Equivalence of ensembles

as follows. It holds that (2.73)

1 N^d

X

x∈T^dN

σ(η⁽⁾(x))− 1 (N)^d

X

y∈T^dN

χ y N

σ(η^l(x+y))

2¹₂

≤ 1

N^d X

x∈T^dN

1 (N)^d

X

y∈T^dN

χ y N

σ(η⁽⁾(x))−σ(η^l(x+y))

2¹₂

+O 1 N

,

where we employed (2.33) and the moment bounds again. Recall that χ(y/(N)) = 0 if

|y|> N. Hence, up to a term O(_N¹ ), it holds that (2.73) is bounded from above by

Ckχk∞

sup

|y|≤N

1 N^d

X

x∈T^dN

η⁽⁾(x)−η^l(x+y)

2¹₂

which is bounded by the two blocks estimate. Note in view of the statement of the two blocks estimate that if we include the the integration overt andη, we take the supremum outside the integration.

Thus it remains to prove the one and two block estimates in order to deduce the replace- ment lemma.

However, we are still missing one important ingredient in order to be able to present the proof of the block estimates. This is the equivalence of ensembles which concerns the closeness of the grand-canonical measures ν_ρ^L and the canonical measures ν^L,K for large L under the condition that the densities are identical. Here L denotes the size of an arbitrary lattice, not necessarily T^dN.

Recall that, with σ given as the inverse function (2.23), the measure ν_ρ^L is the invariant measure with particle densityρ, i.e.

Eν_ρ^L[η(0)] =ρ.

LetH_m denote the m-th Hermite polyomial Hm(λ) = (−1)^me^λ

2 2 d^m

dλ^me^−λ

2 2

for all λ∈R. Define two polynomials q0(λ) = 1

√2πe^−λ

2

2 and

q₁(λ) = 1

√2πe^−λ

2

2 H₃(λ) γ₃(ρ)

6s(ρ)³ = γ₃(ρ) 6√

2πs(ρ)³e^−λ

2

2 (λ³−3λ), where

γ₃(ρ) = Eν_ρ^L

η(0)−ρ3 .

The case of bounded densities

The contents of this section are mostly contained in [47, Appendix 2] and [52]. We include them here for the reader’s convenience. Let us state, without proof, the following uniform central limit theorem. The theorem, including higher order expansions, can be found in [47], see also [73]. The results of this subsection are essentially valid without Assumption 1 (iii).

Lemma 2.6.4. For all 0 < ρ₀ < +∞, there exist finite constants E₁ = E₁(ρ₀) and E₂ =E₂(ρ₀) such that

sup

K≥0

pL^d_∗s(ρ)Pν_ρ^L

P

x∈Λ_Lη(x) = K

−q0(λ)− q₁(λ) pL^d_∗

≤ E₁ L^d_∗s(ρ)² for all ρ≤ρ₀, such that s(ρ)²L^d_∗ ≥E₂. Here we set λ = (K−L^d_∗ρ)/(L^d/2∗ s(ρ)).

The following result is Corollary 1.6 in Appendix 2 of Kipnis and Landim [47].

Lemma 2.6.5. Fix0< ρ₀ <∞, a positive integer l and a cylinder function f :N^Λl →R with finite second moment with respect to ν_ρ^L for all ρ ≤ ρ₀. Then there exist finite

constants E3 =E3(ρ0) and E4 =E4(ρ0), independent of the choice of f, such that

Eν^L,K[f]−Eν^L

K/Ld∗

[f]

(2.74)

≤E₃ l^d_∗ L^d_∗

1 s(ρ)²Eν^L_ρ

h

f−Eν^L_ρ[f] i

+ 1

s(ρ) s

Eν_ρ^L

f−Eν_ρ^L[f]2!

for all L≥2l and all K such that K/L^d_∗ ≤ρ₀ and s(K/L^d_∗)²L^d_∗ ≥E₄. On the right hand side of the inequality, we have set ρ=K/L^d_∗.

Proof. First, we note that there exist constants 0< c₁(ρ₀) and c₂(ρ₀)<∞ such that c₁ ≤ ρ

s(ρ)² ≤c₂ and c₁ ≤ γ3(ρ) s(ρ)² ≤c₂.

Forρbounded away from zero, this is an obvious consequence of continuity in ρ, whereas for small ρ, all moments of η grow linearly in ρ to first order, see equation (2.80) below.

It holds that

q₁(λ)

pL^d_∗ ≤C |λ|

pL^d_∗s(ρ)² ≤C 1

L^d_∗s(ρ)² +|λ|² uniformly in ρ,λ, and L. Consequently, Lemma 2.6.4 yields (2.75) sup

K≥0

pL^d_∗−l^d_∗s(ρ)Pν_ρ^L

P

x∈Λ_L\Λ_lη(x) =K

− 1

√2πe^−λ

2 2

≤C 1

L^d_∗s(ρ)² +|λ|² , where

λ= M(ξ)−l^d_∗ρ p(L^d_∗−l^d_∗)s(ρ)² and M(ξ) =P

x∈Λ_lξ(x). The difference |Eν^L,K[f]−Eν^L_ρ[f]| equals (2.76)

X

ξ∈N^Λl

ν_ρ^l(ξ)

f(ξ)−Eν^l_ρ[f]ν_ρ^L(P

x∈Λ_L\Λ_lη(x) = K−M(ξ)) ν_ρ^L(P

x∈Λ_Lη(x) =K) −1

.

Thus the central limit theorem, see Lemma 2.6.4, yields a bound on the term in braces by

(2.77)

pL^d_∗s(ρ)²

p(L^d_∗ −l^d_∗)s(ρ)²e^−λ

2 2 −1

+C 1

L^d_∗s(ρ)² +|λ|² ,

where in the denominator, we have used that s(ρ)²L^d_∗ ≥E2. We now bound

|λ|² = 1 L^d_∗−l^d_∗

(M(ξ)−l^d_∗ρ)²

s(ρ)² ≤C l_∗^d L^d_∗

(p

l^−d∗ M(ξ)−p l_∗^dρ)² s(ρ)²

and plug this into (2.77). Sinces(ρ)² ≤C₁(ρ₀)ρ₀, it follows that (2.77) is bounded by E₃⁰(ρ₀) l^d_∗

L^d_∗s(ρ)²

1 +p

l^−d∗ M(ξ)−p

l_∗^dρ2 .

By a law of large numbers (a straightforward direct computation) together with bounds similar to the bounds at the beginning of this proof, it holds that

Eν_ρ^l

h X

x∈Λ_l

(η(x)−ρ)⁴i

≤C l^2d_∗ s(ρ)⁴+l^d_∗Eν^l_ρ[(η(x)−ρ)⁴]

≤E₃⁰(ρ₀)l^2d_∗ s(ρ)²

after possibly changing the value of E₃⁰(ρ0). Therefore it holds that Eν_ρ^l

h p

l^−d∗ M(ξ)−p l^d_∗ρ4i

≤E₃⁰(ρ₀)s(ρ)²,

and therefore the Cauchy-Schwarz inequality yields a bound on (2.76) by E₃ l^d_∗

L^d_∗ 1 s(ρ)²Eν^L_ρ

h

f−Eν^L_ρ[f] i

+ 1

s(ρ) r

Eν_ρ^L

h

f −Eν_ρ^L[f]2i

!

for some constantE₃ =E₃(ρ₀).

This lemma is used in [47] to prove the equivalence of ensembles without the lower bound s(ρ)²L^d_∗ ≥ E₄. Since we are interested in obtaining explicit bounds on the rate of con- vergence to the hydrodynamic limit, we need to be a bit more careful in our analysis to identify the dependence on the sizeland not just L. The good news so far is thatE₁ and E₂ do not depend on l and f, and that in our proof of the replacement lemma we shall not need to consider any cylinder functionf, but only the function

(2.78) f(ξ) := 1

l_∗^d X

x∈Λ_l

g(ξ(x))² ≤ (g^∗)² l_∗^d

X

x∈Λ_l

ξ(x)².

Carefully keeping track of the dependence on the integer l, we now prove equivalence of ensembles.

Lemma 2.6.6 (Equivalence of ensembles for bounded densities). Fix 0 < ρ₀ < ∞ and let f as in (2.78). Then there exists a constant E₅ =E₅(ρ₀) such that

Eν^L,K[f]−Eν^L

K/Ld∗

[f]

≤E₅ l^d_∗ L^d_∗

for all L large enough, all L≥2l and all K such that 0≤K/L^d_∗ ≤ρ0.

Proof. The proof follows Corollary 1.7 in Appendix II of [47]. Let E3, E4 as in Lemma 2.6.5 and consider first the case s(K/L^d_∗)²L^d_∗ ≥ E₄. In this case, if ρ=K/L^d_∗ is bounded strictly away from zero, the bracket on the right hand side of the inequality in Lemma 2.6.5 is bounded by a constant since the variance s(ρ)² and all the expectations on the right hand side are continuous with respect to ρ ≤ ρ₀. Hence let us consider ρ close to zero and s(ρ)²L^d_∗ ≥E₄. It holds

Eν_ρ^L[f] =

∞

X

M=0

E^νl,M[f]·ν_ρ^l P

x∈Λ_lξ(x) =M .

Sinceggrows at most linearly, the explicit form (2.78) yields thatf(ξ)≤CM²for particle configurations ξ with M particles. ForM ≥2 particles it holds that

(2.79) E^νl,M[f]·ν_ρ^l P

x∈Λ_lξ(x) =M

≤CM²ρ^M,

where we have used that ν_ρ^l(ξ(x) = n) = σ(ρ)ⁿ/(Z(σ(ρ))g(n)!)≤ Cρ^k since g ≥ δk˜ and ρ≤ρ₀. Furthermore, for M = 0,1 we obtain

ν_ρ^l P

x∈Λ_lξ(x) = M

≤







1

Z(ρ)^l if M = 0, l_g(1)Z(ρ)^ρ l if M = 1.

Hence we can expand

Eν_ρ^l[f] =f(0) + X

x∈Λ_l

{f(dx)−f(0)} ρ

g(1) +O(ρ²)

forρsmall enough, where0denotes the particle configuration inN^Λl without any particles and d_x denotes the configuration containing only a single particle situated at the site x∈Λ_l. Note that (2.79) yields that the term O(ρ²) is bounded independently of l and K for small enough ρ. Likewise we obtain

Eν_ρ^l

f−Eν_ρ^l[f]2

= X

x∈Λ_l

{f(d_x)−f(0)}² ρ

g(1) +O(ρ²) and Eν^l_ρ[f] =

"

X

x∈Λl

{f(d_x)−f(0)}

+X

x∈Λl

|f(d_x)−f(0)|

# ρ

g(1) +O(ρ²).

Of course, in our case these expressions simplify due to f(0) = 0 but we shall not take advantage of this just yet. Replacing f byξ(0) we also see that

(2.80) s(ρ)² = ρ

g(1) +O(ρ²).

Putting all together we can bound the right hand side of inequality (2.74) by E3l_∗^d/L^d_∗. Hence it remains only to consider the case of densitiesK/L^d_∗ such that 0≤s(K/L^d_∗)²L^d_∗ ≤ E₄. Note that as before equation (2.80) implies that

0< c₁ ≤ s(ρ)²

ρ ≤c₂ <+∞

near ρ = 0 and hence for all 0 ≤ ρ ≤ ρ₀. Hence the particle numbers K = ρL^d_∗ under consideration are bounded by K ≤ c₂E₄. In the following we shall use the explicit form of f. Translation invariance of the canonical measures yields that

Eν^L,K[f] =Eν^L,K

"

1 l_∗^d

X

x∈Λ_l

g(ξ(x))²

#

≤ (g^∗)² L^d_∗ Eν^L,K

"

X

x∈Λ_L

ξ(x)²

#

≤ (g^∗K)² L^d_∗ as well as

Eν^L

σ(K/Ld∗)

[f]≤(g^∗)²Eν^L

K/Ld∗

[η(0)²]

= (g^∗)² s(_L^Kd

∗)²+ (_L^Kd

∗)²

≤ (g^∗)²E₂

L^d_∗ +(g^∗K)² L^2d_∗ . Similar computations hold for the expectations of f². Again, this shows that the right hand side of (2.74) is bounded by E5l^d_∗/L^d_∗. This concludes the proof of the equivalence of ensembles in the case of bounded densities.

The case of large densities

Using Assumption 1 (i)-(iii), the following result has been shown in [52]. The proof is similar to the proof of the equivalence of ensembles in the case of bounded densities, but relies on Assumption 1 (iii) in order to obtain estimates on the growth of moments ofη(x) underν_ρ^L.

Lemma 2.6.7. There exist 0< ρ₁ <∞ and constants E₆, L₀ such that

Eν^L,K[f]−Eν^L

K/Ld∗

[f]

≤E₆ l_∗^d L^d_∗

s Eν_ρ^L

f−Eν^L_ρ[f]2

for all l > 0, cylinder functions f : N^Λl → R with finite second moment with respect to ν_ρ^L, L≥max{L₀,2l}, and K such that ρ=K/L^d_∗ ≥ρ₁.

Choosingf as in (2.78), we obtain that Eν_ρ^L

f −Eν_ρ^L[f]2

≤Cρ⁴

for all ρ≥ρ1. Consequently Lemma (2.6.7) yields the equivalence of ensembles for large densities.

Lemma 2.6.8 (Equivalence of ensembles for large densities). Let f as in (2.78). There exist 0< ρ₁ <∞ and constants E₇, L₀ such that

Eν^L,K[f]−Eν^L

K/Ld∗

[f]

≤E₇ l_∗^d L^d_∗ρ²

for all L≥L0 large enough, l < L/2, and K such that ρ=K/L^d_∗ ≥ρ1.

Equivalence of ensembles for arbitrary densities

Combining Lemmas 2.6.6 and 2.6.8, we arrive at the main result of this section.

Theorem 2.6.9. Let f as in (2.78). Then there exist constantsE₈ and L₀ such that

Eν^L,K[f]−Eν^L

K/Ld∗

[f]

≤E₈ l_∗^d L^d_∗

1 + K² L^2d_∗

for all L≥L₀ large enough, l < L/2 and K >0.

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