FROM GLOBAL TO
LOCAL FLUCTUATION THEOREMS
Christian Maes
,Frank Redig
y
and Michel Verschuere
z
July 2,2001
Abstract:The GallavottiCohen uctuation theorem suggests a general symmetry in
the uctuations of the entropy production,a basic concept in the theory of irreversible
processes,based on results in the theory of strongly chaotic maps.We study this
symmetry for some standard models of nonequilibriumsteady states.We give a general
strategy to derive a local uctuation theorem exploiting the Gibbsian features of the
stationary spacetime distribution.This is applied to spin ip processes and to the
asymmetric exclusion process.
Dedicated in honor of Robert Minlos on the occasion of his 70th birthday.
1 Introduction
A basic feature of equilibrium systems is that the restriction to a subsystem is again
in equilibrium and with respect to the same microscopic interaction,for the same
temperature,pressure and chemical potential.We can imagine cutting out a much
smaller but still macroscopic region from our system and we will still nd the same
equilibrium state apart from possible boundary eects.Mathematically,this is ex
pressed via the DLRequation stating that the local conditional probabilities of a
Gibbs measure coincide with the corresponding nite volume Gibbs measures.This
really amounts to the fact that,for Gibbs measures,the ratio of probabilities for
two dierent microscopic congurations that are identical outside a nite volume,is
given by the Boltzmann factor expH for relative Hamiltonian H depending
continuously on the conguration far out.In other words,relative energies make
sense and they can be written as a sum of suciently local interaction potentials.
Instituut voor Theoretische Fysica,K.U.Leuven,Celestijnenlaan 200D,B3001 Leuven,Belgium
 email:Christian.Maes@fys.kuleuven.ac.be
y
T.U.Eindhoven.On leave fromInstituut voor Theoretische Fysica,K.U.Leuven,Celestijnenlaan
200D,B3001 Leuven,Belgium  email:f.h.j.redig@tue.nl
z
Instituut voor Theoretische Fysica,K.U.Leuven,Celestijnenlaan 200D,B3001 Leuven,Belgium
 email:Michel.Verschuere@fys.kuleuven.ac.be
1
Important consequences are found in the theory of equilibrium uctuations and in
the framework of the theory of large deviations;Robert Minlos was among the very
rst to develop a mathematical theory of this Gibbs formalism.In particular in [1],
the description of the thermodynamic limit of Gibbs measures is given.We indeed
usually have in mind here (very large) spatially extended systems which are moni
tored locally.
The idea that time does not enter this equilibrium description is further stimulated
from the fact that at least in classical statistical mechanics,the momenta (entering
only in the kinetic energy part of the Hamiltonian) can be integrated out at once from
the partition function.Time enters already more explicitly in equilibrium dissipative
dynamics such as via Langevin equations (Markov diusion processes) or Glauber
dynamics (for spin relaxation).Yet,under the condition of detailed balance,the sta
tionary dynamics is microscopically reversible and the past cannot be distinguished
from the future.The equilibrium steady state probability distribution on the space
time histories still has a Gibbsian structure with as extra bonus that the restriction to
a spatial layer (at xed time) is still explicitly Gibbsian and in fact,the restriction of
the dynamics to a subregion still satises the detailed balance condition with respect
to it.It was again Robert Minlos who was among the pioneers in the subject of
spacetime Gibbs measures and who,with his great experience in cluster expansion
techniques and his love for eld theory,saw the advantage of the spacetime approach
in the construction of solutions to innite dimensional Markov diusion processes,
see e.g.[2].
While the condition of detailed balance re ects a symmetry (the spacetime distribu
tion is timereversal invariant),the fact that the spacetime distribution is Gibbsian
(at least in some sense) does however not at all depend on it.In other words,the
fact itself that the spatiotemporal probability distribution enjoys Gibbsianness is
much more general and has nothing to do with microscopic reversibility.This can
be checked readily for probabilistic cellular automata,[11,3],but,more generally,it
is the locality of the spacetime interaction that does the job.In the present paper,
we will exploit this fact in going from a global to a local uctuation theorem for the
entropy production in some models of interacting particle systems.At this moment,
we need a second introduction to write about the statistical mechanics of steady state
entropy production and how its uctuations can give interesting information about
the response of the system to perturbations.We refer to [8] for a recent impression.
As we will see and as introduced in [3],once the conceptual framework and the Gibb
sian basis of the uctuation theorem is understood,the transition from a global to
a local uctuation theorem will be merely a technical matter.Physically speaking
however,it is much better for the obvious reason that global uctuations are far too
improbable to be observed,[3,18,19].
2
1.1 Fluctuation theorem
First observed in [13] and later derived in [14,15,23],the GallavottiCohen uctuation
theorem proves a symmetry in the uctuations in time of the phase space contraction
rate for a class of dynamical systems.The dynamics must obey certain conditions;
it is a reversible smooth dynamical system 7!(); 2 K on a phase space K
that is in some sense bounded carrying only a nite number of degrees of freedom (a
compact and connected manifold).The transformation is a dieomorphism on K.
The resulting (discrete) time evolution is obtained by iteration and the reversibility
means that there is a dieomorphism on K with
2
= 1 and =
1
.It is
assumed that the dynamical system satises some technical (ergodic) condition:it is
a transitive Anosov system.This ensures that the system allows a Markov partition
(and the representation via some symbolic dynamics) and the existence of the SRB
measure ,an invariant measure with expectations
(f) = lim
N
1N
N
X
0
f(
n
) (1.1)
corresponding to timeaverages for almost every randomly chosen initial point 2 K
(i.e.,for an absolutely continuous measure with respect to the Riemann volume ele
ment d on K).The change of variables implied by the dynamics denes the Jacobian
determinant J and one writes
_
S lnJ.This is the phase space contraction rate
which GallavottiCohen identify with the entropy production rate via the following ar
gument:Dene the (Shannon) entropy of a probability distribution m(d) = m()d
on K as
S(m) =
Z
dm() lnm() (1.2)
With m
n
as density at time n,under the dynamics,the density at time n +1 is
m
n+1
(v) =
m
n
(
1
v)J(
1
v)
(1.3)
and the change in this entropy is therefore
S(m
N
) S(m
0
) =
Z
dm
0
()
N1
X
0
lnJ(
n
) (1.4)
Dividing by N and taking N to innity,the empirical probability distribution ap
proaches the SRB distribution ,as in (1.1).Therefore,the timeaveraged change
in the entropy of the imagined reservoir is (
_
S),see also [21,22,23].One further
assumes (and sometimes proves) dissipativity:
(
_
S) > 0:(1.5)
3
One is interested in the uctuations of
s
N
()
1(
_
S)N
N=2
X
N=2
_
S(
n
());(1.6)
in the state .Informally,the uctuation theorem then states that s
N
() has a
distribution
N
with respect to such that
lim
N
1N(
_
S)a
ln
N
(a)
N
(a)
= 1 (1.7)
always.In other words,the distribution of the timeaveraged
_
S over long time in
tervals satises some general symmetry property.A more precise phrasing can be
obtained via large deviation theory.For a continuous time version (Anosov ows) we
refer to [20].
The reason why we are interested in the uctuation theorem is because the es
tablished symmetry in uctuations is very general and it may be important for the
construction of nonequilibriumstatistical mechanics beyond linear order perturbation
theory.Now there are various proposals,ideas and results but,at any rate,whatever
the point of view,it is rather natural asking how to establish a local version of the
uctuation theorem.The title of the present paper refers to that problem with the
understanding that local refers to a spacetime window within a much larger spa
tially and temporally extended nonequilibrium system.This was already the subject
of [3,18].We want to understand how general the local uctuation theorem can be
and what form it takes for some standard nonequilibrium models.It turns out that,
as was explicitly pointed out in [3],it is the Gibssian structure of the spacetime
distribution that allows a local uctuation theorem.This was already apparent from
[18,3] but here we add further systematization and explain and illustrate this local
version of the uctuation theorem.
One further question concerns the physical identication of the quantity for which
we are investigating the symmetry in the uctuations.We will call it entropy pro
duction.This name already exists for a physical quantity that appears in close to
equilibrium thermodynamics,and indeed we believe that our choice of words re
ects a generalization.The basic idea is that nonequilibrium steady states are not
timereversal invariant and that the mean entropy production should give a measure
of discriminating between the original spacetime distribution and its timereversal.
That is the relative spacetime entropy density.For the variable entropy production,
we must look up the source of the timereversal symmetry breaking in the spacetime
interaction.It turns out that once it is recognized that the entropy production is the
antisymmetric part of the spacetime interaction under timereversal,the symmetry
in its local uctuations (as expressed in the local uctuation theorem (LFT)) is al
most an immediate consequence of the Gibbsian structure.This we will show.
Of course,the question remains how we wish to use the local uctuation theorem.
4
That is not the subject of the present paper but we refer to [16,17,23,3,8,12,10]
for some ideas.
1.2 Example
We sketch here the nature of a local versus global uctuation theorem via a sim
ple model.We have in mind a (1 + 1)dimensional Ising spin system with formal
Hamiltonian
H() =
X
x;t
t
(x)[
t+1
(x) +b
t+1
(x +1)] (1.8)
where we think x 2as the spatial coordinate and t 2as the (discrete) time;
t
(x) = 1;b 6= 0.
Look at the function
S
n;t
() = b
T
X
t=T
n
X
x=n
t
(x)[
t1
(x +1)
t+1
(x +1)]
of the spins in a spacetime window parametrized by n;T > 0.We are interested in
its uctuations under the probability laws
P
n
,the Gibbs measure on f1;1g
fn;:::;n+1g with respect to the Hamiltonian
H
n
() =
X
t
n
X
x=n
t
(x)[
t+1
(x) +b
t+1
(x +1)]
and
P,any innite volume Gibbs measure on f1;1g2
for the Hamiltonian (1.8).
In both cases we take the counting measure as reference and set the inverse temper
ature = 1.
The dierence is that P
n
is an Ising model on a onedimensional strip (nite spatial
volume with innite timeextension) and P is the corresponding model for innite
spacetime volume.
We start with the statement of a global uctuation theorem;that concerns the law
P
n
.Consider the involution
n;T
by which all spins inside the window
n;T
=
fn;:::;n+1gfT 1;:::;T +1g are re ected over the t = 0 axis:(
n;T
)
t
(x) =
t
(x) if (x;t) 2
n;T
and remains unchanged otherwise.Remark that
S
n;t
(
n;T
) =
S
n;t
() and upon writing H
n
(
n;T
) H
n
() =
S
n;t
() B
n;T
() we nd,after a
simple calculation,that for every function g
Z
dP
n
()g(
S
n;T
()) =
Z
dP
n
()g(
S
n;T
())e
S
n;T
()+B
n;T
()
(1.9)
5
with jB
n;T
()j cn.
As a result,for xed n,for all functions f,
lim
T
1T
lnj
R
dP
n
()f(
S
n;T
()=T)R
dP
n
()f(
S
n;T
()=T)e
S
n;T
()
j = 0 (1.10)
which implies the symmetry expressed in (1.7) with N = T.
Now to a local uctuation theorem;that concerns the law P.A similar calculation
shows that H(
n;T
) H() =
S
n;t
() B
n;T
() F
n;T
() with
jF
n;T
()j cT
so that
Z
dP()g(
S
n;T
()) =
Z
dP()g(
S
n;T
())e
S
n;T
()+B
n;T
()+F
n;T
()
and we conclude that in both order of limits,
lim
n;T
1 nT
lnj
R
dP()f(
S
n;T
()=(nT))R
dP()f(
S
n;T
()=(nT))e
S
n;T
()
j = 0 (1.11)
This is the same symmetry as in (1.7) but for the local uctuations in a spatially
extended system.Of course,(1.11) involves limits but the basic fact behind (1.11) is
that there is a local function R
n;T
= H
n;T
H,antisymmetric under the time
reversal
n;T
that preserves the a priori reference measure,with jR
n;T
()
S
n;T
()j
c
1
n +c
2
T for which
Z
dP()g(R
n;T
) =
Z
dP()g(R
n;T
)e
R
n;T
()
which is an exact local uctuation symmetry.Various things are lacking from this
example.Mathematically,things will be more complicated when the B
n;T
or F
n;T
are not uniformly bounded or when time is not discrete or when the spacetime
Hamiltonian (1.8) is not local or contains hardcore interactions.Physically,the
example above carries no interpretation of
S
n;T
as entropy production.
1.3 Local uctuation theorem
The main theme of the present paper is a general strategy to nd a local uctuation
theorem for the entropy production in a nonequilibrium steady state,in the context
of stochastic interacting particle systems.To get the idea we present the result infor
mally for a typical application.The details and mathematically precise statements
about this model are given in Section 4.The model is a microscopic version of a
reactiondiusion system where the reaction consists of the birth and death of parti
cles on the sites of a regular lattice and the diusion part lets these particles hop to
6
nearest neighbor vacancies subject to an external eld.
Consider the square lattice2
to each site i of which we assign a variable (i) = 0;1,
meaning that site is empty or occupied by a particle.The conguration can change
in two ways:rst,a particle can be created or destroyed at lattice site i:!
i
where
i
is identical to except that the occupation at the site i is ipped.Secondly,
a particle at i can hop to one of the four nearest neighbor sites j under the condition
that j is empty:!
ij
where
ij
is the new conguration obtained by exchanging
the occupations at sites i and j.We make a nonequilibrium dynamics by adding an
external eld E > 0 which introduces a bias for particle hopping in a certain direc
tion.
In formula,rst,a particle is destroyed or created at any given site at xed rates.
The transition from a conguration to the new
i
takes place at rate
c(i;) =
+
(1 (i)) +
(i)
where
+
is the rate for the transition 0!1 and
is the rate for 1!0.Secondly,
the particles on the lattice undergo a diusive motion.To be specic,we choose a
large square V centered around the origin with periodic boundary conditions and
we rst introduce hopping rates over a nearest neighbor pair hiji in the horizontal
direction,i = (i
1
;i
2
);j = (i
1
+1;i
2
):
c(i;j;) = e
E=2
(i)(1 (j)) +e
E=2
(j)(1 (i))
The hopping rate in the vertical direction is constant (put E = 0 in the above if
j = (i
1
;i
2
1)).Taking E large,we expect to see many more jumps of particles
to the right than to the left.In the absence of reaction rates,that is for
= 0,
we recover the so called asymmetric exclusion process and particle number is strictly
conserved.More generally,the Master Equation is
d
t
() dt
=
X
i
[
t
(
i
)c(i;
i
)
t
()c(i;)] +
X
hiji
[
t
(
ij
)c(i;j;
ij
)
t
()c(i;j;)]
For this model,the stationary measure is the product measure with uniformdensity
equal to
+
=(
+
+
) corresponding to a chemical potential ln
+
=
of the particle
reservoir.
For a xed nearest neighbor pair hiji,with j = (i
1
+ 1;i
2
) to the right of i,the
timeintegrated microscopic current over an interval [T;T] is
J
i
T
N
i!j
T
N
j!i
T
with N
i!j
T
the total number of particles that have passed fromsite i to site j.We have
the convention to take this current positive when the net number of particles jumping
to the right (i.e.,in the direction of the external eld) is positive.Multiplying the
sum of all the current contributions in V with the eld E we get
W
V;T
(
s
;s 2 [T;T]) E
X
i2V
J
i
T
(
s
;s 2 [T;T])
7
which is a random variable representing the work done on our system over the time
interval [T;T].Its expectation in the stationary state equals (up to a temperature
factor) the expected heat dissipated in the environment and is given by
hW
V;T
i = 2TjV j Esinh(E=2)
+
(
+
+
)
2
If we now x another square V inside our large system,then
W
;T
E
X
i2:(i
1
+1;i
2
)2
J
i
T
is the randomvariable\work done on the systemin over the timeinterval [T;T]".
That constitutes the main contribution to the local randomvariable\entropy produc
tion in the spacetime window [T;T]".Yet,this is only its bulk contribution.
We have indeed only included in W
;T
the microscopic currents between the sites
strictly inside while particles will of course also hop in and out of via its bound
ary.In other words,the region V n acts as a particle reservoir from which particles
can enter or leave .That also contributes to the entropy production as,quite gen
erally,the change in entropy in the particle reservoir equals the number of particles
transferred to it,multiplied by its chemical potential.Now usually,this chemical po
tential is xed and constant,i.e.,not depending on whatever happens in the system
itself.Here this is not the case.It suces to imagine that almost all particles are
in fact inside with therefore a low density of particles in V n .As a result,the
eective chemical potential for creating or destroying particles at the boundaries of
will depend on time and on whatever happened inside before that time.Moreover
this will contribute to the nonequilibrium condition only for E 6= 0 because only then
will there be a dierent rate of leaving/entering at the right versus the left vertical
boundaries of .This is not the case for the upper versus the lower boundaries but
also there,even when there would not be a eld strictly inside ,the dynamics inside
will be in uenced by the eld outside.This is summarized in the form of the second
contribution to the timeintegrated entropy production and it is a boundary term:
J
@;T
R
`
+R
r
+R
u
+R
d
where the various terms correspond to the reactions taking place at the left,right,
upper and lower boundaries of the square .We will not write all of them down
explicitly but here is for example
R
r
X
i2:j2V n
X
TtT
t
(i) ln
+q
;t
(j;;E)
+q
;t
(j;;E)
+(1
t
(i)) ln
+
+p
;t
(j;;E)
+
+p
;t
(j;;E)
where j = (i
1
+1;i
2
),the sum over times t is over the times when a particle is created
or destroyed at i,and the rates p and q are given by
q
;t
(j;;E) e
E=2
Prob[
t
(j) = 0j
s
(k);k 2 ;s 2 [T;t]]
8
and
p
;t
(j;;E) e
E=2
Prob[
t
(j) = 1j
s
(k);k 2 ;s 2 [T;t]]
where the probabilities refer to the steady state in V.In other words,the external
eld does not only work on the particles in it also creates a gradient in chemical
potential (large at the left boundary and smaller at the right) in .The total random
variable\entropy production in "now reads
S
;T
W
;T
+J
@;T
The result proved in Section 4 is the uctuation theorem symmetry for
S
;T
:
lim
;T
lim
V
1jjT
ln
Prob[
S
;TjjT
= a]Prob[
S
;TjjT
= a]
= a (1.12)
uniformly in the
.
One may wonder whether the work W
;T
satises a similar uctuation symmetry.
That is (1.12) with W
;T
replacing
S
;T
.It remains uncertain however whether that
is true uniformly in the values
#0 but,as we will show,it remains true whenever
6= 0.
The rest of our paper is organized as follows:in Section 2 we give a general
strategy to obtain LFT,which we apply in Section 3 for spin ip processes and in
Section 4 for the asymmetric exclusion process.
2 Abstract setting
We identify the essential mathematical structure,needed to pass from a global to a
local uctuation theorem.Our later specic illustrations will then just be applica
tions of the same theme.
We consider a measurable space (
;F) on which two sequences of probability mea
sures P
n
and P
r
n
.Suppose that
n
is an involution on
such that P
n
and P
n
n
are mutually absolutely continuous and the same for the pairs P
r
n
and P
r
n
n
.We
write
R
n
ln
dP
n dP
n
n
;F
n
R
n
+ln
dP
r
n
ndP
r
n
then,by denition,for all functions f,
Z
dP
n
f(R
n
) =
Z
dP
n
e
R
n
f(R
n
) (2.13)
9
and
Z
dP
r
n
f(R
n
) =
Z
dP
r
n
e
R
n
+F
n
f(R
n
) (2.14)
The identity (2.13) expresses an exact symmetry in the uctuations of R
n
but should
be compared with the global symmetry (1.9,1.10).The next equality (2.14) is very
similar but there is the correction term F
n
.To get rid of it (at least asymptotically
in n) we need extra assumptions.This will then yield the local uctuation theorem.
Before we give a general way of expressing these assumptions,the reader may appre
ciate some more explication concerning our choice of`global'versus'local'as there
is of course no natural interpretation of this within the proposed abstraction.
As we will see in the next sections,we really start from two measures P and P
n
on
where
will be the pathspace of an (innite volume) interacting particle process
on the ddimensional regular latticed
;P will be an innite volume steady state
measure (i.e.,the pathspace measure of a stationary process over some time interval
[T;T]);n will refer to a nite spacetime volume (corresponding to a sequence of
cubes
n
centered around the origin times the interval [T;T]) and
n
will be time
reversal on the spacetime volume
n
[T;T].The process P
n
will be the pathspace
measure of the stationary interacting particle process on this nite
n
[T;T].P
r
n
is the marginal distribution of the trajectories restricted to the spacetime window
n
[T;T] under P.In the context of interacting particle systems,P and P
n
will
be pathspace measures of a Markovian process,whereas P
r
n
will be nonMarkovian.
In the local uctuation theorem it is attempted to recover the global symmetry of R
n
under P
n
also in the restrictions P
r
n
of P to nite volumes
n
.Clearly then,what we
need is that the dierence between P
n
and P
r
n
is a boundary term but this is more or
less implied by having our interacting particle systems enjoy Gibbsianness on space
time.Finally,the meaning of R
n
is that it gives,at least up to spacetime boundary
terms,a statistical mechanical representation of the thermodynamic steadystate en
tropy production.We wish however to refer to [3,4,5,6,7] for explaining this.Still,
it should be kept in mind that the B
n
introduced in the following proposition will
measure the dierence between the true entropy production (denoted there by
S
n
)
and R
n
.
There are in fact various strategies;we present two of them.
Proposition 2.1:Let B
n
be a measurable function so that B
n
n
= B
n
.Dene
S
n
R
n
+B
n
and let (a
n
) be a sequence of positive numbers tending to innity with
n.Assume that P
n
and P
r
n
are mutually absolutely continuous and so that
lim
n
1 a
n
ln
Z
dP
r
n
dP
ndP
r
n
1
e
2
B
n
= 0 (2.15)
for all
1
;
2
2 IR.Suppose that for all z 2 IR
p(z) = lim
n
1 a
n
ln
Z
e
z
S
n
dP
n
(2.16)
10
exist and is nite.Then,whenever
q(z) = lim
n
1a
n
ln
Z
e
z
S
n
dP
r
n
(2.17)
exists,then p(z) = q(z) and q(z) = q(1 z).
Remarks:
1.The symmetry q(z) = q(1 z) is dual to the symmetry as expressed in (1.7).Its
Legendre transformi(a) = sup
z
(q(z)za) satises i(a)i(a) = a.If
S
n
satises
a large deviation principle under P
n
,respectively P
r
n
,then i(a) is the corresponding
rate function,and the symmetry q(z) = q(1z) is equivalent with the large deviation
symmetry i(a) i(a) = a.
2.We will apply the strategy of Proposition 2.1 for obtaining a local uctuation
theorem for spin ip processes in the next Section.
Proof of Proposition 2.1:Since B
n
n
= B
n
,in the same way as for (2.13),
we deduce that
Z
dP
n
f(
S
n
) =
Z
dP
n
e
S
n
+B
n
f(
S
n
) (2.18)
Starting with the left hand side,for f(s) = e
zs
,by the Holder inequality,for 1=a +
1=b = 1 = 1=v +1=w,
ln
Z
dP
n
e
z
S
n
1 a
ln
Z
dP
r
n
(
dP
ndP
r
n
)
a
+
1b
ln
Z
dP
r
n
e
bz
S
n
1 a
ln
Z
dP
r
n
(
dP
ndP
r
n
)
a
+
1bv
ln
Z
dP
n
e
bvz
S
n
+
1 bw
ln
Z
dP
r
n
(
dP
r
ndP
n
)
w1
(2.19)
Dividing this by a
n
and taking limits,we can use condition (2.15) with
2
= 0 to get
p(z)
q(bz)b
p(bvz)bv
Again by the Holder inequality,both functions p and q are convex,and hence con
tinuous.Therefore we can take the limit for b;v!1 to conclude that p(z) = q(z).
The right hand side of (2.18) can be treated in the same way:
ln
Z
dP
n
e
(1z)S
n
+B
n
1 a
ln
Z
dP
r
n
(
dP
ndP
r
n
)
a
e
aB
n
+
1b
ln
Z
dP
r
n
e
b(1z)
S
n
1 a
ln
Z
dP
r
n
(
dP
ndP
r
n
)
a
e
aB
n
+
1bv
ln
Z
dP
n
e
bv(1z)
S
n
+B
n
+
1 bw
ln
Z
dP
r
n
(
dP
r
ndP
n
)
w1
e
wB
n
=v
(2.20)
11
which,again after taking limits n"+1,and using B
n
=
S
n
R
n
,gives
p(z) = q(z)
q(b(1 z))b
q(bv(1 z) +1)bv
and we can take the limits b;v!1 to get the desired q(z) = q(1 z).Proposition 2.2:Let B
n
be a measurable function such that B
n
n
= B
n
and
dene
S
n
= R
n
+B
n
.Let (a
n
) be a sequence of positive numbers tending to innity
with n so that for all 2 IR
lim
n
1a
n
ln
Z
dP
r
n
e
(B
n
+F
n
)
= 0 (2.21)
Suppose that for all z 2 IR
q(z) = lim
n
1a
n
ln
Z
e
z
S
n
dP
r
n
(2.22)
exists and is nite.Then,q(z) = q(1 z).
Proof of Proposition 2.2:By denition of F
n
,we have
Z
dP
r
n
f(
S
n
) =
Z
dP
r
n
e
S
n
+F
n
+B
n
f(
S
n
)
We thus leave the left hand side and apply a similar chain of inequalities to the right
hand side as was used in the proof of Proposition 2.1:
ln
Z
dP
r
n
e
(1z)S
n
+F
n
+B
n
1 a
ln
Z
dP
r
n
e
aF
n
+aB
n
+
1b
ln
Z
dP
r
n
e
b(1z)
S
n
1 a
ln
Z
dP
r
n
e
a(F
n
+B
n
)
+
1bv
ln
Z
dP
r
n
e
bv(1z)
S
n
+F
n
+B
n
+
1 bw
ln
Z
dP
r
n
e
w(F
n
B
n
)=v
(2.23)
We may thus again divide by a
n
and take limits rst n"+1to reach
q(z)
q(b(1 z))b
q(bv(1 z) +1)bv
By convexity we can take the limits b;v#1 to obtain the desired conclusion.Remarks:
1.Of course,if it happens that jF
n
+ B
n
j=a
n
!0 uniformly,then,for all positive
functions f,
lim
n
1a
n
ln
R
dP
r
n
f(
S
n
)R
dP
r
n
e
S
n
f(
S
n
)
= 0
12
without further ado.
2.The dierence between Proposition 2.1 and Proposition 2.2 is that in the rst
we suppose that P
n
and P
r
n
are mutually absolutely continuous while in the latter,
we need that P
r
n
and P
r
n
n
are mutually absolutely continuous.We will follow the
second strategy in Section 4 for the asymmetric exclusion process.
3.The condition that the limits dening p(z) and q(z) exist is natural in the context
where we have a large deviation principle for
S
n
under P
n
and P
r
n
resp.However if
we dene p
+
;p
;q
+
;q
by the corresponding limsup,resp.liminf,then we still have
convexity of p
+
;q
+
(the limsups),but not necessarily of p
;q
.We can still conclude
however the equality p
+
(z) = q
+
(z),and q
+
(z) = q
+
(1 z).
3 LFT for spin ip processes
We start our study with the,for physical applications,less interesting case of pure
spin ip processes.For details on the construction of spin ip processes,we refer to
[25].
The conguration space is K = f+1;1gd
(spins on the ddimensional regular lat
tice) and the path space is
= D(K;[T;T]) the set of rightcontinuous trajectories
having left limits,parametrized by time t 2 [T;T];T > 0 and having values!
t
2 K.
Our processes are specied in terms of spin ip rates c(x;);x 2d
; 2 K for which
our rst most important assumption is that they are positive and bounded:there
are constants b
1
> 0;b
2
< +1 so that b
1
< c(x;) < b
2
for all x;.For convenience
we assume that c(x;) only depends on the neighboring spins (y) with jy xj 1.
Thirdly,we assume the rates to be translation invariant:c(x;) = c(0;
x
).Here
and afterwards we put
n
= [n;n]
d
\d
n
denotes timereversal on
n
dened
by (
n
!)
t
(x) !
t
(x) if x 2
n
,and (
n
!)
t
(x) !
t
(x) if x =2
n
.On the jump
times we adapt
n
!so that it becomes rightcontinuous,and thus obtain
n
as an
involution on
We dene
?
n
fx 2
n
;c(x;) = c(x;
0
) for all ;
0
2 K with (y) =
0
(y);y 2
n
g for the subset of sites where the spin ip rates do not depend on the conguration
outside
n
.
We rst describe the sequence of processes P
n
corresponding to P
n
in the previous
abstract setting.For this we x a boundary condition 2 K and we dene spin ip
rates
c
n
(x;) I[x 2
n
]c(x;
n
c
n
);x 2d
; 2 K (3.24)
where I[] is the indicator function and
n
c
n
2 K coincides with on
n
and equals
on the complement
c
n
d
n
n
.P
n
is the stationary process on
with generator
L
n
f()
X
x
c
n
(x;)[f(
x
) f()]
13
corresponding to a spin ip process in
n
with rates c(x;) and boundary condition
.We call the (unique) stationary measure
n
:
R
d
n
L
n
f = 0.We always assume
that for all 2 f1;+1g
n
;
n
() b
1
exp[b
2
j
n
j].We can compute the density of
P
n
with respect to P
n
n
via a Girsanov formula for point processes,e.g.in [9,26].
For given!2
we let N
x
s
(!);s 2 [T;T];x 2d
denote the number of spin ips at
x up to time s;that is,N
x
s
(!) jft 2 [T;s];!
t
(x) = !
t
(x)gj;then,
R
n
ln
dP
n dP
n
n
=
X
x2
n
Z
T
T
dN
x
s
(!) ln
c
n
(x;!
s
)c
n
(x;!
s
)
+ln
n
(!
T
)
n
(!
T
)
As a consequence,the distribution of R
n
as induced from P
n
satises immediately
the global uctuation symmetry (2.13).
We can also consider the observable
S
n
X
x2
?
n
Z
T
T
dN
x
s
(!) ln
c(x;!
s
) c(x;!
s
)
which is measurable inside
n
.This is not an arbitrary choice but there is too little
physics here to call R
n
or
S
n
the entropy production;we will not elaborate on this.
The dierence B
n
S
n
R
n
is mainly a sum over x 2
n
n
?
n
.
The other process P
r
n
we need to look at is very similar but it is in general not Marko
vian.To dene it,we take a stationary process P on
and we take its restriction to
n
.We write for the corresponding stationary measure and we let
n
be its restric
tion to
n
.We always assume that for all 2 f1;+1g
n
;
n
() a
1
exp[a
2
j
n
j].
Being more explicit,we let P be an innite volume stationary process with formal
generator
Lf()
X
x
c(x;)[f(
x
) f()]
and put P
r
n
the unique pathspace measure such that
1.The distribution of f!
t
(x):x 2
n
;t 2 [T;T]g under P
r
n
and P coincide.
2.Under P
r
n
,!
t
(x) = (x) for all x 62
n
,t 0.
Theorem 3.1 [LFT for spin ip processes] For all z 2 IR,
lim
n;T
1 n
d
T
ln
R
dPe
z
S
nR
dPe
(1z)
S
n
= 0
Proof of Theorem 3.1:
Even though P
r
n
is not Markovian (in general),it remains a jump process and the
jumpintensities can be computed from the original spin ip rates.In order to have a
Gibbsian structure these intensities must be the same in the bulk of
n
as they were
14
for the innite volume process P.As the rates are local,the process P
r
n
restricted
to
n
indeed has the same intensities as the process P
n
except at the sites of the
boundary
n
n
?
n
.This is a consequence of the following generally stated
Lemma 3.2:
Suppose N
t
is a point process with intensity c
s
,i.e.,M
t
= N
t
R
t
0
c
s
ds is a martingale
for the ltration F
t
.Suppose that F
0
t
F
t
is a subltration of F
t
,and dene
N
0
t
= IE[N
t
jF
0
t
] (3.25)
Then N
0
t
is a point process with intensity
c
0
s
= IE[c
s
jF
0
s
] (3.26)
Proof of Lemma 3.2:
It is easy to see that M
0
t
= N
0
t
IE(
R
t
0
c
s
dsjF
0
t
) = IE[M
t
jF
0
t
] is a F
0
t
martingale.Hence,
it suces to show that
B
t
= IE[
Z
t
0
c
s
dsjF
0
t
]
Z
t
0
IE[c
s
jF
0
s
]ds (3.27)
is a F
0
t
martingale.This is a consequence of the following equalities:
IE[B
t
jF
0
s
]
= B
s
+IE
Z
t
s
(c
r
IE[c
r
jF
0
r
])drjF
0
s
= B
s
+IE[
Z
t
s
c
r
drjF
0
s
] IE
Z
t
s
IE[c
r
jF
0
r
]jF
0
s
= B
s
(3.28)Therefore,the rates of the restricted process on
n
are given by
~c
s
(x;!) = IE[c(x;
s
)j
(y) =!
(y);T s;y 2
n
]
where the expectation is with respect to P.
Or,for all x 2
n
,N
x
t
(!)
R
t
0
~c
s
(x;!) is a martingale under P
r
n
.As a consequence,
~c
s
(x;!) = c(x;) when x 2
?
n
and!
s
= .
Just as for the pair P
n
;P
n
n
,the absolutely continuity of P
r
n
n
with respect to P
r
n
and
vice versa is guaranteed by the positivity of the spin ip rates inside
n
.We are thus
ready to apply Proposition 2.1.We must rst verify the corresponding assumption
(2.15).We nd
B
n
=
S
n
R
n
=
X
x2
n
n
?
n
Z
dN
y
s
(!) ln
c
n
(x;!
s
)c
n
(x;!
s
)
+ln
n
(!
T
)
n
(!
T
)
(3.29)
15
and
ln
dP
r
ndP
n
= ln
(!
T
)
n
(!
T
)
+
X
x2
n
n
?
n
Z
dN
x
s
(!) ln
~c
s
(x;!)c
n
(x;!
s
)
Z
T
T
ds[~c
s
(x;!) c
n
(x;!
s
)]
(3.30)
Clearly,both jB
n
j and j lndP
r
n
=dP
n
j are bounded by c
1
N([T;T];
n
n
?
n
) +c
2
j
n
j +
c
3
Tj
n
n
?
n
j for some constants c
1
;c
2
;c
3
< 1,where N([T;T];
n
n
?
n
) is the
number of spin ips that have occurred in the spacetime window [T;T] (
n
n
?
n
).
It remains thus to show for all
lim
n"1
1 n
d
ln
Z
dP
r
n
e
N([T;T];
n
n
n
)
= lim
n"1
1n
d
ln
Z
dP
0
dP
r
ndP
0
n
e
N([T;T];
n
n
n
)
= 0
where we inserted the reference process P
0
(and its restriction P
0
n
to
n
) correspond
ing to the product process of independent spin ips (rate 1).In particular,
lim
n"1
1 n
d
ln
Z
dP
0
e
N([T;T];
n
n
n
)
= lim
n"1
j
n
n
n
jn
d
2T(e
1) = 0
and we can apply the same argument as in Proposition 2.1.Finally,the condition
(2.16) of Proposition 2.1 is a consequence of the large deviation results of [24].4 LFT for the asymmetric exclusion process
The conguration space is nowK = f0;1g2
(occupation variables on the 2dimensional
regular lattice) and the pathspace
= D(K;[T;T]) is essentially unchanged from
that in the previous section.For 2 K,(x) = 1;0 indicates the presence,respec
tively absence of a particle at the site x 2 d
.This hopping dynamics will be modeled
by an asymmetric exclusion process.This is a bulk driven diusive lattice gas.The
hopping rates for vertical (v) and horizontal bonds (h) depend on the direction in
the following way:
c
v
(x;)
1 2
[(x)(1 (x +e
2
)) +(x +e
2
)(1 (x))]
c
h;E
(x;)
1 2
[e
E=2
(x)(1 (x +e
1
)) +e
E=2
(x +e
1
)(1 (x))]
where e
1
;e
2
are the unit vectors in the positive horizontal and vertical direction.
In addition,for the moment,we allow for the possibility of particle creation and
destruction.We put the birth/death rate c(x;) independent of the conguration
16
and the site x.
The formal Markov generator L to the innite volume process is then found as the
sum
Lf()
X
x
[f(
x
) f()] +
X
hxyi
c(x;y;)[f(
xy
) f()]
where
x
is the new conguration after changing the occupation at x,
xy
is the
new conguration after exchanging the occupations at x and y and c(x;y;) is given
by (4.31) for nearest neighbors x;y = x e
i
.We can allow more general reaction
diusion processes (e.g.with extra interaction,speed change,etc.) but we will stick
here to this choice.What is simpler here is that the Bernoulli measure with density
1/2 is a nonreversible stationary measure.The corresponding pathspace measure
over the timeinterval [T;T] is P = P
E
and we put P
E
n
the process restricted to
the nite square
n
This P
E
n
will now play the role of P
r
n
of Section 2.For a given
trajectory!2
we let N
xy
s
(!);s 2 [T;T] be the number of hopping times where
the occupation at the nearest neighbor sites hxyi was exchanged.Since this model
has a clear physical interpretation we can dene the variable entropy production in
n
.
The rst contribution comes from the work done by the external eld
W
n
E
X
hxyi;y=x+e
1
2
n
Z
T
T
dN
xy
s
(!)[!
s
(x)(1 !
s
(y)) !
s
(y)(1 !
s
(x))]
(4.31)
This is of the form eld (E) times current.There is a second contribution from
dierences in the reaction rates at the boundary:particles enter or leave at dierent
rates at the various boundaries;this contribution is present even in the case where
no external eld is applied inside
n
:
J
n
(!)
X
x2@
X
y=xe
1
2
c
n
Z
T
T
dN
x
s
(!)[!
s
(x) ln
2 +
E
y
(!;s)e
E
j
=22 +
E
y
(!;s)e
E
j
=2
+(1 !
s
(x)) ln
2 +
E
y
(!;s)e
E
j
=22 +
E
y
(!;s)e
E
j
=2
]
+
X
x2@
X
y=xe
2
2
c
n
Z
T
T
dN
x
s
(!)[!
s
(x) ln
2 +
E
y
(!;s)2 +
E
y
(!;s)
+(1 !
s
(x)) ln
2 +
E
y
(!;s) 2 +
E
y
(!;s)
] (4.32)
where the second sum is over all (external) neighbors y of x and @
n
is the interior
boundary of
n
.Here,the additional rates are
E
y
(;t) IE
E
[1
t
(y)j
s
(z);z 2
n
;s 2 [T;t]]
17
and
E
y
(;t) IE
E
[
t
(y)j
s
(z);z 2
n
;s 2 [T;t]]
for E
j
= E if y = x e
1
;E
j
= 0 if y = x e
2
and the expectations are in the
process P = P
E
.The variable entropy production is put
S
n
=
W
n
+J
n
The symmetry in the uctuations of
S
n
is given by
Theorem 4.1 [LFT for the aymmetric exclusion process] For all (including = 0),
for all z 2 R,
lim
n;T
1n
2
T
ln
R
dPe
z
S
nR
dPe
(1z)
S
n
= 0
Proof of Theorem 4.1 We start by noting that for the timereversal
n
,
P
E
n
n
= P
E
n
Obviously then,for a function f measurable in
n
[T;T],
Z
dPf(!) =
Z
dP
E
n
f(!) =
Z
dP
E
n
dP
E
ndP
E
n
f(
n
!)
and we must investigate express the density dP
E
n
=dP
E
n
via a Girsanov formula.This
is the strategy of Proposition 2.2.The Girsanov formula gives
ln
dP
E
n dP
E
n
=
S
n
+F
n
(4.33)
with the following correction term:
F
n
(!) sinh(E=2)
X
x;y=x+e
1
2
n
Z
T
T
ds[!
s
(y)(1 !
s
(x)) !
s
(x)(1 !
s
(y))]
+
X
x2@
X
y=xe
1
;xe
2
2
c
n
Z
T
T
ds[!
s
(x)[
E
y
(!;s))
E
y
(!;s)]
+(1 !
s
(x))[
E
y
(!;s))
E
y
(!;s)] (4.34)
Now,jF
n
j cj@jT because the (rst) bulk term in (4.34) telescopes to a boundary
term.We can thus apply Remark 1 after Proposition 2.2 to nish the proof.Next,we investigate whether the variable work W
n
of (4.31) itself satises the
same local symmetry as the entropy production.
18
Theorem 4.2 [LFT for the work done] For all > 0,for all z 2 IR,
lim
n;T
1n
2
T
ln
R
dPe
z
W
nR
dPe
(1z)
W
n
= 0
Proof of Theorem 4.2 Clearly,since > 0,jJ
n
j cN([T;T];
n
n
n
),that is
bounded,up to a constant,by the number of ips in the trajectory on sites x 2
n
n
n
,
for times t 2 [T;T].We can therefore verify condition (2.21) in the same way as
we did for Theorem 3.1.Finally,the large deviation results of [24] remain valid for
> 0,so that we can nish the proof along the lines of Proposition 2.2.5 Remarks
1.It is clear from the preceding analysis that the reasons for having a global
or local uctuation theorem do not in any way depend on the
n
being time
reversal.Thus,the same results will be reproduced in exactly the same formfor
any other involution.Of course,the symmetry breaking part in the pathspace
action functional will be the variable for which the uctuation symmetry holds
(replacing entropy production corresponding to timereversal symmetry break
ing).As an example,if for a spin ip process,the rates are not even under a
global spin ip (by the presence of a bias or magnetic eld),then a local uctu
ation theorem will be established for the variable magnetization.Furthermore,
we may consider the composition of two or more involutions in this way,we
could e.g.obtain a local uctuation theorem for the odd part (under spin ip)
of the variable entropy production.Finally,we can even go beyond the case
of involutions and consider instead the generators of the symmetry group for
the unperturbed dynamics.In this case,the precise form of the uctuation
symmetry is not preserved but its modication presents no real problem.
2.We restricted our discussion to interacting particle systems where the evolution
is Markovian.Within the Gibbsian spacetime picture,this means that the
interaction is\nearest neighbor"in the time direction (the jump intensity at
time t depends only on the conguration at time t
).However,this restriction is
not at all necessary.If the jump intensities are local in space and bounded from
above and from below,then we can still apply the Girsanov formula for point
processes to obtain the local uctuation theorem from the global uctuation
theorem.
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