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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: [email protected]
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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