solves the limit equation (2.3). The notation is fur- thermore justified on noting that we expect f_t → f_∞ as t → ∞, cf. Lemma 2.4.15.

Furthermore we denote the pressure by

(2.60) p(λ) = logZ(e^λ),

whereZ is the partition function given in equation (2.7). Then we define the macroscopic entropy as

(2.61) H^∞(f_t) :=

Z

T^d

h(f_t(u))du−h(f∞), where the function h is given by

h(ρ) =ρlogσ(ρ)−p logσ(ρ) .

Let us find the corresponding macroscopic Fisher information by differentiating in time.

It holds that d

dtH^∞(f_t) = Z

T^d

∂_tf_tlogσ(f_t) +f_tσ⁰(ft)

σ(f_t)∂_tf_t−p⁰(logσ(f_t))σ⁰(ft) σ(f_t)∂_tf_t

du.

Since σ is the inverse function of ρ∂_ρlogZ(ρ), we find thatp⁰(λ) = σ⁻¹(e^λ) and hence

(2.62) d

dtH^∞(f_t) =− Z

T^d

|∇σ(f_t(u))|²

σ(ft(u)) du=:−D^∞(f_t),

where D^∞(f_t) is called the macroscopic Fisher information. Next we establish a micro- scopic analogue of equation (2.62), relating the microscopic entropy H^N(µ^N_t |ν_ρ^N) and its

Fisher information D^N(µ^N_t |ν_ρ^N), to be defined presently. Let f_t^N ∈ Cb(XN) denote the density of µ^N_t ∈ P(X_N) with respect to the grand-canonical measure ν_f^N_∞ ∈ P(X_N), i.e.

set

(2.63) f_t^N(η) := dµ^N_t

dν_f^N_∞(η).

The microscopic Fisher information is then defined as D^N(µ^N_t |ν_f^N_∞) :=

Z

XN

q

f_t^NN⁻²G^N q

f_t^Ndν_f^N_∞ (2.64)

= q

f_t^N, N⁻²G^N q

f_t^N

L²(ν_f∞^N )

.

Remark 2.5.1. Abusing notation, we shall sometimes refer toD^N(µ^N_t |ν_f^N_∞) byD^N(f_t^N|ν_f^N_∞), wheref_t^N is the density defined in (2.63). Also note that we have left out a factor ofN² as opposed to the natural (macroscopic) time-scaling, i.e. the time scale of the microscopic Fisher information is the microscopic time scale.

As a first result we show the equivalence of the convergence of the entropy and entropic chaos, by which we mean

Nlim→∞

1

N^dH^N(µ^N_t |ν_f^N_t(·)) = 0.

.

Lemma 2.5.2. Under assumption 2, it holds that 1

N^dH^N(µ^N_t |ν_f^N_∞) =H^∞(f_t) + 1

N^dH^N(µ^N_t |ν_f^N_t(·)) +O1

N +r_HL(N) .

In particular, the microscopic entropy N^−dH^N(µ^N_t |ν_f^N_∞) converges to the macroscopic en- tropy H^∞(f_t) if and only if there is entropic chaos.

Proof. It holds that 1

N^dH^N(µ^N_t |ν_f^N_∞) = 1 N^d

Z

XN

log

dµ^N_t dν_f^N_∞

dµ^N_t

= 1 N^d

Z

XN

log

dµ^N_t dν_f^N

t(·)

dµ^N_t + Z

XN

log dν_f^N

t(·)

dν_f^N_∞

dµ^N_t .

By definition, it also holds that

(2.65) dν_f^N

t(·)

dν_f^N_∞ (η) = Y

x∈T^dN

Z(σ(f∞)) Z(σ(f_t(_N^x)))

σ(f_t(_N^x)) σ(f∞)

η(x)

.

Consequently, the second term equals 1

N^d Z

XN

log dν_f^N

t(·)

dν_f^N_∞

dµ^N_t

= 1 N^d

X

x∈T^d_N

Z

XN

log Z(σ(f∞))

Z(σ(f_t(_N^x))) +η(x) logσ(f_t(_N^x)) σ(f_∞)

dµ^N_t (η).

By Assumption 2, the macroscopic solution f_t is differentiable, and the hydrodynamic limit yields that the right hand side converges to

Z

T^d

f_t(u) logσ(f_t(u))du− Z

T^d

p logσ(f_t(u))

du−f∞logσ(f∞) +p logσ(f∞) as N → ∞. Thus, in view of (2.61), we have shown that

1

N^dH^N(µ^N_t |ν_f^N_∞) = 1

N^dH^N(µ^N_t |ν_f^N_t(·)) +H^∞(f_t) +O1

N +r_HL(N) . which concludes the proof.

The main result of this section is the following theorem.

Theorem 2.5.3. Under Assumption 2, let the initial microscopic entropy converge, i.e.

let (2.66)

1

N^dH^N(µ^N₀ |ν_f^N_∞)−H^∞(f₀)

≤r_H,0(N)

for some rate function r_H,0(N). Then both the microscopic entropy and the time aver- age of the Fisher information converge towards the corresponding macroscopic quantities.

Specifically, it holds that

1

N^dH^N(µ^N_t |ν_f^N_∞)−H^∞(f_t)

≤r_H,0(N) +C + 1

N + 1

^3+dr_HL(N) +t r_RL(, N) , and

Z t 0

4N^2−dD^N(µ^N_s |ν_f^N_∞)ds− Z t

0

D^∞(f_s)ds

≤r_H,0(N) +C + 1

N + 1

^3+dr_HL(N) +t r_RL(, N) for all > 0, N ∈ N, and t ≥ 0 (note that this bound is not uniform in time). In

particular, it holds that

N→∞lim 1

N^dH^N(µ^N_t |ν_f^N_∞) = H^∞(f_t) and

N→∞lim Z t

0

4N^2−dD^N(µ^N_s |ν_f^N_∞)ds= Z t

0

D^∞(f_s)ds

for all t≥0.

The key to the proof of Theorem 2.5.3 is the following lemma.

Lemma 2.5.4. Under the assumptions of Theorem 2.4.6, it holds that 1

N^dH^N(µ^N_t |ν_f^N_∞)≥H^∞(f_t)−C1

N +r_HL(N) , as well as

Z t 0

4N^2−dD^N(µ^N_s |ν_f^N_∞)ds≥ Z t

0

D^∞(f_t)ds−C1

N ++ 1

^3+dr_HL(N) +tr_RL(, N) .

In particular, it holds that lim inf

N→∞

1

N^dH^N(µ^N_t |ν_f^N_∞)≥H^∞(f_t)and lim inf

N→∞

Z t 0

4N²

N^dD^N(µ^N_s |ν_f^N_∞)ds ≥D^∞(f_t) for all t≥0.

Proof. The bound on the relative entropy is a direct consequence of Lemma 2.5.2 and the fact that all entropies H^N are non-negative. As a warm-up for the bound on the Fisher-information, let us give an alternative proof. The well-known variational formula for the relative entropy, see [47], yields

H^N(µ^N_t |ν_f^N_∞) = sup

f∈C_b(XN)

nhµ^N_t , fi −loghν_f^N_∞, e^fio ,

where the supremum is (formally) obtained by taking f = logdµ^N_t /dν_f^N_∞. In view of the hydrodynamic limit and Yau’s results using the relative entropy method, we expect that µ^N_t is close to the local Gibbs state ν_f^N

t(·). Thus we choose f(η) = logdν_f^N

t(·)

dν_f^N_∞ (η).

Then we obtain a lower bound 1

N^dH^N(µ^N_t |ν_f^N_∞)≥ 1 N^d

Z

XN

log dν_f^N

t(·)

dν_f^N_∞ (η)dµ^N_t (η).

Equation (2.65) thus yields a lower bound on the entropy of the form 1

N^d X

x∈T^d_N

Z

XN

η(x) logσ f_{t N}^x

σ(f∞) + log Z(σ(f∞)) Z σ ft x

N

dµ^N_t (η).

As in the proof of Lemma 2.5.2, since the macroscopic solution ft is differentiable, the hydrodynamic limit yields a bound from below by

H^∞(f_t)−C1

N +r_HL(N) .

To prove a similar estimate for the Fisher information, we need the following variational formula, cf. [47]. It holds that

D^N(µ^N_t |ν_f^N_∞) = sup

f

− Z

XN

N⁻²G^Nf(η)

f(η) dµ^N_t (η)

,

where the supremum is taken over all positivef ∈Cb(XN) such thatf is strictly bounded away from zero. The supremum is (formally) obtained at the function f =q

dµ^N_t /dν_f^N_∞, so here we choose

f(η) =

sdν_f^N

t(·)

dν_f^N_∞ (η).

After cancelling factors, this corresponds to taking f(η) = Y

x∈T^d_N

r σ

f_t(x N

^η(x) .

We obtain that

N⁻²G^Nf(η) = X

x∈T^dN

X

|e|=1

g(η(x)) s

σ ft x+e N

σ f_{t N}^x −1

f(η),

and hence

G^Nf(η)

N²f(η) = X

x∈T^d_N

X

|e|=1

g(η(x)) q

σ f_t(_N^x r

σ f_t x+e N

− r

σ f_t x N

.

Hence it holds that

4N^2−dD^N(µ^N_t |ν_f^N_∞)≥ −4 Z

XN

1 N^d

X

x∈T^dN

g(η(x)) q

σ f_t(_N^x

∆N

r σ ft

x N

dµ^N_t (η),

where we recall that ∆_N denotes the discrete Laplacian (2.22). Even thoughf as chosen

above is not strictly bounded away from zero (uniformly inη), this can be made rigorous by a standard approximation argument. Since f_t ∈ C³(T^d) is bounded away from zero, we replace g(η(x)) by (g ◦η)⁽⁾(x), up to an error of O(). Now the replacement lemma yields

Z t 0

4N^2−dD^N(µ^N_s |ν_f^N_∞)ds ≥ −4 Z t

0

Z

XN

1 N^d

X

x∈T^d_N

σ(η⁽⁾(x)) q

σ f_s(_N^x

×∆_N r

σ f_s x N

!

dµ^N_s (η)ds−C(+tr_RL(, N)).

It holds that η⁽⁾(x) = hα^N_η , δ⁽⁾x

Ni and the following bounds are satisfied:

kδ⁽⁾x

N k_C³ ≤^−3−d as well as kδ⁽⁾x

N k²_H1 ≤^−2−d. Hence the hydrodynamic limit yields

4N^2−dD^N(µ^N_t |ν_f^N_∞)≥ −4 Z

T^d

pσ(f_t(u))∆_Np

σ(f_t(u))du

−C(+tr_RL(, N) +^−3−dr_HL(N)).

Finally we replace, up to an error O(N⁻¹), the discrete Laplacian ∆_N by its continuous version ∆ and note that

−4 Z

T^d

pσ(f_t(u))∆p

σ(f_t(u))du= Z

T^d

|∇σ(f_t(u))|² σ(f_t(u)) du.

This completes the proof of Lemma 2.5.4.

Proof of Theorem 2.5.3. Equation (2.62) yields

(2.67) H^∞(f_t) +

Z t 0

D^∞(f_s)ds=H^∞(f₀).

Next, we establish a corresponding microscopic relation. First we note that f_t^N defined in (2.63) satisfies the forward Kolmogorov equation

∂_tf_t^N(η) =G^Nf_t^N(η).

SinceG^N is self-adjoint in L²(ν_f^N_∞) and in particular hG^Nf_t^N,1i_L2(ν_f∞^N ) = 0, it holds that

(2.68) d

dtH^N(µ^N_t |ν_f^N_∞) = d dt

Z

XN

f_t^Nlogf_t^Ndν_f^N_∞ = Z

XN

G^Nf_t^N

logf_t^Ndν_f^N_∞.

Now we note that

D^N(µ^N_t |ν_f^N_∞) =q

f_t^N, N⁻²G^N q

f_t^N

L²(ν_f∞^N )

= 1 2

Z

XN

X

x∈T^dN

X

|e|=1

g(η(x)) q

f_t^N(η^x,x+e)− q

f_t^N(η) 2

dν_f^N_∞,

see for example [47, Appendix 1]. Polarization yields Z

XN

N⁻²G^Nf_t^N

logf_t^Ndν_f^N_∞ =hN⁻²G^Nf_t^N,logf_t^Ni_L2(ν^N_f∞)

= 1 2

Z

XN

X

x∈Td

|e|=1N

g(η(x)) f_t^N(η^x,x+e)−f_t^N(η)

logf_t^N(η^x,x+e)−logf_t^N(η) dν_f^N_∞.

Therefore the elementary inequality (√ a−√

b)² ≤ (a−b)(loga−logb)/4, which holds for all a, b > 0, yields

Z

XN

N⁻²G^Nf_t^N

logf_t^Ndν_f^N_∞ ≥4 Z

XN

q

f_t^NN⁻²G^N q

f_t^Ndν_f^N_∞ = 4D^N(µ^N_t |ν_f^N_∞).

Therefore time-integration of equation (2.68) yields (2.69) H^N(µ^N_t |ν_f^N_∞) + 4N²

Z t 0

D^N(µ^N_s |ν_f^N_∞)ds≤H^N(µ^N₀ |ν_f^N_∞).

Note that this inequality holds for any ρ > 0 replacing f∞ > 0. Since the microscopic entropy converges initially and Lemma 2.5.4 yields lower bounds on each term on the left hand side, a simple technical lemma completes the proof, i.e. we set

a= 1

N^dH^N(µ^N_t |ν_f^N_∞)−H^∞(f_t) andb = Z t

0

4N^2−dD^N(µ^N_s |ν_f^N_∞)−D^∞(f_s) ds in Lemma 2.5.5.

Lemma 2.5.5. Let a, b∈R, and r_i >0, i= 1,2,3 be any reals such that

|a+b| ≤r₁, a≥ −r₂, and b≥ −r₃. Then it holds that

|a| ≤r1+r2+r3 and |b| ≤r1+r2+r3.

Proof. For any λ∈R, we set

λ₊ = max{0, λ} and λ−= max{0,−λ},

i.e. λ=λ+−λ−, |λ|=λ++λ−. The first inequality of the assumptions yields a+b=a₊−a−+b₊−b−≤ |a+b| ≤r₁.

On the other hand, the other two assumptions yield a− ≤r₂ and b−≤r₃ and hence

a₊+b₊ ≤r₁+r₂ +r₃.

Since both terms on the left hand side are positive and we already established bounds on a− and b−, this shows the hypothesis.

As a consequence of Lemma 2.5.2 and Theorem 2.5.3, we recover Yau’s result on entropic chaos. Of course, our proof is very different in spirit, since we assumed the hydrodynamic limit in order to arrive at entropic chaos. In other words, we conclude the conservation of the hydrodynamic limit in the strong form of Theorem 2.2.3 from the conservation of the hydrodynamic limit in the weak form of Theorem 2.2.2.

Corollary 2.5.6. Under Assumption 2, suppose that entropic chaos holds initially, i.e.

Nlim→∞

1

N^dH^N µ^N₀ |ν_f^N_0(·)

= 0.

Then this property is conserved along the evolution of the process, i.e.

N→∞lim 1

N^dH^N µ^N_t |ν_f^N_t(·)

= 0 for all t≥0.

Note that so far we have not proved that the rate of convergence of the microscopic entropy is uniform in time. This should be done in future work.

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