Proposition 2.3. Suppose that the above couple of entropy - flux pairs satisfies 2.10, then any limit distribution ˆ
P
θ
∈ ˆ P
θ
of the Young measure is Dirac, i.e. it is concentrated on a set of measurable functions.
As a direct consequence we obtain 1.22 from Proposition 2.1, while 2.11 turns into the weak entropy condition 1.23, which imply uniqueness of the limit by Main Theorem of [CR00].
The verification of the conditions of Proposition 2.2 is mainly based on inequalities involving relative entropy and the associated Dirichlet form; the basic ideas go back to [GPV88], see also [Var94] and
[Yau97] with further references. These computations are supplemented with an extension of the method of relaxation schemes to microscopic systems, see Lemma 3.6 and Sections 5.4 and 5.6.
Since we are going to treat hydrodynamic limit in infinite volume, we have to control entropy flux by means of its rate of production, cf. [Fri90], [Fri01] and [FN06] for some earlier results in this
direction. This a priori bound allows us to apply the robust LSI and the relaxation scheme of the microscopic process, see Lemma 3.3, Lemma 3.4 and Lemma 3.6. Most technical details of this
estimation procedure have been elaborated in [Fri01], [Fri04] and [FT04]; [FN06] is our basic reference. First we substitute the microscopic time derivative
L
o
hˆ u
ǫ
with the spatial gradient of a mesoscopic flux depending on the block averages ¯
η
l,k
and ¯ ω
l,k
. The replacement of ¯ η
l,k
with its empirical estimator F ¯
ω
l,k
is based on the microscopic relaxation scheme, see Lemma 3.6 below, and also Section 5.6.
3 Entropy, Dirichlet Form, LSI and Relaxation
In this section we derive some fundamental estimates based on entropy and the associated Dirichlet forms. The parameters
β, σ and l are almost arbitrary here, their dependence on the scaling pa- rameter
ǫ 0 is not important. We only need β 0, σ ≥ 1 and l ∈ N . We follow calculations of [FN06] with slight modifications.
3.1. Entropy and its temporal derivative: If
µ and λ are probability measures on the same space, then entropy of
µ relative to λ is defined by S[µ|λ] := µlog f if µ ≪ λ and f := dµdλ . A frequently used entropy inequality,
µϕ ≤ S[µ|λ] + log λe
ϕ
follows by convexity, and we have another definition of relative entropy:
S[ µ|λ] := sup
ϕ
µϕ − log λe
ϕ
: λe
ϕ
+∞ ;
3.1 ϕ = log f is the condition of equality. Note that
f logg f = 2 f log
p g
f ≤ 2 p
g f − 2 f = g − f −
p g
− p
f
2
, 3.2
whence another useful inequality, E
λ
p f
− 1
2
≤ S[µ|λ] follows immediately. Given a Markov generator
A , the Donsker - Varadhan rate function of large deviations is defined as D[
µ|A ] := − inf ¨Z
A ψ ψ
d µ : 0 ψ ∈ DomA
« ;
3.3 D[
µ|A ] = −λ p
f A
p f if
A is self-adjoint in L
2
λ and f = dµdλ . Of course, λϕA ϕ = λϕG ϕ if G denotes the symmetric part of A . In view of their variational characterizations 3.1
242
and 3.3, both S and D are lower semi - continuous, convex functionals of µ , and the definitions
and relations above extend to conditional distributions and densities, too. As a reference measure we can choose any of the equilibrium product measures
λ = λ
∗ u
with −1
u 1 fixed, say u = 0 . At a level ǫ 0 of scaling, µ
ǫ,t
denotes the evolved measure, µ
ǫ,t,n
is the distribution of the variables
{ω
k
: |k| ≤ n} , and f
n
= f
ǫ,t,n
:= d µ
ǫ,t,n
dλ . Entropy in the box Λ
n
:= [ −n, n] ∩ Z is defined as
S
n
t := S[µ
ǫ,t,n
|λ] = Z
log f
ǫ,t,n
d µ
ǫ,t
. Local versions of the Dirichlet forms for
L
o
, S and G
∗
at ϕ =
p f
ǫ,t,n
can easily be computed; in the first line below ˜
c
b
ω := 12η
k
+ η
k+1
if b = k, k + 1 . D
o n
t := 1
2 X
b ⊂Λ
n
Z ˜
c
b
ω Æ
f
ǫ,t,n
ω
b
− p
f
ǫ,t,n
ω
2
λdω ,
D
s n
t := 1
2 X
b ⊂Λ
n
Z Æ
f
ǫ,t,n
ω
b
− p
f
ǫ,t,n
ω
2
λdω , D
∗ n
t := 1
2 X
b ⊂Λ
n
Z Æ
f
ǫ,t,n
ω
b ∗
− p
f
ǫ,t,n
ω
2
λdω , respectively. By convexity, S
n
and any of D
q n
, q ∈ {o, ∗, s} are nondecreasing sequences.
The Kolmogorov equation yields ∂
t
S
n
t = Z
∂
t
+ L log f
ǫ,t,n
ω µ
ǫ,t
dω =
Z f
ǫ,t,n+1
ω L log f
ǫ,t,n
ω λdω = β
X
b ∈Z
∗
Z f
ǫ,t,n+1
ω log f
ǫ,t,n
ω
b ∗
f
ǫ,t,n
ω λdω
+ X
b ∈Z
∗
Z c
b
ω + σ f
ǫ,t,n+1
ω log f
ǫ,t,n
ω
b
f
ǫ,t,n
ω λdω .
Taking into account inequality 3.2, it is easy to recover the local Dirichlet forms from the segments −n ≤ k n of the corresponding sums, the rest gives the so called boundary terms.
3.2. Entropy flux: Our basic a priori bound on local entropy and Dirichlet forms is the content of Lemma 3.1. If
σ ≥ 1 and β ≤ σ then we have a universal constant C such that
S
n
t + β Z
t
D
∗ n
τ dτ + σ Z
t
D
n
τ dτ ≤ C
t + p
n
2
+ σt
for any initial distribution µ
ǫ,0
, n ∈ N and t 0 .
243
Proof. We have to estimate the boundary terms of ∂
t
S
n
by means of S
n
and its associated Dirichlet forms D
q n
. Following the lines of the proof of Lemma 3.1 in [FN06], we arrive at a system ∂
t
S
n
t + 2β D
∗ n
t + 2σD
n
t ≤ K
1
S
n+1
t − S
n
t + β K
1
p S
n+1
t − S
n
t p
D
∗ n+1
t − D
∗ n
t + σK
1
p S
n+1
t − S
n
t p
D
n+1
t − D
n
t , 3.4
of differential inequalities with some universal constant K
1
. The cases of boundary term induced by L
o
and S are the same as in [FN06], the asymmetric L
o
yields the flux S
n+1
− S
n
on the right hand side. When we estimate boundary terms induced by
S , we are exploiting its following properties. The jump rates are constant, and the elementary actions,
ω → ω
b
all preserve λ in a reversible way.
Since G
∗
possesses all of these features, replacing ω
b
with ω
b ∗
in the computations concerning the boundary terms induced by
G
∗
, we get 3.4. This system can explicitly be solved by means of the following lemma, which is a simple generalization of Lemma 3 in [Fri90].
Lemma 3.2. Suppose that s
n+1
t ≥ s
n
t ≥ 0 , u
n+1
t ≥ u
n
t ≥ 0 , v
n+1
t ≥ v
n
t ≥ 0 for t ≥ 0 and n
∈ N , moreover ds
n
d t + 2βu
n
+ 2σv
n
≤ Cs
n+1
− s
n
+ β K p
s
n+1
− s
n
u
n+1
− u
n
+ σK p
s
n+1
− s
n
v
n+1
− v
n
, where 0
β = O σ , C, K 0 and s = u
= v = 0 . Then we have a constant, M depending on C
and K such that for all t ≥ 0 , n ∈ N we have
s
n
t + β Z
t
u
n
s ds + σ Z
t
v
n
s ds ≤ M
R
+∞
X
m=0
s
m
0 exp
− m
R
, where R := M
t + n
2
+ σt
1 2
.
Proof. Just as in [Fri90], a clever cutoff function g
n
r can be defined by g
n
r = Z
∞ −∞
gx rgn − x d x
for n ∈ Z
+
and r 0 , where gx = 1 if |x| ≤ 1 , g
′
x = −gxsign x if |x| ≥ 3 and g
′
x = −12gxx − sign x otherwise. It is easy to check that 0 g
n
r − g
n+1
r ≤ 2rg
n+1
r if r
≥ 1 , moreover g
n
r − g
n+1
r ≤ 2 min {g
′ n
r, g
′ n+1
r} . Introduce now st, r := ξt, r if ξ
n
t = s
n
t , ut, r := ξt, r if ξ
n
t = u
n
t and vt, r := ξt, r if ξ
n
t = v
n
t , where ξt, r =
+∞
X
n=0
g
n
r − g
n+1
r ξ
n
t =
+∞
X
n=0
g
n+1
r ξ
n+1
t − ξ
n
t .
Using βu
n
and σv
n
to estimate the corresponding square roots on the right hand side, we get ∂ st, r
∂ t + βut, r + σvt, r ≤ M
1
1 +
σ r
∂ st, r
∂ r 244
because β = O σ , consequently
st, rt + β
Z
t
us, rs ds + σ
Z
t
vs, rs ds ≤ s0, r0 ,
provided that the decay of rt is fast enough. More precisely, let r solve r d r d t + M
1
r + M
1
σ = 0 with terminal condition rt = n . Since r0 =
O t + n
2
+ σt
1 2
in this case, and g
n
r ≥ e
−nr
if r ≥ 1 , this completes the proof by a direct calculation.
Now we are in a position to complete the proof of Lemma 3.1. Since our reference measure λ
∗
is the uniform distribution on
{0, 1, −1}
2n+1
, S
n
0 ≤ 2n + 1 log 3 . Therefore choosing s
n
= S
n
, u
n
= D
∗ n
and v
n
= D
n
we see that 3.4 really implies Lemma 3.1. This lemma is the fundamental a priori bound we need to materialize hydrodynamic limit in infinite
volume. Since ǫσ → 0 , t ≈ τǫ and n ≈ rǫ in its following consequences, r + τǫ is the order
of the bound, and r + τ ≤ rτ if r, τ ≥ 1 . To simplify formulae, from now on we are assuming that
ǫσ ≤ 1 and ǫl
3
≥ σ ≥ l ≥ 1 .
3.3. One block and two blocks estimates: The first replacement lemma for microscopic currents is based on the logarithmic Sobolev inequality for stirring
S . Given ¯ ω
l,k
= u and ¯ η
l,k
= ρ , let ¯ λ
l,k ρ,u
denote the conditional distributions of ω
k
, ω
k+1
, ..., ω
k+l −1
with respect to λ . In view of Proposition
4 of [FT04], we have a universal constant, ℵ such that
S[ ¯ µ|¯λ
l,k ρ,u
] ≤ ℵ l
2
X
b ⊂[k,k+l
Z p
f ω
b
− p
f ω
2
¯ λ
l,k ρ,u
dω 3.5
whenever ¯ µ ≪ ¯λ
l,k ρ,u
is a probability measure and f := d ¯ µd ¯
λ
l,k ρ,u
. It is very important that ℵ does
not depend on ρ, u, l and f . This LSI allows us to estimate canonical expectations via the basic
entropy inequality. The moment generating part is also a conditional expectation, consequently its bound should be independent of the conditions. Let us remark that Lemmas 3.3, 3.4 and 3.6 are
consequences of the local entropy bound Lemma 3.1, therefore methods of [FN06] work also in case of the creation - annihilation process, and the results are valid for the spin - flip dynamics, too.
Lemma 3.3. Let j ω
, ω
1
denote a given function, Jρ, u := λ
ρ,u
j and j
k
ω := jω
k
, ω
k+1
. We have a threshold l
∈ N and a universal constant C
1
depending only on C and j such that
ǫ
2
X
|k|rǫ
Z
τǫ
Z ¯j
l −1,k
− J ¯ η
l,k
, ¯ ω
l,k
2
d µ
ǫ,t
d t ≤ C
1
ǫrτl
2
σ whenever r,
τ ≥ 1 and ǫl
3
≥ σ ≥ l ≥ l .
Proof. In view of Lemma 3.1 the statement is more or less a direct consequence of the first inequality of Proposition 1 in [FT04], where notation is slightly different from ours. Lemma 3.2 of [FN06]
treats the case of j
k
= j
ηo k
, our problem is essentially the same. This is a sharp form of the so called One Block Lemma of [GPV88], the explicit rate due to LSI is
needed for the evaluation of microscopic currents in the expression of the Lax entropy production X
ǫ
. The following comparison of block averages of type ˆ ξ and ¯
ξ follows also via LSI in much the same way as the previous lemma did.
245
Lemma 3.4. We have a threshold l ∈ N and a universal constant C
2
such that if r, τ ≥ 1 and
ǫl
3
≥ σ ≥ l ≥ l , then
ǫ
2
X
|k|rǫ
Z
τǫ
ξ
2 l,k
d µ
ǫ,t
d t ≤ C
2
r τǫl
2
σ where
ξ
l,k
= ˆ ω
l,k
− ¯ ω
l,k
, or ξ
l,k
= ˆ η
l,k
− ¯ η
l,k
, or ξ
l,k
= ˆ ω
l,k+l
− ˆ ω
l,k
. The statement on block averages of
η is a direct consequence of Lemma 3.3 in [FN06], the bound for ξ
l,k
= ˆ ω
l,k
− ¯ ω
l,k
can be proven in the same way. The argument works even if ξ
l,k
= ¯ ω
l,k+m
− ¯ ω
l,k
, or
ξ
l,k
= ¯ η
l,k+m
− ¯ η
l,k
, but an integration by parts trick yields better results in these cases when m is large. By means of the Cauchy inequality the deviation of blocks of different size can also be
estimated.
Lemma 3.5. We have universal constants l
∈ N and C
3
+∞ such that if l ≤ l ≤ m but ǫl
3
≥ σ ≥ l and r,
τ ≥ 1 , then ǫ
2
X
|k|rǫ
Z
τǫ
Z ¯
ξ
m,k
− ¯ ξ
l,k 2
d µ
ǫ,t
d t ≤ C
3
r τǫm
2
σ with
ξ = ω or ξ = η . For block averages of
η the statement is proven in [FN06], see Lemma 3.4 and Section 6 there, the case of
ω is the same. To estimate the deviation of consecutive block averages, set m = 2l . We see that m =
δ p
σǫ , δ → 0 is the maximal size of the bigger block, but ǫσ → 0 , thus large microscopic block averages can not be replaced with small macroscopic ones: a strong compactness argument in
not available.
3.4. The first relaxation inequality: To complete the evaluation of entropy production, we have to replace ¯
η
l,k
with its empirical estimator F ¯ ω
l,k
.
Lemma 3.6. For all r, τ ≥ 1 we have universal constants l
∈ N and C
4
+∞ such that ǫ
2
X
|k|rǫ
Z
τǫ
Z
¯ η
l,k
− F ¯ ω
l,k
2
d µ
ǫ,t
d t ≤ C
4
rτσ β l
2
+ r
τǫl
2
σ
whenever τ, r ≥ 1 and ǫl
3
≥ σ ≥ l ≥ l .
Proof. Let us consider the evolution of ¯
Ht, ψ :=
Z
∞ −∞
ψt, xH¯ v
ǫ
t, x d x along the modified empirical process
¯ v
ǫ
t, x = ¯ ρ
ǫ
t, x, ¯ u
ǫ
t, x :=
¯
η
l,k
tǫ, ¯ ω
l,k
tǫ
if |x − ǫk| ǫ2 ,
where 0 ≤ ψ ∈ C
1 c
R
2
, and Hρ, u := 12 ρ − Fu
2
is our Liapunov function. Of course, the evolution of ¯
v
ǫ
is governed by L , thus the Kolmogorov equation yields
¯ Ht,
ψ = ¯ H0,
ψ + ¯It, ψ
′ t
+ 1
ǫ Z
t
L ¯ Hs,
ψ ds + ¯ M
ǫ
t, ψ , 246
where ¯ M
ǫ
t, ψ is a martingale and It, ψ
′ t
:= R
t
¯ Hs,
ψ
′ t
ds , whence E ¯
H ∞, ψ = 0 = E ¯
H0, ψ + E¯I∞, ψ
′ t
+ 1
ǫ Z
∞
E L ¯
Ht, ψ d t .
3.6 Here, and also later on, most calculations are done at the microscopic level. Since ¯
v
ǫ
is a step function, the integral mean,
ψ
k
t := 1
ǫ Z
ǫk+ǫ2 ǫk−ǫ2
ψǫt, x d x of
ψ appears in such expressions. We write V
k
t := V ¯ η
l,k
t, ¯ ω
l,k
t whenever V is a function of the empirical process; for instance H
′ u,k
t = H
′ u
¯ v
ǫ
ǫt, ǫk . In the forthcoming calculations notation and facts from Sections 1.2 and 1.3 may be used without any reference.
The first and second terms, E ¯ H0,
ψ and E¯I∞, ψ
′ t
on the right hand side of 3.6 are bounded; to evaluate the third one, let us consider its decomposition:
1 ǫ
Z
∞
L ¯ Ht,
ψ d t = ¯L
o ǫ
ψ + σ¯L
s ǫ
ψ + β ¯L
∗ ǫ
ψ , cf.
L = L
o
+ βG
∗
+ σS , where ¯L
o ǫ
ψ := ǫ X
k ∈Z
Z
∞
ψ
k
tL
o
H
¯ η
l,k
t, ¯ ω
l,k
t
d t , ¯L
s ǫ
ψ and ¯L
∗ ǫ
ψ are defined analogously: we have to replace L
o
by S or G
∗
, respectively. L
o
H ¯ η
l,k
, ¯ ω
l,k
= X
b ∈Z
∗
c
b
ω H ¯
η
b l,k
, ¯ ω
b l,k
− H
¯ η
l,k
, ¯ ω
l,k
;
¯ ω
b l,k
and ¯ η
b l,k
are block averages of the sequences ω
b j
and η
b j
= ω
b 2
j
, respectively. Next we expand
S H by means of the Lagrange theorem. From H ¯
η
b l,k
, ¯ ω
b l,k
− H ¯ η
l,k
, ¯ ω
l,k
= H
′ ρ,k
t ¯ η
b l,k
− ¯ η
l,k
+ H
′ u,k
t ¯ ω
b l,k
− ¯ ω
l,k
+ B
k
¯ η
b l,k
− ¯ η
l,k
, ¯ ω
b l,k
− ¯ ω
l,k
, where B
k
is a quadratic form with bounded coefficients, we get ¯L
s ǫ
= R
s ǫ
+ Q
s ǫ
, R
s ǫ
ψ := ǫ X
k ∈Z
Z
∞
ψ
k
t H
′ ρ,k
t∆
1
¯ η
l,k
+ H
′ u,k
t∆
1
¯ ω
l,k
d t , and Q
s ǫ
ψ is a quadratic form of the differences ¯ η
b l,k
− ¯ η
l,k
and ¯ ω
b l,k
− ¯ ω
l,k
. Remember now that G
∗
H ¯ η
l,k
, ¯ ω
l,k
= X
b ∈Z
∗
H ¯ η
b ∗
l,k
, ¯ ω
b ∗
l,k
− H ¯ η
l,k
, ¯ ω
l,k
and H ¯ η
b ∗
l,k
, ¯ ω
b ∗
l,k
− H ¯ η
l,k
, ¯ ω
l,k
= H ¯ η
b ∗
l,k
, ¯ ω
b ∗
l,k
− H ¯ η
b ∗
l,k
, ¯ ω
l,k
+ H ¯ η
b ∗
l,k
, ¯ ω
l,k
− H ¯ η
l,k
, ¯ ω
l,k
, where ¯
ω
b ∗
l,k
and ¯ η
b ∗
l,k
are again block averages, those of ω
b ∗
and η
b ∗
, respectively. Moreover, H ¯
η
b ∗
l,k
, ¯ ω
l,k
−H ¯ η
l,k
, ¯ ω
l,k
= H
′ ρ
¯ η
l,k
, ¯ ω
l,k
¯ η
b ∗
l,k
− ¯ η
l,k
+ 12 ¯ η
b ∗
l,k
− ¯ η
l,k 2
247
and H
′ ρ
ρ, u = ρ − Fu , finally from 1.12 G
∗
¯ η
l,k
= X
b ∈Z
∗
¯ η
b ∗
l,k
− ¯ η
l,k
= 1
l X
b ⊂[k,k+l]
c
+ b
ω − c
× b
ω + c
+ b
−
ω − c
× b
−
ω
. Having in mind also Lemma 3.3, we now decompose ¯L
∗ ǫ
as ¯L
∗ ǫ
= G
u ǫ
+ Q
ρ ǫ
+ Γ
ρ ǫ
+ R
ρ ǫ
, where G
u ǫ
ψ := X
k ∈Z
ǫ X
b ∈Z
∗
Z
∞
ψ
k
t H ¯
η
b ∗
l,k
, ¯ ω
b ∗
l,k
− H ¯ η
b ∗
l,k
, ¯ ω
l,k
,
Γ
ρ ǫ
ψ := ǫ X
k ∈Z
Z
∞
ψ
k
t
¯ η
l,k
− F ¯ ω
l,k
C
∗ l,k
ω d t , C
∗ l,k
ω := 2
l − 1
X
b ⊂[k,k+l
c
+ b
ω − c
× b
ω
, R
ρ ǫ
ψ := ǫ X
k ∈Z
Z
∞
ψ
k
t
¯ η
l,k
− F ¯ ω
l,k
G
∗
¯ η
l,k
− C
∗ l,k
ω d t ,
finally Q
ρ ǫ
ψ is the contribution of the squared differences 12 ¯ η
b ∗
l,k
− ¯ η
l,k 2
. Γ
ρ ǫ
ψ is the critical term here, it looks much larger than the others: its order seems to be 1ǫ . However, due to some cancelations we have
− EΓ
ρ ǫ
ψ ≤ Kψ 1 +
β ǫβ l
+ σ
ǫβ l
2
≤ Kψ 1
ǫl +
2 σ
ǫβ l
2
, 3.7
where K ψ +∞ is a constant depending only on ψ . To prove this, observe first that
¯ ω
b l,k
− ¯ ω
l,k
= ¯ η
b l,k
− ¯ η
l,k
= ¯ ω
b ∗
l,k
− ¯ ω
l,k
= 0 unless b = k
− 1, k or b = k + l − 1, k + l , while any of | ¯ ω
b l,k
− ¯ ω
l,k
| , | ¯ η
b l,k
− ¯ η
l,k
| , | ¯ ω
b ∗
l,k
− ¯ ω
l,k
| and
| ¯ η
b ∗
l,k
− ¯ η
l,k
| is bounded by 2l , finally ¯ η
b ∗
l,k
− ¯ η
l,k
= 0 if b ⊂ −∞, k or b ⊂ [k + l, +∞ . The derivation of 3.7 reduces to these deterministic bounds by simple computations.
Indeed, ¯ H
ǫ
0, ψ and ¯ H
ǫ
∞, ψ
′ t
are uniformly bounded, while ¯L
o ǫ
ψ = O 1ǫl , G
u ǫ
ψ = O 1ǫl , Q
s ǫ
ψ = O ǫ
−1
l
−2
, Q
ρ ǫ
ψ = O 1ǫl and R
ρ ǫ
ψ = O 1ǫl . In the case of R
s ǫ
we do discrete integration by parts to get
R
s ǫ
ψ : = −ǫ X
k ∈Z
Z
∞
∇
1
ψ
k
tH
′ ρ,k
t ∇
1
¯ η
l,k
t d t − ǫ
X
k ∈Z
Z
∞
∇
1
ψ
k
tH
′ u,k
t ∇
1
¯ ω
l,k
t d t , where
∇
1
ξ
k
= ξ
k+1
− ξ
k
, whence R
s ǫ
ψ = O 1ǫl
2
. Summarizing these computations we get 3.7 because l
≤ σ and β ≤ σ . Remember now again that
λ
ρ,u
C
∗ l,k
= 2Cρ, u = 32ρ − F
∗
uρ − Fu , 248
and ρ − F
∗
u ≤ −23 . That is why we set W
ρ ǫ
ψ : = 3
ǫ 2
X
k ∈Z
Z
∞
ψ
k
t
¯ η
l,k
− F
∗
¯ ω
l,k
¯
η
l,k
− F ¯ ω
l,k
2
d t ≤ −ǫ
X
k ∈Z
Z
∞
ψ
k
t
¯ η
l,k
− F ¯ ω
l,k
2
d t , and consider 2Γ
ρ ǫ
− W
ρ ǫ
= 2Γ
ρ ǫ
− 2W
ρ ǫ
+ W
ρ ǫ
. From 2Γ
ρ ǫ
− 2W
ρ ǫ
= 2ǫ X
k ∈Z
Z
∞
ψ
k
t
¯ η
l,k
− F ¯ ω
l,k
D
k
t d t , where D
k
:= C
∗ l,k
ω − 2C ¯ η
l,k
, ¯ ω
l,k
we obtain 2Γ
ρ ǫ
− W
ρ ǫ
≤ ǫ X
k ∈Z
Z
∞
ψ
k
tD
2 k
t d t . Now we are in a position to apply Lemma 3.3 to conclude 2EΓ
ρ ǫ
ψ − EW
ρ ǫ
ψ ≤ K
′
ψl
2
σ , whence by 3.7
−EW
ρ ǫ
ψ ≤ ¯ K
ψ 1
ǫl +
2 σ
ǫβ l
2
+ l
2
σ
≤ 2 ¯ K
ψ
σ ǫβ l
2
+ l
2
σ
, where K
′
ψ and ¯ K
ψ depend only on ψ in a simple way. Choosing ψ such that ψt, x = 1 if ≤ t ≤ τ and |x| ≤ 1 + r , while ψt, x = 0 if t τ + 1 and |x| 2 + r , we obtain the statement
of the lemma by a direct computation. The full power of our tools has not been exploited in the proof of 3.7, it is possible to improve
Lemma 3.6, thus also Theorem 1.1 a bit. Namely, the lower bound ǫσ
2
ǫβ
2
ǫ → +∞ can be replaced with
ǫσ
2
ǫβ
3
ǫ → +∞ as ǫ → 0 . We do not go into details because by means of a clever Lax entropy - flux pair a much better condition,
σǫβǫ → +∞ can be proven, see Sections 5.5 and 5.6.
4 Estimation of Entropy Production
The main part of this section is devoted to the verification of the conditions of Proposition 2.2, first of all the components L
ǫ
, M
ǫ
, J
ǫ
and N
ǫ
of entropy production X
ǫ
have to be evaluated. We are assuming that h, J
∈ C
2
R with bounded first and second derivatives; J
′
u = F
′
u − 2uh
′
u is the relation of h and J . Our calculations are based on the a priori bounds of Section 3, the
argument follows [FN06] with some modifications. Although h and J here are now functions of ˆ
ω , considerable differences are coming only from the replacement of αG
κ
with βG
∗
. Spin - flips are simple because
G
κ
η
k
= 0 , and the linear expression, G
κ
ω
k
= −2κη
k
− ω
k
is also easily controlled;
G
∗
η
k
and G
∗
ω
k
are more complicated, cf. Section 1.3. Nevertheless, the main lines of our computations are essentially the same as in [FN06], just Lemma 3.6 of this paper is used instead
249
of Lemma 3.5 of [FN06] at the final evaluation of entropy production. Among others, we have to show that due to reversibility of
G
∗
, the microscopic current j
ω∗
vanishes in the limit. The a priori bounds 2.8 and 2.9 we need for compensated compactness are localized by a smooth
function φ ∈ C
2 co
R
2 +
of compact support, thus H
ǫ
0, φψ, h = 0 , see 2.6 also for the basic decomposition of X
ǫ
ψ, h . Since ˆ u
ǫ
t, x is a step function of x ∈ R , the integral mean, ψ
k
t := 1
ǫ Z
ǫk+ǫ2 ǫk−ǫ2
ϕǫt, x d x of
ϕt, x := φt, xψt, x appears quite frequently in our equations. Finally,
∇
ǫ
ϕx := ǫ
−1
ϕx + ǫ − ϕx for functions, while in the case of sequences we write ∇
l
ξ
k
:= l
−1
ξ
k+l
− ξ
k
, ∇
∗ l
ξ
k
:= l
−1
ξ
k −l
− ξ
k
, and ∆
l
ξ
k
:= −∇
∗ l
∇
l
ξ
k
. Note that ∇
∗ l
is the adjoint of ∇
l
in ℓ
2
Z , ∇
1
ˆ ξ
l,k
= ∇
l
¯ ξ
l,k+1 −l
and ∇
∗ 1
ˆ ξ
l,k
= ∇
∗ l
¯ ξ
l,k
. For ∇
1
ψ
k
we have an identity: ∇
1
ψ
k
t = 1
ǫ Z
ǫ −ǫ
ǫ − |x|ϕ
′ x
ǫt, ǫk + x + ǫ2 d x , where
ϕ = φψ , whence by the Schwarz inequality ∇
1
ψ
k
t
2
≤ 2
ǫ 3
Z
ǫk+3ǫ2 ǫk−ǫ2
ϕ
′2 x
ǫt, x d x . A similar bound of
∇
l
ψ
k 2
follows easily because ∇
l
ψ
k
= ∇
1
¯ ψ
l,k
, thus ∇
l
ψ
k
t
2
≤ 1
l
k+l −1
X
j=k
ψ
j+1
t − ψ
j
t
2
. Such estimates are frequently used in the following calculations to obtain bounds in terms of
kψk
+1
. From now on we are assuming that the parameters
σǫ , βǫ and lǫ of our problem are specified as in Theorem 1.1 and before 2.2.
4.1. The numerical error: This is the easiest case, by a direct calculation
