Weak coupling limits in a stochastic model of heat conduction

0.15 MB·English

by  Frank Redig

Weak coupling limits in a stochastic model of heat
conduction
1
1
0
Frank Redig(a), Kiamars Vafayi(b)
2
a) IMAPP, Radboud Universiteit Nijmegen
n
a Heyendaalse weg 135, 6525 AJ Nijmegen, The Netherlands
J
[email protected]
4
(b) Mathematisch Instituut Universiteit Leiden
1
Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

]
h [email protected]
c
e January 17, 2011
m
-
t
a
t Abstract
s
.
t We study the Brownian momentum process, a model of heat con-
a
m duction, weakly coupled to heat baths. In two different settings of
weak coupling to the heat baths, we study the non-equilibrium steady
-
d state and its proximity to the local equilibrium measure in terms of
n the strength of coupling. For three and four site systems, we obtain
o the two-point correlation function and show it is generically not mul-
c
tilinear.
[

1
Keywords: weakcouplinglimit,localequilibrium,Brownianmomen-
v
tum process, inclusion process, duality.
7
7
8
2 1 Introduction
.
1
0 In the study of non-equilibrium systems, exactly solvable models can serve
1
as test-cases with which general statements about non-equilibrium, such as
1
: in [3], [11] can be tested. Recently, in [6], [7], [8], we studied the Brownian
v
momentum process (BMP) and showed that this models is exactly solvable
i
X
via duality with a particle system, the symmetric inclusion process. In
r this paper, we look at the close-to-equilibrium states of the BMP. First,
a
we consider a close-to-equilibrium scenario where the temperature of the
right heat bath is close to the temperature of the left heat bath, and show
that the distance between the local equilibrium measure and the true non-
equilibrium steady state is of order at most the square of the temperature
difference, in agreement with the theory of Mc Lennan ensembles, see [11].
Next, we consider a situation where the linear chain is coupled weakly to
1
heatbathstoleftandrightends(withfixedanddifferenttemperatures),and
study which equilibrium measure is selected in the limit where the coupling
strength λ tends to zero, as well as how far the true non-equilibrium steady
stateisfromthelocalequilibriummeasureforsmallcouplingstrengths. The
temperature profile can be computed for all values of λ and is only linear in
the chain including the extra sites associated to the heat baths for λ = 1,
and linear if these sites are not included for all values of λ > 0. Finally,
we explicitly compute the two-point correlation for all λ > 0 for a three
and four sites system and show that the multilinear ansatz of the two-point
function introduced in [6], see also [3], [4] fails for a system of four sites,
except when λ = 1.
2 The model
The Brownian momentum process on a linear chain 1,...,N coupled at
{ }
theleftandrightendtoaheatbathisaMarkov process x(t) : t 0 onthe
{ ≥ }
state space Ω = R{1,...,N}. The configuration x(t) = x (t) :i 1,...N is
N i
∈ { }
interpretedasmomentaassociatedtothesitesi 1,...,N . Theprocessis
∈ { }
definedviaitsgeneratorworkingonthecoreofsmoothfunctionsf : Ω R
N
→
which is given by
N
L = λB +λB + p(i,j)L (1)
1 N i,j
i,j
X
with
L = (x ∂ x ∂ )2
i,j i j j i
−
and where ∂ is shorthand for ∂ . The underlying random walk transition
j ∂xj
rate p(i,j) is chosen to be symmetric and nearest neighbor, i.e., p =
i,i+1
p = 1,i 1,...,N 1 , p(i,j) = 0 otherwise. Since L = L the
i+1,i i,j j,i
∈ { − }
symmetry of p(i,j) is no loss of generality.
The boundary operators B ,B model the contact with the heat baths,
1 N
and are chosen to be Ornstein-Uhlenbeck generators corresponding to the
temperatures of the left and right heat bath, i.e.,
B = T ∂2 x ∂
1 L 1 − 1 1
B = T ∂2 x ∂
N R N N N
−
Finally, λ > 0 measures the strength of the coupling to the heat baths.
The process with generator (1) is abbreviated as BMP .
λ
IfT = T = T,then,forallλ > 0, theuniquestationary measureof the
L R
process x(t) : t 0 is the product of Gaussian measures with mean zero
{ ≥ }
and variance T. If T = T there exists a unique stationary measure; the
L R
6
2
so-called non-equilibriumsteadystatedenotedbyµλ . Theexistenceand
TL,TR
uniqueness of the measure µλ follows from duality (see next section).
TL,TR
We will look at two different close-to-equilibrium scenarios:
1. λ = 1, T = T +ǫ and ǫ 0,
R L
→
2. T = T , and λ 0.
L R
6 →
In both cases we look at the behavior of the measure µλ , in case two, as
TL,TR
λ 0, and in case one as ǫ 0. Since for λ = 0, the system has infinitely
→ →
many equilibrium measures, in the second case it is of interest to find out
which of these measure is selected in the limit λ 0. Both in the first
→
and second case, we want to understand how close the true non-equilibrium
steady state is to the local equilibrium measure.
3 Duality
The BMP can be analyzed via duality. The dual process is an interacting
λ
particlesystem,theso-calledsymmetricinclusionprocess[8],whereparticles
arejumpingonthelattice 0,1,...,N,N+1 andinteractingby“inclusion”
{ }
(i.e., particles at site i can attract particles at site j). The “extra sites”
0,N +1 -associated to the heat baths- are absorbing. I.e., a dual particle
configuration is a map
ξ : 0,...,N +1 N
{ } →
specifyingateachsitethenumberofparticlespresentatthatsite. Thespace
of dual particle configurations is denoted by Ωd For ξ Ωd , ξi,j denotes
N ∈ N
the configuration obtained from ξ by removing a particle from i and putting
it at j.
The generator of the dual process then reads
L φ(ξ) =2λξ [φ(ξ1,0) φ(ξ)]+
d 1
−
N−1
+ p(i,j) 2ξ (2ξ +1)[φ(ξj,i) φ(ξ)]+2ξ (2ξ +1)[φ(ξi,j) φ(ξ)]
j i i j
− −
i,j=1
X (cid:0) (cid:1)
+2λξ [φ(ξN,N+1) φ(ξ)] (2)
N
−
In words, this means particles at site i jump to j at rate 2p(i,j)(2ξ +1).
j
At the boundary site 1 (resp. N) particles can jump at rate 2λ to the site 0
(resp. N +1) where they are absorbed. Absorbed particles do not interact
with non-absorbed ones. The dual process is abbreviated as SIP . The
λ
duality functions for duality between BMP and SIP are independent of
λ λ
λ and given by
N x2ξi
D(ξ,x) = Tξ0TξN+1 i
L R (2ξ 1)!!
i
i=1 −
Y
3
for ξ Ωd a dual particle configuration, and x Ω .
∈ N ∈ N
The duality relation then reads
LD(ξ,x) = L D(ξ,x) (3)
d
where L works on x and L on ξ. By passing to the semigroup, from (3) we
d
obtain the duality relation
E D(ξ,x(t)) = EdD(ξ(t),x) (4)
x ξ
where E is expectation in BMP starting from x Ω , and Ed is expec-
x λ ∈ N ξ
tation in SIP starting from ξ Ωd .
λ ∈ N
For ξ Ωd we denote ξ = N+1ξ the total number of particles in
∈ N | | i=0 i
ξ. Since eventually all particles in a particle configuration ξ Ωd will be
P ∈ N
absorbed, we have a unique stationary distribution µλ with
TL,TR
D(ξ,x)µλ (dx) = TkTlPd(ξ(t = )= kδ +lδ ) (5)
TL,TR L R ξ ∞ 0 N+1
Z k,l:k+l=|ξ|
X
where ξ(t = ) denotes the final configuration when all particles are ab-
∞
sorbedandkδ +lδ theconfiguration withk particles at 0andl particles
0 N+1
at N +1.
4 Temperature profile
The local temperature at site i 1,...,N is defined as
∈ { }
T = x2µλ (dx)
i i TL,TR
Z
and by definition T = T ,T = T . We say that the temperature profile
0 L N+1 R
is linear in the lattice interval [K,L] if there exist a,b R with T = ai+b,
i
∈
for all i [K,L]. For the computation of the temperature profile we only
∈
need a single dual walker, which performs a continuous-time random walk
with rates 2p(i,j) and absorption at rate 2λ from the sites 1,N.
Indeed, using (5) we have
T = T Pd (ξ( ) = δ )+T 1 Pd (ξ( ) = δ ) (6)
i L δi ∞ 0 R − δi ∞ 0
(cid:16) (cid:17)
From this expression, one obtains the following equations for the tem-
4
perature profile:
N
p(i,1)T = T λ(T T )
i 1 L 1
− −
i=1
X
N
p(i,k)T = T
i k
i=1
X
N
p(i,N)T = T λ(T T ) (7)
i N R N
− −
i=1
X
The second equation expresses that the temperature profile is a harmonic
function of the transition probabilities, whereas the first and third equation
are boundary conditions. In the case λ = 1 and p corresponding to the
simple nearest neighbor random walk, the equation for T ,i = 0,...,N is
i
the discrete Laplace equation, which gives a linear temperature profile in
[0,N +1].
REMARK 4.1. In this paper we restrict to the symmetric nearest neighbor
walk kernel p(i,j). The equations (7) hold for general symmetric p(i,j).
However, in the cases where it is not translation-invariant and/or not near-
est neighbor, the temperature profile will not be linear.
We have the following theorem that follows immediately from the equa-
tions (7).
THEOREM 4.1. For all λ >0, the temperature profile is linear in [1,N] and
is given by
T = ai+b (8)
i
i= 1,...,N with
λ(T T )
R L
a = −
λ(N 1)+2
−
T +T +λ(NT T )
L R L R
b = −
λ(N 1)+2
−
We can now look at different limiting cases:
1. In the case λ = 1 we recover the result from [6]:
T T
R L
T = T + − i
L
N +1
In this case (only) the temperature profile is linear in [0,N +1].
2. In the limit λ 0 we obtain for all i 1,...,N
→ ∈ { }
T +T
(λ) L R
lim T =
λ→0 i 2
5
3. In the limit λ we obtain T = T ,T = T and the profile is
1 L N R
→ ∞
linear in [1,N], similar to a system with λ = 1 and N 2 sites.
−
4. In the limit N , such that i/N r [0,1] fixed,
→ ∞ → ∈
lim T = T +r(T T )
i L R L
N→∞,i →r −
N
This means that the macroscopic profile is linear and does not depend
on λ.
REMARK 4.2. The expectation of the heat current in the steady state in the
system is J = T T . Heat conductivity κ is defined via the equation J =
i+1 i
−
κ∆T. From Theorem 4.1 it follows that κ = λ which is independent
λ(N−1)+2
of the temperature (i.e. the system obeys the Fourier’s law for all values of
λ > 0).
5 The stationary measure for ǫ 0
→
We consider the first weak coupling setting, i.e, λ = 1, T = T +ǫ. We will
R L
prove that up to corrections of order ǫ2, the stationary measure is given by
a product of Gaussian measures corresponding to the temperature profile,
i.e., the local equilibrium measure.
Let us denote this local equilibrium measure
ν = N G (x )dx
TL,TR ⊗i=1 Ti i i
with T given by (8),
i
1
G (x) = exp( x2/2T)
T
√2πT −
and µ the true non-equilibrium steady state (with λ = 1). Then we
TL,TL+ǫ
have the following result.
THEOREM 5.1. The true equilibriummeasure and the local equilibriummea-
sure are at most order ǫ2 apart, i.e., there exists ǫ > 0 such that for all
0
ξ Ωd there exists a constant C = C(ξ) < such that for all 0 ǫ ǫ
∈ N ∞ ≤ ≤ 0
we have
D(ξ,x)µ (dx) D(ξ,x)ν (dx) C(ξ)ǫ2 (9)
TL,TL+ǫ − TL,TL+ǫ ≤
(cid:12)Z Z (cid:12)
(cid:12) (cid:12)
PROOF.(cid:12) For the local equilibrium measure we have (cid:12)
(cid:12) (cid:12)
N
D(ξ,x)ν (dx) = Tξi (10)
TL,TL+ǫ i
Z i=1
Y
6
expanding this up to order ǫ we find,
ǫi ξi ǫ
Tξi = T + = T|ξ| 1+ iξ +O(ǫ2)
i L N +1 L T (N +1) i
L !
i i (cid:18) (cid:19) i
Y Y X
Start now from (5) and expand up to order ǫ:
D(ξ,x)µ (dx)
TLTL+ǫ
Z
ǫ
= T|ξ| 1+ lPd(ξ( ) = kδ +lδ ) +O(ǫ2) (11)
L  T ξ ∞ 0 N+1 
L
k,l:k+l=|ξ|
X
 
Upon identification of (10) and (11) we see that we have to prove
lPd(ξ( ) = kδ +lδ ) = E (ξ (N +1))
ξ 0 N+1 ξ ∞
∞
k,l:k+l=|ξ|
X
N+1
1
= iξ =:ψ(ξ) (12)
i
(N +1)
i=0
X
The function φ(ξ) := E (ξ (N +1)) is the harmonic function for the dual
ξ ∞
process, i.e.,
L φ= 0
d
which satisfies the boundary conditions
N N
φ kδ + ξ δ +lδ = φ ξ δ +l (13)
0 i i N+1 i i
! !
i=1 i=1
X X
Therefore, it suffices to show that
N+1
1
iξ =:ψ(ξ)
i
(N +1)
i=0
X
both satisfies
L ψ = 0
d
and the boundaryconditions (13). That ψ satisfies the boundaryconditions
is immediately clear. The fact that ψ is harmonic follows from explicit
7
computation:
L ψ(ξ) = 2ξ [ψ(ξ1,0) ψ(ξ)]
d 1
−
N−1
+ 2ξ (2ξ +1)[ψ(ξi+1,i) ψ(ξ)]+2ξ (2ξ +1)[ψ(ξi,i+1) ψ(ξ)]
i+1 i i i+1
− −
i=1
X (cid:0) (cid:1)
+ 2ξ [ψ(ξN,N+1) ψ(ξ)]
N
−
1
= 2ξ [ 1]+
1
N +1 −
(cid:16)
N−1
+ (2ξ (2ξ +1)[ 1]+2ξ (2ξ +1)[+1])
i+1 i i i+1
−
i=1
X
+ 2ξ [+1]
N
(cid:17) N−1
1
= 2ξ [ 1]++2 (ξ ξ )+2ξ [+1]
1 i i+1 N
N +1 − −
!
i=1
X
and since N−1(ξ ξ )= ξ ξ we indeed have
i=1 i− i+1 1− N
P L ψ(ξ) = 0
d
6 The case λ 0
→
Next, we consider the second weak coupling setting, i.e., we fix T = T
L R
6
and study the behavior of the measure µλ as a function of λ.
TL,TR
In this case, the local equilibrium measure is the product of Gaussian
measures correspondingto the temperature profile (8), i.e., we have to com-
pare µλ with νλ where
TL,TR TL,TR
νλ = N G (x )(dx )
TL,TR ⊗i=1 Tiλ i i
where Tλ is given by (8). Denote
i
φ(ξ) = D(ξ,x) µλ (dx) (14)
TL,TR
Z
thenφistheharmonicfunctionofthedualgeneratorsatisfyingtheboundary
conditions
φ(ξ∗ = ξ+kδ +lδ ) = ψ(ξ).ψ(kδ +lδ ) =TkTlψ(ξ)
0 N+1 0 N+1 L R
On the other hand if we put
ψ(ξ) := D(ξ,x) νλ (dx) = TkTl (T(λ))ξi (15)
TL,TR L R i
Z i
Y
8
then we see immediately that ψ satisfies the boundary conditions.
We will now first prove
LEMMA 6.1. There exists λ > 0 such that for all ξ Ωd there exists
0 ∈ N
A(ξ) > 0 such that for all 0< λ λ we have
0
≤
(L ψ)(ξ) λ2A(ξ)
d
| | ≤
In particular, since there is only a finite number of dual particle configura-
tions with total number of particles equal to K, we have, for all 0 < λ λ
0
≤
sup (L ψ)(ξ) C(K)λ2
d
| | ≤
ξ:|ξ|=K
for some C(K)> 0
PROOF. Compute
T T
L R
L ψ(ξ) = 2ψ(ξ) λξ 1 +λξ 1
d 1 N
T − T −
(cid:18) (cid:18) 1 (cid:19) (cid:18) N (cid:19)(cid:19)
N−1
T T
i i+1
+ 2ψ(ξ) ξ (2ξ +1) 1 +ξ (2ξ +1) 1
i+1 i i i+1
T − T −
i=1 (cid:18) (cid:18) i+1 (cid:19) (cid:18) i (cid:19)(cid:19)!
X
Put T T = T T =:γ
R N 1 L
− −
γ γ
L ψ(ξ) = 2ψ(ξ) λξ − +λξ
d 1 N
T T
(cid:18) (cid:18) 1 (cid:19) (cid:18) N(cid:19)(cid:19)
N−1 (T T )2 ξ ξ
i i+1 i+1 i
+ 2ψ(ξ) 2ξ ξ − +(T T ) (16)
i+1 i i i+1
T T − T − T
i=1 (cid:18) i i+1 (cid:18) i+1 i(cid:19)(cid:19)!
X
Remember from Theorem 4.1 that T = λai+b, hence T T = λα,
i i i+1
− −
with
λ(T T )
R L
λα = −
λ(N 1)+2
−
T +T +λ(NT T )
L R L R
b = −
λ(N 1)+2
−
We find
T T
R L
γ = − = α
λ(N 1)+2
−
and hence, from (16)
N−1
α α ξ ξ ξ ξ
L ψ(ξ) = 2ψ(ξ) λξ [− ]+λξ [ ]+ 2λ2α2 i i+1 λα i+1 i
d 1 N
T T T T − T − T
1 N i=1 (cid:18) i i+1 (cid:18) i+1 i(cid:19)(cid:19)!
X
9
We then see that the first order terms form a vanishing telescopic sum:
N−1
ξ ξ ξ ξ
i i+1 1 N
=
T − T T − T
i=1 (cid:18) i i+1(cid:19) 1 N
X
and therefore;
N−1
ξ ξ
L ψ(ξ) = 4λ2a2ψ(ξ) i+1 i
d
T T
i=1 (cid:18) i i+1(cid:19)
X
Given this result, we will prove that the measures νλ and µλ are at
TL,TR TL,TR
most order O(λlog(1/λ)) apart as λ 0.
→
THEOREM 6.1. Let φ,ψ be the functions defined in (14) and (15), then we
have the following. There exists λ > 0, such that for all ξ Ωd there is
0 ∈ N
C(ξ) > 0, such that for all 0 < λ λ
0
≤
1
φ(ξ) ψ(ξ) C(ξ)λlog (17)
| − | ≤ λ
as a consequence,
lim µλ = N G (x )dx
λ→0 TL,TR ⊗i=1 TL+2TR i i
i.e., in the limit λ 0, the equibrium measure corresponding to temperature
→
(T +T )/2 is selected.
L R
PROOF. We start with the following lemma
LEMMA 6.2. For all ξ Ωd a (dual) particle configuration, there exists
∈ N
c = c(ξ) > 0, a = a(ξ) > 0 such that for all λ > 0, and for all t > 0
E D(ξ,x ) νλ (dx) D(ξ,x) µλ (dx) ce−λat
x t TL,TR − TL,TR ≤
(cid:12)Z Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
PROOF. Using duality between BMP and SIP , and (5)
(cid:12) λ λ (cid:12)
E D(ξ,x ) νλ (dx) D(ξ,x) µλ (dx)
x t TL,TR − TL,TR
(cid:12)Z Z (cid:12)
(cid:12) N (cid:12)
(cid:12) (cid:12)
= (cid:12)E (T(λ))ξi(t)Tξ0(t)TξN+1(t) E Tξ0(∞)TξN(cid:12)+1(∞)
(cid:12) ξ i L R !− ξ L R (cid:12)
(cid:12) Yi=1 (cid:16) (cid:17)(cid:12)
(cid:12)(cid:12)C(ξ)Pdξ(ξ(t) = ξ( )) (cid:12)(cid:12)
≤ (cid:12) 6 ∞ (cid:12)
C(ξ)Pd( there exist particles that are not absorbed at time t)
ξ
≤
C(ξ)e−aλt
≤
10