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
N
ǫ
φψ, h = ǫ X
k ∈Z
Z
∞
∇
1
ψ
k
t − ǫ∇
ǫ
ϕǫt, ǫk − ǫ2 J
k
t d t . 4.1
Since J
′
is bounded, without any modification of the argument of the proof of Lemma 4.1 of [FN06], we obtain that the numerical error, N
ǫ
satisfies 2.8 with a vanishing bound.
4.2. The martingale: We estimate the H
−1
norm of M
ǫ
∞, ψ, h as follows. The stochastic dif- ferential dhˆ
u
ǫ
= ǫ
−1
L hˆu
ǫ
+ d m
ǫ
defines a martingale m
ǫ
= m
ǫ
t, x for each x ∈ R such that m
ǫ
t, x = m
ǫ
t, ǫk if |x − ǫk| ǫ2 and M
ǫ
t, φψ, h = Z
∞ −∞
Z
t
ψs, x φs, x m
ǫ
ds, x d x . 4.2
250
The martingale m
ǫ
is identified by the intensity q
ǫ
of its quadratic variation: q
ǫ
t, x : = 1
ǫ
L h
2
ˆ u
ǫ
− 2hˆu
ǫ
L hˆu
ǫ
=
1 ǫ
X
b ∈Z
∗
c
b
ω + σǫ h ˆ
ω
b l,k
− h ˆ ω
l,k 2
+ βǫ
ǫ X
b ∈Z
∗
h ˆ ω
b ∗
l,k
− h ˆ ω
l,k 2
if |x − ǫk| ǫ2 , where ˆ
ω
b ∗
l,k
denotes the block average of ω
b ∗
.
Lemma 4.1. M
ǫ
∞, φψ, h satisfies 2.8 with a vanishing bound. Proof. Let ˙
m
ǫ
t, x denote the time derivative of m
ǫ
in the H
−1
sense, we have to show that E
kϕ ˙ m
ǫ
k
2 −1
→ 0 as ǫ → 0 . Since φ ˙ m
ǫ
= ∂
t
φm
ǫ
− φ
′ t
m
ǫ
in H
−1
, we have |M
ǫ
| ≤ kψk
+1
kφ ˙ m
ǫ
k
−1
≤ kψk
+1
kφm
ǫ
k
2
+ kφ
′ t
m
ǫ
k
2
,
consequently we have to estimate E m
2 ǫ
t, x = Z
t
E q
ǫ
τ, x dτ . However,
| ˆ ω
b l,k
− ˆ ω
l,k
| ≤ 2l
2
and | ˆ
ω
b ∗
l,k
− h ˆ ω
l,k
| ≤ 1l
2
, thus independently of the configuration we have q
ǫ
t, x = O σ + βǫl
3
ǫ , which completes the proof.
4.3. The microscopic current: The starting point of the estimation of L
ǫ
is an identity, L h ˆ
ω
l,k
= h
′
ˆ ω
l,k
L ˆ ω
l,k
+ 1
2 X
b ∈Z
∗
h
′′
˜ ω
b k
c
b
ω + σǫ ˆ ω
b l,k
− ˆ ω
l,k 2
+ βǫ
2 X
b ∈Z
∗
h
′′
˜ ω
b ∗
k
ˆ ω
b ∗
l,k
− ˆ ω
l,k 2
, 4.3
where ˜ ω
b k
and ˜ ω
b ∗
k
are intermediate values. The contribution of the quadratic remainders vanishes in the space of measures in an obvious way, cf. 2.9, because
O l
−2
is the order of both differences, and
σǫl
3
→ 0 . Therefore we are facing with the resultant of
L ˆ ω
l,k
= ∇
∗ l
¯j
ωo l,k
+ βǫ∇
∗ l
¯j
ω∗ l,k
+ σǫ∇
∗ l
¯ ω
l,k
, see Section 1.3 for the definition of currents. Let us consider first the easy case of
L
ωs ǫ
φψ, h := ǫσǫ X
k ∈Z
Z
∞
ψ
k
th
′ k
t∆
1
ˆ ω
l,k
t d t , 4.4
251
it is the contribution of σS . We have L
ωs ǫ
= Y
ωs ǫ
+ Z
ωs ǫ
, where Y
ωs ǫ
φψ, h := −ǫσǫ X
k ∈Z
Z
∞
∇
1
ψ
k
th
′ k
t∇
1
ˆ ω
l,k
t d t ,
Z
ωs ǫ
φψ, h := −ǫσǫ X
k ∈Z
Z
∞
ψ
k+1
t∇
1
h
′ k
t∇
1
ˆ ω
l,k
t d t . Since the entropic Dirichlet form of
S also has a factor σǫ in our fundamental a priori bound, Lemma 3.1, we have
Lemma 4.2. Y
ωs ǫ
satisfies 2.8 with a vanishing bound, while Z
ωs ǫ
satisfies 2.9. The bound of Z
ωs ǫ
does not vanish, and Z
ωs ǫ
≤ 0 if h is convex and φψ ≥ 0 . Proof. It is exactly the same as that of the second part of Lemma 4.2 in [FN06]. First we separate the
factors by means of the Cauchy - Schwarz inequality to let Lemma 3.5 work. For example, suppose that
φ is supported in the rectangle [0, τ] × [−r − 1, r + 1] , then |Y
ωs ǫ
| ≤ σǫkh
′
k p
Ψ
1
Q
1
, where Ψ
1
:= ǫ
2
X
k ∈Z
Z
∞
∇
1
ψ
k
t
2
d t , Q
1
:= ǫ
2
X
|k|rǫ
Z
τǫ
∇
1
ˆ ω
l,k 2
d t . It is plain that Ψ
1
≤ ǫ
2
kφk
2
kψk
2 +1
, while Q
1
= O ǫσ follows by Lemma 3.5 because ∇
1
ˆ ω
l,k
= ∇
l
¯ ω
l,k
. This trick will be used several times in the next coming computations. By means of the one and two blocks estimates the contributions of the microscopic currents j
ωo
and j
ω∗
can be reduced as follows. Let L
ωo ǫ
φψ, h := ǫ X
k ∈Z
Z
∞
ψ
k
th
′ k
t∇
∗ l
¯j
ωo l,k
ωt d t , 4.5
L
ω∗ ǫ
φψ, h := ǫβǫ X
k ∈Z
Z
∞
ψ
k
th
′ k
t∇
∗ l
¯j
ω∗ l,k
ωt d t , 4.6
and introduce their mesoscopic counterparts: V
ωo ǫ
φψ, h := ǫ X
k ∈Z
Z
∞
ψ
k
th
′ k
t∇
∗ l
J
uo
¯ η
l,k
t, ¯ ω
l,k
t d t , 4.7
V
ω∗ ǫ
φψ, h := ǫβǫ X
k ∈Z
Z
∞
ψ
k
th
′ k
t∇
∗ l
J
u ∗
¯ η
l,k
t, ¯ ω
l,k
t d t , 4.8
where J
uo
ρ, u = λ
ρ,u
j
ωo k
and J
u ∗
ρ, u = λ
ρ,u
j
ω∗ k
, see Section 1.3. Now we split the corre- sponding differences by doing discrete integration by parts such that L
ωo ǫ
− V
ωo
= Y
ωo ǫ
+ Z
ωo ǫ
and L
ω∗ ǫ
− V
ω∗
= Y
ω∗ ǫ
+ Z
ω∗ ǫ
, where Y
ωo ǫ
φψ, h := ǫ X
k ∈Z
Z
∞
∇
l
ψ
k
h
′ k
t¯j
ωo l,k
− J
uo
¯ η
l,k
, ¯ ω
l,k
d t , 252
Z
ωo ǫ
φψ, h := ǫ X
k ∈Z
Z
∞
ψ
k+l
t∇
l
h
′ k
¯j
ωo l,k
− J
uo
¯ η
l,k
, ¯ ω
l,k
d t , Y
ω∗ ǫ
φψ, h := ǫβǫ X
k ∈Z
Z
∞
∇
l
ψ
k
h
′ k
t¯j
ω∗ l,k
− J
u ∗
¯ η
l,k
, ¯ ω
l,k
d t , Z
ω∗ ǫ
φψ, h := ǫβǫ X
k ∈Z
Z
∞
ψ
k+l
t∇
l
h
′ k
¯j
ω∗ l,k
− J
u ∗
¯ η
l,k
, ¯ ω
l,k
d t . The following bounds are more or less direct consequences of Lemma 3.3 and Lemma 3.4 or 3.5;
β can not be too large here.
Lemma 4.3. Y
ωo ǫ
and Y
ω∗ ǫ
satisfy 2.8 while Z
ωo ǫ
and Z
ω∗ ǫ
satisfy 2.9; all bounds vanish as ǫ → 0 .
Proof. It follows the argument of the first part of Lemma 4.2 in [FN06]. First we separate ¯j
ωo l,k
− J
uo
¯ η
l,k
, ¯ ω
l,k
and ¯j
ω∗ l,k
− J
u ∗
¯ η
l,k
, ¯ ω
l,k
by means of the Cauchy inequality from their factors, then we can use Lemma 3.3 and Lemma 3.4 or Lemma 3.5 in both cases. The procedure is terminated by
the elementary computation of ∇
l
ψ
k
. We get E
|Y
ωo ǫ
| = kψk
+1
O
φ
l p
ǫσ ,
E |Y
ω∗ ǫ
| = kψk
+1
O
φ
β l p
ǫσ and E
|Z
ωo ǫ
| = kψk O
φ
lσ , while E|Z
ω∗ ǫ
| = kψk O
φ
β lσ , which complete the proof as β l
p ǫσ → 0 and β lσ → 0 as ǫ → 0 , cf. 2.3.
So far we have replaced the dominant parts of the microscopic currents of ˆ ω
k,l
with their canonical expectations, when ¯
η
k,l
and ¯ ω
l,k
are given. The crucial step of the whole proof follows right now, it is the replacement of ¯
η
l,k
with F ¯ ω
l,k
.
4.4. Relaxation in action: As we have indicated above, the last step of the evaluation of entropy production consists in a comparison of J
ǫ
and V
ωo ǫ
, see 2.7 and 4.7. Indeed, as total charge is preserved by the creation - annihilation mechanism, we expect that V
ω∗ ǫ
vanishes in the limit, while J
ǫ
and V
ωo ǫ
cancel each other. By a direct calculation we get J
ǫ
φψ, h = −ǫ X
k ∈Z
Z
∞
ψ
k
t∇
∗ 1
J
k
t d t , 4.9
where J
k
t = J ˆ ω
l,k
t , h
′ k
= h
′
ˆ ω
l,k
t and J
′
u = h
′
u f
′
u with f u := F u − u
2
, see 1.6 for the definition of F . We are going to replace J
k −1
− J
k
with h
′ k
∇
∗ l
f ¯ ω
l,k
. We have J
k −1
− J
k
− h
′ k
∇
∗
f ¯ ω
l,k
= h
′ k
f
′ k
∇
∗ l
¯ ω
l,k
− h
′ k
f
′
˜ ω
l,k
∇
∗ l
¯ ω
l,k
= h
′ k
f
′′
˜ ω
′ l,k
ˆ ω
l,k
− ˜ ω
l,k
∇
∗ l
¯ ω
l,k
with some intermediate values ˜ ω
l,k
and ˜ ω
′ l,k
such that the quadratic remainders on the right hand side can be neglected. More precisely, the contribution of these remainders to J
ǫ
satisfies 2.9 with a vanishing bound; this follows by Lemma 3.4 and Lemma 3.5 in the usual way.
Therefore J
jo ǫ
φψ, h := −ǫ X
k ∈Z
Z
∞
ψ
k
th
′ k
t∇
∗ l
f ¯ ω
l,k
t d t , 4.10
253
is the essential component of J
ǫ
, we split J
jo
− V
ωo ǫ
into two parts: Y
jo ǫ
φψ, h := ǫ X
k ∈Z
Z
∞
∇
l
ψ
k
h
′ k
t
J
uo
¯ η
l,k
, ¯ ω
l,k
− f ¯ ω
l,k
d t ,
Z
jo ǫ
φψ, h := ǫ X
k ∈Z
Z
∞
ψ
k+l
t∇
l
h
′ k
J
uo
¯ η
l,k
, ¯ ω
l,k
− f ¯ ω
l,k
d t ,
i.e. J
jo
− V
ωo ǫ
= Y
jo ǫ
+ Z
jo ǫ
. Similarly, V
ω∗
= Y
j ∗
ǫ
+ Z
j ∗
ǫ
, where Y
j ∗
ǫ
φψ, h := ǫβǫ X
k ∈Z
Z
∞
∇
l
ψ
k
h
′ k
tJ
u ∗
¯ η
l,k
, ¯ ω
l,k
d t ,
Z
j ∗
ǫ
φψ, h := ǫβǫ X
k ∈Z
Z
∞
ψ
k+l
t∇
l
h
′ k
J
u ∗
¯ η
l,k
, ¯ ω
l,k
d t . Since J
uo
ρ, u − f u = ρ − Fu and J
u ∗
ρ, u = −34ρ − Fuρ − F
∗
u , see Section 1.3, we are now in a position to apply Lemma 3.6.
Lemma 4.4. Y
jo ǫ
and Y
j ∗
satisfy 2.8, while Z
jo ǫ
and Z
j ∗
satisfy 2.9; all bounds do vanish. Proof. In much the same way as in the proof of Lemma 4.3, by means of Lemma 3.6 we get
E |Y
jo ǫ
| = kψk
+1
O
φ
p σβ l
2
+ ǫl
2
σ and
E |Y
i ∗
ǫ
| = kψk
+1
O
φ
p βσl
2
+ ǫβ
2
l
2
σ .
Finally, as in the case of Z
ω ǫ
, Lemma 3.4 and Lemma 3.6 imply E
|Z
jo ǫ
| = kψk O
φ
p ǫβ l
2 −1
+ l
2
σ
2
and E
|Z
i ∗
ǫ
| = kψk O
φ
p βǫl
2
+ β
2
l
2
σ
2
. Therefore we need
lim
ǫ→0
σ β l
2
= lim
ǫ→0
βσ l
2
= lim
ǫ→0
ǫβ
2
l
2
σ = lim
ǫ→0
1 ǫβ l
2
= lim
ǫ→0
β ǫl
2
= lim
ǫ→0
β
2
l
2
σ
2
= 0 , which complete the proof as l
2
≈ σ p
ǫ , see also 2.2 and 2.3. The results of this section can be summarized as follows. We have decomposed entropy production
X
ǫ
ψ, h in a correct way, therefore Proposition 2.2 applies. Apart from Z
s ǫ
, all terms of the decom- position vanish, while Z
s ǫ
ψ, h ≤ 0 if h is convex and ψ ≥ 0 , thus lim sup X
ǫ
ψ, h ≤ 0 in probability as
ǫ → 0 holds true in this case.
254
Proof of Theorem 1.1: First we prove Proposition 2.1. In fact we have to evaluate X
ǫ
ψ, h when hu = u , which is easy. Non - gradient analysis is not needed at all, and Z
s ǫ
is missing from the decomposition of X
ǫ
. For ψ ∈ C
1 c
R
2
by Kolmogorov lim
t →∞
Z
∞ −∞
ψt, xˆ u
ǫ
t, x d x = 0 = Z
∞ −∞
ψ0, xˆ u
ǫ
t, x d x + M
ǫ
∞, ψ +
Z
∞
Z
∞ −∞
ψ
′ t
t, xˆ u
ǫ
t, x d x d t + Z
∞
Z
∞ −∞
∇
l ǫ
ψt, x ¯j
ǫ
t, x d x d t , where M
ǫ
is the terminal value of a martingale, and ¯j
ǫ
is a block average, ¯j
ǫ
t, x := ¯j
ωo l,k
tǫ + βǫ¯j
ω∗ l,k
tǫ + σǫ¯j
ωs l,k
tǫ if |x − ǫk| ǫ2 , see Section 1.3. In view of Lemma 4.2, EM
2 ǫ
→ 0 as
ǫ → 0 , and the replacement of ¯j
ǫ
with f ¯ u
ǫ
is a consequence of Lemma 3.3 and Lemma 3.6; remember that f u = F u
− u
2
is just the flux of 1.16. Finally, Lemma 3.4 implies that ¯ u
ǫ
− ˆu
ǫ
also vanish in the limit, thus we have E |X
ǫ
ψ, u| → 0 as ǫ → 0 , which completes the proof because the distributions ˆ
P
ǫ,θ
of the Young measures form a tight family. Now we are in a position to finish the proof of Theorem 1.1. Proposition 2.1 and Proposition
2.3 imply that any limit distribution, ˆ P
θ
of the Young measures is concentrated on a set of weak solutions. On the other hand, lim sup X
ǫ
ψ, h ≤ 0 in probability if ψ ≥ 0 and h is convex, thus first we get 2.11, whence Proposition 2.3 yields 1.23 almost surely with respect to any ˆ
P
θ
. Therefore the uniqueness of the limiting solution follows by the Main Theorem of [CR00] on uniqueness of
entropy solutions. Since the limit is deterministic, for ψ ∈ C
c
R
2
we have lim
ǫ→0
Z
t
Z
∞ −∞
ψt, xˆ u
ǫ
t, x d x d t = Z
t
Z
∞ −∞
ψt, xut, x d x d t in probability. The space integral on the left hand side is actually a sum, thus the block average can
be transposed on ψ , which completes the proof of Theorem 1.1 because ψ is uniformly continuous.
5 Concluding Remarks
Here we summarize some improvements and explanations of our main result including further re- marks on the method of relaxation schemes.
5.1. Strong convergence: The last step of the argument yields a stronger form of Theorem 1.1, cf. [FT04]. Let ¯
u
ǫ
denote the empirical process of Section 3.4, ¯ u
ǫ
t, x = ¯ ω
l,k
tǫ if |x − ǫk| ǫ2 , l = l
ǫ as in 2.2. This version is certainly more natural than ˆ u
ǫ
, which has been introduced because of technical reasons: l
∇
∗ 1
ˆj
ωo
is a difference of block averages, and ¯j
ωo
is well controlled by Lemma 3.3;
∇
∗ 1
¯j
ωo
is more singular. The second statement of the following theorem is a consequence of Lemma 3.4.
Theorem 5.1. Under conditions of Theorem 1.1, for τ, r 0 we have
lim
ǫ→0
E Z
τ
Z
r −r
|ut, x − ˆu
ǫ
t, x| d x d t = lim
ǫ→0
E Z
τ
Z
r −r
|ut, x − ¯u
ǫ
t, x| d x d t = 0 ,
255
where u denotes the unique entropy solution to 1.16 with initial value u .
5.2. Microscopic block averages: By means of Lemma 3.5 we can fill in the gap between large microscopic and small macroscopic block averages of the evolved configuration, see [FN06]. Let
¯ u
ǫ,l
t, x := ¯ ω
l,k
tǫ if |x − ǫk| ǫ2 , where l ∈ N does not depend on ǫ , then for all τ, r 0 we have
lim
l →∞
lim sup
ǫ→0
E Z
τ
Z
r −r
|¯u
ǫ
t, x − ¯u
ǫ,l
t, x| d x d t = 0 , whence
lim
l →∞
lim sup
ǫ→0
E Z
τ
Z
r −r
|ut, x − ¯u
ǫ,l
t, x| d x d t = 0 .
5.3. Measure - valued solutions: Convergence of the empirical process ¯ u
