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Monotonicity of the dynamical activity

Christian Maes,1 Karel Netocny,2 and Bram Wynants3

1Instituut voor Theoretische Fysica, K.U.Leuven, Belgium

2Institute of Physics AS CR, Prague, Czech Republic

3Institut de Physique Theorique, CEA-Saclay, France∗

We find a new Lyapunov function for the time-evolution of probability distribu-

tions evolving under a Master equation, possibly relevant for the characterization of

steady nonequilibria. This function is non-entropic when the system is driven away

from detailed balance; it rather measures an excess in dynamical activity. Our proof

of monotone behavior works under two conditions: (1) the initial distribution is close

enough to stationarity, or equivalently for our context, we look at large times, and

(2) a certain spectral sector–condition equivalent to “normal” linear-response behav-

ior is satisfied. The deeper reason for its monotone behavior is that our Lyapunov

function is the dynamical fluctuation functional for the occupation times.

I. INTRODUCTION — MAIN FINDING

Dynamical activity (DA) refers to a combination of properties of a statistical mechanical

system that are related to its reactivity and the ability to escape from its present state.

The notion originates in kinetic theory, and has also been generally described as “noise” or

“dither” in nonequilibrium modeling. There are connections with ideas of Landauer (1975)

who emphasized the nonequilibrium kinetics for determining the relative state occupation;

standard thermodynamic state functions or potentials only do not suffice, [11]. More

recently, DA has been studied in connection with glassy behavior and the glass transitions;

kinetically constrained models show a reduced dynamical activity over an extensive number

of states which leads to dynamical phase transitions, [3, 7–9]. Finally, DA has appeared in

fluctuation and response theory for steady nonequilibria. The point is that as a function on

trajectories, the dynamical activity is time-symmetric and therefore provides an essential

∗Electronic address: bram.wynants@cea.fr

2

complement to entropic considerations for understanding transport and response beyond

the linear regime around equilibrium, [12, 15]; in [1, 2] DA was also called frenesy.

So far little systematic studies have been devoted to the DA. The present text gives

a mathematical introduction to some of its basic properties in the set-up of Markov

jump processes. We concentrate really on one new finding: that there is a functional on

probability distributions capturing the excess in dynamical activity, that shows monotone

behavior back to steady nonequilibrium. This result mirrors the monotone behavior of

the relative entropy which is associated to the static fluctuations of the system. Our

DA-functional is much related to dynamical fluctuations, in particular as large deviation

functional of the occupation times for a time-ergodic Markov process. Further explorations

and numerical evidence for monotone behavior of our functional is collected in [13], but

more simulations will be needed to better understand the limitations of our proposal.

While much of its physical interpretation still awaits further exploration we believe that

this DA-functional can give essential information about the system. In particular it involves

the inverse problem of finding the conservative force that, when added to the original

system, produces the sought for stationary behavior. Such a procedure of course stems

from dynamical large deviation theory as pioneered by Donsker and Varadhan (1975), but

it can get a kinetic-operational meaning when, additional to the given environment, extra

channels of energy exchange are opened that bring the system to a different stationary

statistics. Our DA-functional measures then the excess or the change in escape rates and

gets minimal in the true stationary distribution.

The next section reminds us of the monotone behavior of the relative entropy, to push us

off to consider a new functional, the DA-functional, which also shows monotone behavior at

least under some further conditions. Section III collects the definitions and main properties,

with our result on monotone behavior. Proofs are collected in Section IV.

II. RELATIVE ENTROPY

It is well known that the relative entropy is monotone under stochastic transformations.

A useful realization is found for Markov processes where the dynamics itself defines the

3

transformation. To be specific we consider a Markov jump process on a finite state space

K with states x, y, . . . and transition rates k(x, y). Probability distributions on K will be

denoted by ρ, µ, ν, . . .. The backward generator on functions f is

Lf(x) :=∑

y∈K

k(x, y) [f(y)− f(x)]

and its transpose generates the Master equation

d

dtµt(x) +

∑

y∈K

jµt(x, y) = 0, jν(x, y) := k(x, y) ν(x)− k(y, x) ν(y) (II.1)

for the evolution on probabilities µt starting from some initial µ0 = µ on K. We assume

that the Markov process is irreducible with unique stationary probability distribution ρ, i.e.,

ρ(x) > 0 solves∑

y jρ(x, y) = 0 for all x ∈ K. Then, the relative entropy

s(µt | ρ) :=∑

x∈K

µt(x) logµt(x)

ρ(x)≥ 0

is monotone in time t ≥ 0, converging to zero exponentially fast.

The relative entropy governs the fluctuations of an ensemble of N independent copies of the

system, in the sense that asymptotically for N → ∞,

Probρ

[ 1

N

N∑

i=1

δxi,x ≃ µ(x), ∀x]

= e−N s(µ | ρ) (II.2)

to be understood as a large deviation principle for independent variables (xi) under identical

distribution ρ; see e.g. [4, 5]. The above mathematics is true under very general conditions,

but we wish to specify that only in the case of detailed balance the relative entropy allows

for a straightforward thermodynamic interpretation. That means that, decorating the rates

and the stationary law in that case with the subscript ‘o’, there is a function U on K for

which

ko(x, y) e−U(x) = ko(y, x) e

−U(y), ρo(x) ∝ e−U(x) (II.3)

Then, the relative entropy becomes

s(µ | ρo) =∑

x∈K

µ(x)U(x)− s(µ)− F(ρo) = F(µ)− F(ρo)

for Shannon entropy s(µ) := −∑

x µ(x) logµ(x) and free energy functional F(µ) :=∑

x µ(x)U(x) − s(µ) ≥ F(ρo) = − log∑

x exp [−U(x)]; we set the inverse temperature

4

β = 1 throughout the whole text. As a consequence, always in the case of a detailed balance

dynamics, the monotonicity of the relative entropy is the decrease of the free energy func-

tional to its minimal value, the equilibrium free energy F(ρo). Away from detailed balance,

we mostly have no systematic knowledge of the stationary distribution ρ and we have no

clear interpretation of the relative entropy in terms of work or heat.

III. DYNAMICAL ACTIVITY FUNCTIONAL

We embed the original dynamics into a larger family of processes with transition rates,

kW (x, y) := k(x, y) expW (y)−W (x)

2(III.1)

parameterized by functions W on K. Clearly, under detailed balance (II.3), the new sta-

tionary distribution for (III.1) has potential U −W . When (II.3) is violated however, the

new stationary distribution can be qualitatively very different from the original one even for

small W . That effect is sometimes referred to as an instance of the “blowtorch theorem”

where Landauer (1975) emphasized that in nonequilibrium the relative weights of states

are strongly influenced by the kinetics of a process and not by thermodynamic information

alone, [10, 11]. For spatially extended systems, where the state x contains some spatial

degrees of freedom, adding such a potential W as in (III.1) can have very nonlocal effects,

cf. [14].

Here we consider potentials that are directly connected with a probability distribution. For

that we need a solution of the inverse stationary problem.

Proposition III.1. For an arbitrary probability distribution µ > 0 there exists a function

V = Vµ on K such that µ is invariant under the modified dynamics with transition rates

kV (x, y). The function Vµ is unique up to an additive constant for irreducible dynamics.

In other words, for arbitrary µ > 0 we can always find a function V so that

∑

y∈K

[

kV (x, y)µ(x)− kV (y, x)µ(y)]

= 0, x ∈ K (III.2)

The inverse problem remarkably simplifies close-to-stationarity where it can be approxi-

mately linearized. To specify distributions µε close to the stationary distribution ρ, we take

5

a function g on K with zero mean,∑

x ρ(x)g(x) = 0 and for small ε > 0 with ε g(x) ≥ −1,

x ∈ K, we make

µε := ρ(1 + ε g)

We claim that to these µε there correspond Vµε = Vε, verifying (III.2), for which necessarily

Vε = εv +O(ε2), for v :=2

LsL∗g (III.3)

Here, we have introduced the generator L∗ of the time-reversed process,

L∗f(x) :=∑

y∈K

ρ(y) k(y, x)

ρ(x)[f(y)− f(x)]

and the symmetric part of the generator Ls := L+L∗. Note for (III.3) that ρ is also invariant

under the time-reversed process and under the (detailed balanced) process generated by Ls.

Hence,∑

x ρ(x)L∗g(x) = 0 and L∗g is in the domain of the (Drazin) pseudo-inverse (Ls)

−1.

The proof of (III.3) follows directly from expanding the equation (III.2) (with µ = µε) to

first order in ε. More generally a function V satisfying (III.2) will be called a blowtorch

potential (for µ) and the corresponding stationary process is the blowtorched process.

We now come to our main player, which measures the excess in escape rate between the

original and the blowtorched process. The escape rate from state x ∈ K for a Markov jump

process is the sum∑

y∈K k(x, y). Its expectation under a given probability µ estimates the

dynamical activity under µ. We then define what we call the dynamical activity functional

for irreducible dynamics as, cf. Proposition III.1,

D(µ) :=∑

x,y∈K

µ(x) [k(x, y)− kV (x, y)] with V = Vµ (III.4)

Without the irreducibility assumption, one needs a more general definition,

D(µ) := supV

∑

x,y∈K

µ(x) [k(x, y)− kV (x, y)] (III.5)

see Section IVA for details of the connection between the two constructions. For simplicity,

in the sequel we always assume the irreducibility.

Formula (III.4) defines a nonlinear functional of µ that has a number of immediate prop-

erties that we list in the following

Proposition III.2. The DA-functional (III.4) satisfies

6

1. D(µ) ≥ 0 with equality only when µ = ρ;

2. D(µ) is strictly convex in µ;

3. Under the condition of detailed balance (II.3),

D(µ) = −∑

x

ρo(x)

√

µ(x)

ρo(x)

(

Lo

√

µ

ρo

)

(x) (III.6)

in terms of the Dirichlet form for generator Lo of the Markov process with rates

ko(x, y).

The main “unifying” reason for the above properties is the relation between D(µ) and

the fluctuation functional for the large deviations of the occupation times. We give separate

and self–consistent proofs in Section IVC. However, our main finding is that D(µt) is

also monotone under the evolution (II.1) when close enough to stationarity. i.e., for large

enough times t.

Define the real–space scalar product (f, g) :=∑

x f(x)g(x) ρ(x) so that (f, Lg) = (L∗f, g).

Theorem III.3. If (Lsf, Lf) ≥ 0 for all functions f on K, then for all initial probability

distributions µ on K there is a time to > 0 so that

d

dtD(µt) ≤ 0 for all times t ≥ to

The sufficient condition (Lsf, Lf) ≥ 0 on the stationary process can be interpreted in two

main ways, first, more algebraically, as a sector condition on the generator, and secondly,

more physically, as a condition on the short-time behavior of a generalized susceptibility.

To explain we start with the sector-condition. The inequalities

(Lf, (L+ L∗)f) ≥ 0 ⇔ (f, L2f) ≥ −(Lf, Lf) (III.7)

for functions f on K, are easily seen to be equivalent with

([Ls, La]f, f) ≥ −(L2sf, f) (III.8)

where now also the antisymmetric part of the generator L enters as La := L − L∗. The

right-hand side of (III.8) is −(L2sf, f) = −(Lsf, Lsf) ≤ 0. It means that (III.8) is verified

7

when La is “relatively small,” e.g. close to satisfying detailed balance (II.3), but also when

the commutator [Ls, La] is “small.” Furthermore,

([Ls, La]f, f) = (Lf, Lf)− (L∗f, L∗f)

so that (Lf, Lf) ≥ (L∗f, L∗f) implies (III.8). A sufficient condition is

(f, L2f) ≥ 0 (III.9)

The inequality (III.9) is a sector condition on the eigenvalues of the generator L; their

imaginary part should not be too big when their (negative) real part is small.

The second interpretation of the condition (III.7) for Theorem III.3 is in terms of the

generalized susceptibility for the linear response around the stationary probability ρ. For a

function B on K consider perturbed transition rates

k(s; x, y) := k(x, y) ehs2[B(y)−B(x)], s ≥ 0 (III.10)

with small time-dependent amplitude hs. That new time-dependent process is started from

time zero in the distribution ρ which is stationary for h ≡ 0. At a later time t > 0, in

the process with rates (III.10), when taking the expectation of a function G, we see the

difference

〈G(xt)〉h −

∑

x

G(x)ρ(x) =

∫ t

0

ds ht−s χGB(s) +O(h2)

which defines the generalized susceptibility χGB. There is an explicit formula for that func-

tional derivative, extending the standard fluctuation–dissipation theorem, see e.g. [15]:

χGB(t) :=δ

δh0

〈G(xt)〉h∣

∣

∣

h=0= −

1

2

[ d

dt

⟨

B(x0)G(xt)⟩

ρ+⟨

LB(x0)G(xt)⟩

ρ

]

(III.11)

with right-hand expectations in the original stationary process. It is then a simple compu-

tation, that we do in (IV.7), to conclude that for t ↓ 0,

d

dtχff(t = 0) = χLf,f(t = 0) = −

1

2(Lsf, Lf) (III.12)

The negativity of the right-hand side of (III.12) for all functions f on K is the sufficient

condition in Theorem III.3. In other words, the DA-functional is monotone for large times

whenever the linear response χff (t) at small times t ≥ 0 to a small perturbation at time zero

decays as a function of t. Physically we have in mind that we apply a delta-perturbation in

8

the form of a potential −h0f at time zero; that is (III.10) for B = f and with hs peaked at

s = 0. Let us define the effective inverse temperature βf (t) by

χff (t) =: −βf(t)d

dt〈f(x0)f(xt)〉 = −βf (t) 〈f(x0)Lf(xt)〉

We have normalized the formulæ in such a way that under detailed balance, we would have

βf(t) ≡ 1, see also (II.3). From (III.11),

χff(t) = −1

2〈f(x0)Lf(xt)〉 −

1

2〈Lf(x0)f(xt)〉

Observe that βf(0) = 1 and χff (t = 0) = −〈f(x0)Lf(x0)〉 ≥ 0. On the other hand, the

time derivative of the response function is

χff (t) = −βf (t) 〈f(x0)Lf(xt)〉 − βf (t)⟨

f(x0)L2f(xt)

⟩

At time zero this gives

χff (0) = −βf (0) 〈fLf〉 −⟨

fL2f⟩

which is negative (and implies (III.7)) when

βf(0) ≤ −〈fL2f〉

〈fLf〉

From condition (III.9), we certainly get monotone decay of our DA-functional whenever

βf(0) ≥ 0 for all functions f , which is like the effective temperature not going up for initial

times.

Remark: It is interesting to compare with the time-dependence of the relative entropy

s(µt | ρ) =∑

x

µt(x) logµt(x)

ρ(x)

for µt = ρ(1 + εgt), under (II.1). To leading order in ε, s(µt | ρ) = ε2 s(gt) + o(ε2) for

s(g) = (g, g). The first derivative is s(gt) = 2(gt, Lgt) ≤ 0. The second derivative is

d2

dt2s(gt) = s(gt) = 2(gt, LLs gt)

Using the quadratic form Q(f) := (LLsf, f) we have thus found s(gt) = 2Q(gt). The

quadratic forms Q(f) := (Lsf, Lf) (appearing in (III.7)) and Q are equal when the generator

is normal, LL∗ = L∗L. In a way, the forms Q and Q are each others time-reversal. For

fixed function f they cannot be both negative because their sum is always non-negative:

Q(f) +Q(f) = (Lsf, Lsf) ≥ 0.

9

IV. PROOFS

A. Existence and uniqueness of the blowtorch

We prove here Proposition III.1. For arbitrary µ > 0 we consider the auxiliary functional

Yµ(W ) :=∑

x,y∈K

µ(x) kW (x, y) =∑

x,y∈K

µ(x) k(x, y) exp[W (y)−W (x)

2

]

, W ∈ C0(K)

(IV.1)

defined on the collection C0(K) of all functions that are equal to zero on a fixed “root”

x0 ∈ K. This functional is nonnegative and convex,

Yµ(λW1 + (1− λ)W2) ≤ λYµ(W1) + (1− λ)Yµ(W2) (IV.2)

by convexity of each contribution µ(x) kW (x, y). Below in Lemma IV.1 we prove that under

the irreducibility assumption, Yµ(W ) is actually strictly convex and that it attains inside

C0(K) a unique minimum at some W = Wµ. Hence, Wµ is also a minimizer (unique up to

additive constant) on the unconstrained space of all functions on K, implying that for all

x ∈ K,

0 =δYµ

δW (x)

∣

∣

∣

W=Wµ

=1

2

∑

y∈K

[µ(y) kW (y, x)− µ(x) kW (x, y)] (IV.3)

which is just the stationarity of µ for the dynamics with rates kW (x, y), i.e., V = Vµ of

Proposition III.1 does exist and equals Wµ. This also proves formula (III.5) as

D(µ) =∑

x,y∈K

µ(x) k(x, y)− Yµ(Wµ) = supV

∑

x,y∈K

µ(x) [k(x, y)− kV (x, y)] (IV.4)

A general reducible dynamics can be decomposed into irreducible components (including

isolated sites) and for each of them the above argument holds true, i.e., the supremum

on the right-hand side of (III.5) is attained on a function Vµ, which is also a solution of

the inverse stationarity problem and which is unique up to a constant within each component.

Again turning to irreducible dynamics, we have

Lemma IV.1. For any µ > 0, Yµ|C0(K) is strictly convex and has a unique minimum.

Proof. By irreducibility, there exists a cyclic sequence of states (x0, x1, ..., xn = x0) that cov-

ers the whole space K and such that for all consecutive pairs of states, µ(xi−1) k(xi−1, xi) ≥ δ

10

with some δ > 0. If W1 and W2 are such that the relation (IV.2) becomes an equality, then,

for all i = 1, . . . , n, W1(xi)−W1(xi−1) = W2(xi)−W2(xi−1), by using that the exponential is

strictly convex. From W1(x0) = W2(x0) = 0 then follows W1 = W2, identically. This proves

the strict convexity and hence the uniqueness of the minimum for Yµ|C0(K).

To prove that the minimum exists, we consider the compact sets

Ca0 (K) := {W ∈ C0(K); |W (x)| ≤ a for all x}, a > 0

and defineMµ := Yµ(0) =∑

x,y∈K µ(x) k(x, y). By construction, for anyW ∈ C0(K)\Ca0 (K)

there exists i such that W (xi)−W (xi−1) > a/n and hence Yµ(W ) > δ ea/(2n). Fix now some

a so that δ ea/(2n) > Mµ. By compactness, Yµ on the set Ca0 (K) attains the minimum, which

then coincides with the minimum of Yµ|C0(K).

B. Monotonicity close-to-stationarity

We take the initial conditions from a family µ = µε of distributions smoothly param-

eterized with ε > 0 measuring the distance from stationarity, so that we can expand

µε = ρ(1+ εg+ . . .). By the irreducibility assumption, the dynamics has a spectral gap, and

hence both µεt and V ε

t are also smooth in ε. (In the latter case, one needs to slightly refine the

argument of the previous section and to use the implicit function theorem; we skip details.)

Then we can also write perturbation series µεt = ρ(1 + εgt + . . .) and V ε

t = εvt + . . . with gt

and vt, respectively, the time-dependent first-order expansion coefficients. Furthermore, the

relaxation to stationarity is exponentially fast and uniform in ε around ε = 0. That allows

to do the exchange of limits ε ↓ 0 and t ↑ +∞ or, equivalently, to analyze the long-time

asymptotics by a more convenient linearization in ε.

Lemma IV.2.

D(µ) = −ε2

4(v, Lv) + o(ε2) and (v, Lv) = (g, Lv)

Proof. We only need to expand (III.4) around ε = 0. The function V is the solution of

L+V µ = 0 where L+

V is the forward generator of the blowtorched dynamics with rates kV (x, y),

see (III.2). The function v is obtained from expanding that equation L+V µ = 0 to first order

in ε, to get

Lsv = L∗g (IV.5)

11

As a consequence,

(v, Lv) = (v, Lsv) = (g, Lv)

We now consider the time evolution which upon linearization reads µt = ερ gt + O(ε2)

and L+µt = ερL∗gt +O(ε2) for the forward generator L+ of the original dynamics (referred

to as the transpose of L in (II.1)). Hence, gt = L∗gt. The corresponding linearized time-

evolved blowtorch potential is Vt = εvt+O(ε2) is then obtained via (IV.5). Writing D(µt) =

−ε2I(gt)/4 + o(ε2), we have the following identities:

I(gt) := (vt, Lvt) = (gt, Lvt) = (vt, gt) = (gt,1

Lsgt)

We take the derivative

d

dtI(gt) = 2(vt, Lsvt) = 2(gt, Lvt) = 2(Lvt, Lsvt) = 2Q(vt) (IV.6)

for the quadratic form Q(f) := (Lf, Lsf) = (f, L∗Lsf) = (LsLf, f).

We conclude from (IV.6) that (III.7)–(III.8) imply the monotone decay of the dynamical

activity, D(µt) = −ε2Q(vt)/2 + o(ε2), at least for the leading term in the expansion around

ρ. However, as indicated above, this is sufficient to have the monotone decay also for all

large enough times.

We still verify (III.12), in computing

χff (t) = −1

2

[ d

dt

⟨

f(x0)f(xt)⟩

ρ+⟨

Lf(x0)f(xt)⟩

ρ

]

= −1

2

⟨

(L+ L∗)f(x0)f(xt)⟩

ρ

d

dtχff (t)

∣

∣

∣

∣

t=0

= −1

2(Lsf, Lf) (IV.7)

Comparing (IV.6) with (IV.7) we find

d

dtχff (t)

∣

∣

∣

∣

t=0

= −Q(f) (IV.8)

Similarly, we look at the response of the observable Lf to find

χLf,f(t) = −1

2

[ d

dt

⟨

f(x0)Lf(xt)⟩

ρ+⟨

Lf(x0)Lf(xt)⟩

ρ

]

= −1

2

⟨

(L∗ + L)f(x0)Lf(xt)⟩

ρ

χLf,f(0) = −1

2(Lsf, Lf) (IV.9)

12

The equality (IV.7) = (IV.9) indicates that we can commute the time-derivative and the

derivative with respect to the perturbation.

C. Relation with Donsker-Varadhan theory

1. Dynamical activity in equilibrium

Taking V (x) = logµ(x) − log ρo(x) yields detailed balance for the blowtorched process

with rates

ko,V (x, y) = ko(x, y)

√

ρo(x)µ(y)

ρo(y)µ(x)

We can thus compute from (III.4),

D(µ) =∑

x,y

µ(x)ko(x, y)(

1−

√

µ(y) ρo(x)

ρo(y)µ(x)

)

=

√

µ(x)

ρo(x)

∑

x,y

ρo(x)ko(x, y)(

√

µ(x)

ρo(x)−

√

µ(y)

ρo(y)

)

(IV.10)

and the dynamical activity equals the Dirichlet form

D(µ) = −∑

x

ρo(x)

√

µ(x)

ρo(x)

(

Lo

√

µ

ρo

)

(x) = −(

√

µ

ρo, Lo

√

µ

ρo

)

which proves (III.6).

2. Dynamical activity as relative entropy

The dynamical activity (III.4) is a relative entropy on path-space between the blowtorched

process PV and the original process P , both started from ρ at time zero, over the time-interval

[0, T ]. The subscript V will refer to the blowtorched process with rates kV (x, y) as in (III.1)

satisfying (III.2), started from ρ. We show that

D(µ) = limT↑+∞

1

T

∫

dPV (ω) logdPV

dP(ω) (IV.11)

The derivation starts from a Girsanov formula for the relative path-space densities

logdPV

dP(ω) =

∑

t

logkV (xt− , xt)

k(xt− , xt)−

∫ T

0

ds∑

y

[kV (xs, y)− k(xs, y)] (IV.12)

13

The first sum is over all jump times over [0, T ] in the path ω. Combining (III.1) with (IV.12)

gives

logdPV

dP=

1

2[V (xT )− V (x0)] +

∫ T

0

ds∑

y

[k(xs, y)− kV (xs, y)]

The first term vanishes when dividing by T ↑ ∞ so that

limT

∑

x,y

〈pT (ω, x)〉V [k(x, y)− kV (x, y)]

gives the right-hand side of (IV.11), where

pT (ω, x) :=1

T

∫ T

0

δxs,x ds, x ∈ K (IV.13)

are the empirical occupation fractions. Under the blowtorched process these become equal

to µ(x) for large T from whatever state you start.

An immediate consequence of the equality (IV.11) is that the dynamical activity functional

is always non-negative, D(µ) ≥ 0. Of course it is zero only when µ = ρ, the stationary

distribution of the original process with rates k(x, y).

3. Large deviations of occupation times

The dynamical activity functional D(µ) has a prominent role in dynamical fluctuation

theory. The latter was developed by Donsker and Varadhan in 1975, [6]. We recall the main

setting.

Consider the stationary process P = Pρ started in ρ and with rates k(x, y). The empirical

time that the system spends in state x ∈ K over times [0, T ] is (IV.13). By the assumed

ergodicity, pT → ρ for τ ↑ +∞. For the fluctuations around that law of large times, there is

a principle of large deviations, abbreviated as

P [pT ≃ µ] ≃ e−TI(µ), T ↑ +∞

in the usual logarithmic and asymptotic sense T ↑ +∞; see e.g. [4, 5]. There is a variational

expression for the large deviation functional I,

I(µ) = supg>0

{

−∑

x

µ(x)

g(x)

∑

y

k(x, y) [g(y)− g(x)]}

(IV.14)

14

over positive functions g. The actual supremum is exactly reached for g = exp(V/2), with

the blowtorch potential V that makes µ stationary, as in (III.2). Hence,

I(µ) = D(µ)

Furthermore, D(µ) is a convex functional. That relation is of course the deeper truth behind

Proposition III.2.

Acknowledgments

C.M. benefits from the Belgian Interuniversity Attraction Poles Programme P6/02. K.N.

acknowledges the support from the Academy of Sciences of the Czech Republic under Project

No. AV0Z10100520.

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