József Fritz
2. The anharmonic chain
2.3. Derivation of the Euler equations in a smooth regime
Here we are going to outline Yau’s method for the anharmonic chain with random exchange of velocities. The argument is similar but much more transparent than that of Olla–Varadhan–Yau (1993). The derivation of (2.8) is easier, its main steps are also included in the next coming calculations. Since the noise is not strong enough to control the flux of the relative entropy, we have to formulate the problem in a periodic setting: pk(0) =pk+n(0)andrk(0) =rk+n(0)for allk with somen∈N. The evolved configuration remains periodic for all times, which means that the system can be considered on the discrete circle of length n→+∞. The coefficientσ >0can be kept fixed during the procedure of scaling because the only role of the exchange mechanism is to ensure the strong ergodic hypothesis. At a levelε= 1/n of scaling letµt,n denote the evolved measure, and consider thelocal equilibrium distributions λt,n of type λβ,π,γ with parameters depending on space and time: β =βk(t, n),π=πk(t, n)andγ=γk(t, n).
Theorem 2.4. Suppose that (1/n)S[µ0,n|λ0,n] →0 asn →+∞, and the related initial values determine a smooth solution(u, v, h)to (2.5)on the interval[0, T]of time such that β=−Sχ0(χ, v)remains strictly positive. Then
n→∞lim Z∞
−∞
ψ(x)zn(t, x)dx= Z∞
−∞
ψ(x)z(t, x)dx
in probability for all continuous ψ with compact support if t ≤ T, where (zn, z) is any of the couples (un, u), (vn, v), (hn, h), and un(t, x) := pk(tn), vn(t, x) :=
rk(tn),hn(t, x) :=Hk(tn) if|k−xn|<1/2.
In view of the argument we have sketched before Theorem 2.3, we have to show that if the parameters of λt,n are defined by means of the smooth solution, then (1/n)S[µτ n,n|λτ n,n] → 0 as n → +∞ for all τ ≤ T, consequently the empirical processes converge in a weak sense to that solution of (2.5).
Calculation of entropy: Letft,n:=dµt,n/dλt,n and consider the time evolution of S[µt,n|λt,n] =R
logft,ndµt,n. In the next coming calculations we are assuming that the evolved densityft,n(ω)>0 is a continuously differentiable function. This hypothesis can be relaxed by means of a standard regularization procedure, see e.g. Fritz–Funaki–Lebowitz (1994). The required regularity of the parameters is a consequence of their construction via discretizing the macroscopic system (2.5).
By a formal computation
∂tS[µt,n|λt,n] = Z
(∂t+L0+σGep) logft,ndµt,n≤ Z
(∂t+L0)ft,ndλt,n
because R
ft,ndλt,n ≡1, L0logft,n = (1/ft,n)L0ft,n, and the contribution of Gep is certainly not positive. Moreover, as L0 is antisymmetric with respect to the Lebesgue measure, we have
Z
(∂tft,n+ft,n∂tloggt,n)dλt,n = Z
(L0ft,n+ft,nL0loggt,n)dλt,n= 0,
wheregt,n denotes the Lebesgue density ofλt,n, consequently
S[µt,n|λt,n]≤S[µ0,n|λ0,n]− Zt
0
Z
(∂s+L0) loggs,ndµs,nds. (2.10) On the other hand, as
loggs,n=
n−1X
k=0
(γkrk−βkIk−F(βk, γk)),
whereIk:=I(pk, rk|πk) = (pk−πk)2/2 +V(rk), by a direct calculation we obtain that
∂sloggs,n=
n−1
X
k=0
γ˙k(rk−vk) +βkπ˙k(pk−πk)−β˙k(Ik−χk) , where "dot" indicates differentiation with respect to time.
There is a fundamental relation between the parametersβ, π, γ ofλn,t, namely
n−1X
k=0
((γk−1−γk)πk+ (βkπk−βk+1πk+1)Jk+ (βk+1−βk)πk+1Jk) = 0.
As it is explained by Tóth–Valkó (2003), this identity is due to the conservation of the thermodynamic entropy in a smooth regime, which is a basic feature of all models with a proper physical motivation. On the other hand, it is a necessary requirement when we evaluate the rate of production of S in order to conclude (2.9). Indeed, we get
L0loggs,n=
n−1X
k=0
(γk−1−γk)(pk−πk)
+
nX−1 k=0
(βkπk−βk+1πk+1)(V0(rk)−Jk) (2.11)
+
n−1X
k=0
(βk+1−βk)(pk+1V0(rk)−πk+1Jk), where vk := R
rkdλt,n, πk := uk = R
pkdλt,n and χk := R
Ikdλt,n, finallyJk = J(χk, vk) =γk/βk:=R
V0(rk)dλt,n. Notice that the local equilibrium mean of any of the last factors on the right hand sides of (2.11) above does vanish: for instance R(V0(rk)−Jk)dλt,n= 0.
The crucial step: The microscopic time t is as big as t = nτ, thus there is a danger of explosion on the right hand side of (2.10) as n →+∞. However, due to the smoothness of the macroscopic solution, the nonlinear functions appearing in the sums above can be substituted by their block averages, and the celebrated
One-block Lemma, which is the main consequence of strong ergodicity, allows us to approximate the block averages by theircanonical equilibrium expectations, see Lemma 3.1 in Guo–Papanicolau–Varadhan (1988) or Theorem 3.5 of Fritz (2001).
The wave equation: The case of (2.8) is quite simple becauseβk ≡1 then, thus Vk0 = V0(rk) is the only nonlinear function we are facing with. Block averages
¯
ηl,k:= (1/l)(ηk+ηk−1+· · ·+ηk−l+1) of sizel ∈Nare also periodic functions of k∈Zwith periodn. SinceR
Vk0dλ1,π,γ=S0(vk) =Jkifvk is the local equilibrium mean of rk, V¯l,k0 ≈S0(¯rl,k) is the desired substitution, which is valid asl →+∞ after n → ∞. Presupposing |πk+1−πk| =O(1/n)and |vk+1−vk|=O(1/n) we write
n−1X
k=0
(πk−πk+1)(V0(rk)−S0(vk))≈
n−1X
k=0
(πk−πk+1)( ¯Vl,k0 −S0(vk))
≈
n−1
X
k=0
(πk−πk+1)(S0(¯rl,k)−S0(vk))≈
n−1
X
k=0
(πk−πk+1)S00(vk)(rk−vk).
The remainders including the squared differences coming from the expansion of S0(¯rl,k)−S0(vk) are estimated by means of the basic entropy inequality and the related large deviation bound; let us omit these technicalities. Comparing the leading terms we see that
˙
γ=S00(vk)(πk+1−πk) and π˙k=γk−γk−1
is the right choice of the parameters because then there is a radical cancelation on the right hand side of (2.10). Since γk =S0(vk), this system is just a lattice approximation to (2.8), thus our conditions on the regularity of the parameters are also justified. Summarizing the calculations above, we get a bound
S[µτ,n|λt,n]≤S[µ0,n|λ0,n] +K n
Zt 0
S[µs,n|λs,n]ds+Rn(T, l) (2.12) such that Rn(T, l) →0 as n→ +∞ and then l →+∞, whence S[µτ n,n|λτ n,n] = o(n)follows by the Grönwall inequality ifτ≤T.
The general case: It is a bit more complicated then the case of the p-system, the required substitutions read as
V0(rk)≈J( ¯Il,k,¯rl,k)≈Jk+Jχ0(χk, vk)( ¯Il,k−χk) +Jv0(χk, vk)(¯rl,k−vk), and
pk+1V0(rk)≈p¯l,k+1J( ¯Il,k,r¯l,k)≈πk+1J(χk, vk) +J(χk, vk)(¯pl,k+1−πk+1)
+πk+1Jχ0(χk, vk)( ¯Il,k−χk) +πk+1Jv0(χk, vk)(¯rl,k−vk).
These steps are justified by the strong ergodicity of the dynamics (One-block Lemma), provided thatV0(rk)and πk+1V0(rk)can be replaced by their block av-erages. This second condition turns out to be a consequence of the smoothness of the macroscopic solution, see the construction below. The second order quadratic terms of the expansions above are estimated by means of the entropy inequality, we only need standard large deviation bounds.
To minimizeS[µt,n|λt,n], the parameters ofλt,n should be defined by means of a discretized version of the Euler equations. In fact we set
πk=uk, γk =Sv0(χk, vk), βk=−Sχ0(χk, vk), where
˙
vk =uk+1−uk, u˙k =J(χk+1, vk+1)−J(χk, vk) andχ˙k=J(χk, vk)(uk+1−uk), whence
βkπ˙k = (γk−γk−1) + (βk−1−βk)Jk−1,
˙
γk = (βk+1πk+1−βkπk)Jv0(χk, vk) + (βk−βk+1)Jv0(χk, vk), β˙k = (βkπk−βk+1πk+1)Jχ0(χk, vk) + (βk+1−βk)Jχ0(χk, vk) follow by a direct computation.
As a consequence of these calculations, we see the expected cancelation of the sum of all leading terms on the right hand side of (2.10), while the remainders can be estimated by means of the entropy inequality. The summary of these computations results in (2.12), thus the proof can be terminated as it was outlined in the previous two paragraphs.