PHYSICAL REVIEW E 85, 041125 (2012)
Stochastic thermodynamics under coarse graining
Massimiliano Esposito
Complex Systems and Statistical Mechanics, University of Luxembourg, L-1511 Luxembourg, Luxembourg
(Received 22 December 2011; published 17 April 2012)
A general formulation of stochastic thermodynamics is presented for open systems exchanging energy and
particles with multiple reservoirs. By introducing a partition in terms of “mesostates” (e.g., sets of “microstates”),
the consequence on the thermodynamic description of the system is studied in detail. When microstates within
mesostates rapidly thermalize, the entire structure of the microscopic theory is recovered at the mesostate level.
This is not the case when these microstates remain out of equilibrium, leading to additional contributions to the
entropy balance. Some of our results are illustrated for a model of two coupled quantum dots.
DOI: 10.1103/PhysRevE.85.041125 PACS number(s): 05.70.Ln, 05.40.−a
I. INTRODUCTION
The past decade has witnessed major progress in nonequi-
librium statistical mechanics. It is becoming increasingly clear
that a consistent theory of nonequilibrium thermodynamics
can be constructed for physical systems described by a
stochastic Markovian dynamics. This so-called theory of
stochastic thermodynamics generalizes the phenomenological
formulation of nonequilibrium thermodynamics developed for
systems locally close to equilibrium more than half a century
ago [1–3]. Early developments in this field were restricted
to the ensemble-averaged level and focused on steady-state
situations [4–8]. The crucial conceptual breakthrough came
later and consisted in identifying the central thermodynamic
quantities at the level of single stochastic trajectories [9–28].
The discovery of fluctuation theorems has played a major role
in this regard [29–33]. The second law of thermodynamics,
traditionally expressed as an inequality at the ensemble-
averaged level, is now understood as resulting from a universal
equality at the level of the full probability distribution of
the entropy production defined at the trajectory level [19,34].
These new theoretical developments are particularly important
for the study of small systems subjected to sufficiently large
and measurable fluctuations [35–37]. They coincided with an
unprecedented development in the experimental techniques
used to manipulate small systems and triggered a great deal
of experimental studies in a variety of contexts, such as single
molecule stretching experiments [35,38–40], nanomechanical
oscillator work measurements [41], spectroscopic measure-
ment of trajectory entropies [42,43], and electronic current
fluctuations in full counting statistics experiments [44]. Since
stochastic thermodynamics combines kinetics and thermody-
namics, it has also proved extremely useful to describe the
finite-time thermodynamics (e.g., efficiency at finite power)
of various nanodevices operating as thermodynamic machines
[45–54]. Overall, this theory is becoming a fundamental tool
for the study of nanosciences.
The stochastic description underlying stochastic thermody-
namics relies on a time-scale separation between system and
reservoirs. The slow degrees of freedom entering the stochastic
description constitute the “system.” They may be controlled by
an external time-dependent force, but are also stochastically
driven by hidden degrees of freedom which constitute the
“reservoirs.” These latter are so fast that they can be assumed to
always remain at equilibrium. They can thus be characterized
statistically by a temperature and a chemical potential. The
systematic procedures to perform the elimination of these fast
degrees of freedom (starting from a Hamiltonian description
of the complete set of degrees of freedom) are nowadays
well known [55–58]. While very fast equilibrated degrees of
freedom constituting the reservoirs are ubiquitous at small
scales (without them, the very existence of a thermodynamic
description is compromised), nontrivial differences may exist
between system degrees of freedom. This is particularly true in
modeling biological systems where each level of description
hides a significant underlying complexity. Although some of
these system degrees of freedom can be faster than others, they
may be maintained out-of-equilibrium, therefore preventing us
from modelling them as reservoirs. Alternatively, one might
only observe a subset of the true system degrees of freedom,
since correctly identifying the system states is not always an
easy task [59]. It is therefore important to understand how
to formulate stochastic thermodynamics at a coarse-grained
level of description. This is the central topic of this paper.
For convenience, we are going to call the true system states
“microstates,” and states which lump together multiple true
system states will be called “mesostates.” While recent studies
have investigated various aspects of such coarse-graining at
the level of the stochastic description [60–64], as well as
some of its implications at the level of the thermodynamic
description [65–69], the present paper analyzes the effect of
coarse-graining on stochastic thermodynamics as a whole. A
rewarding outcome of this study is that the coarse-graining
procedure a posteriori unambiguously clarifies the implicit as-
sumptions made when formulating stochastic thermodynamics
at the microscopic level.
This paper is organized as follows. A general formulation
of stochastic thermodynamics for open systems is presented
in Sec. II. The case of multiple reservoirs as well as the
limit of a single reservoir is considered. In the latter case,
standard equilibrium thermodynamics is recovered in the
reversible limit. In Sec. III, the coarse-graining procedure is
applied to the dynamics. A natural approximation scheme is
presented for situations in which the partition is motivated
by a time-scale separation between the microlevel and the
mesolevel. In Sec. IV, the effect of coarse-graining on the
entropy balance is discussed for various scenarios. In Sec. V,
applications to a double quantum dot model are presented.
Conclusions are drawn in Sec. VI.
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MASSIMILIANO ESPOSITO PHYSICAL REVIEW E 85, 041125 (2012)
II. STOCHASTIC THERMODYNAMICS
IN THE GRAND-CANONICAL ENSEMBLE
A. Multiple reservoirs
We consider a system described by a set of states i with a
given system energy
i
, number of particles n
i
, and equilibrium
entropy s
i
. Transitions between these states are induced
by multiple reservoirs ν with a given chemical potential
μ
(ν)
and temperature T
(ν)
. Each state has therefore a given
grand potential (or Landau potential) ω
(ν)
i
with regard to the
reservoir ν,
ω
(ν)
i
=
i
− μ
(ν)
n
i
− T
(ν)
s
i
. (1)
We assume that the system energy, number of particle, and
entropy of a level i, as well as the reservoirs’ chemical
potentials and temperatures (i.e., all terms in ω
(ν)
i
), may be
controlled in a time-dependent manner by an external agent.
We will refer to this process as external driving. Without loss
of generality, we parametrized this time dependence through
λ so that ˙ω
(ν)
i
=
˙
λ∂
λ
ω
(ν)
i
. The dynamics resulting from the
stochastic transitions between system states is ruled by the
Markovian master equation
˙
p
i
=
j
W
ij
p
j
. (2)
The rate matrix, which may depend on time due to the external
driving, satisfies
i
W
ij
= 0 and is assumed irreducible. The
unique stationary distribution p
st
i
is thus obtained by solving
j
W
ij
p
st
j
= 0. Since transitions can be due to different
reservoirs, the rate matrix is decomposed in their respective
contributions,
W
ij
=
ν
W
(ν)
ij
. (3)
Because reservoirs are assumed to always remain at equilib-
rium, the rate matrix satisfies local detailed balance,
W
(ν)
ij
W
(ν)
ji
= exp
−
ω
(ν)
i
− ω
(ν)
j
k
b
T
(ν)
, (4)
where k
b
is the Boltzmann constant. This property guarantees
that a system in contact with a single reservoir (or equivalently
in contact with multiple reservoirs with identical temperatures
and chemical potentials) and in the absence of external driving
will eventually reach the grand-canonical equilibrium,
p
eq
i
= exp
−
ω
i
−
eq
k
b
T
. (5)
This distribution satisfies the detailed balance condition
W
(ν)
ij
p
eq
j
= W
(ν)
ji
p
eq
i
, ∀ ν,i,j, (6)
which defines equilibrium and indicates that all currents vanish
in the system.
The system energy and number of particle are naturally
given by the ensemble averages
E =
i
i
p
i
,N=
i
n
i
p
i
. (7)
However, the system entropy is not simply the ensemble
average of the entropy of each state i but also contains an
information (Shannon-like) contribution
S =
i
[s
i
− k
b
ln p
i
]p
i
. (8)
We note that this defines entropy out-of-equilibrium.
The change in energy and number of particle can be
expressed as
˙
E =
i
i
˙
p
i
+
i
˙
i
p
i
=
ν
I
(ν)
E
+
˙
λ∂
λ
E, (9)
˙
N =
i
n
i
˙
p
i
+
i
˙
n
i
p
i
=
ν
I
(ν)
N
+
˙
λ∂
λ
N. (10)
The second contribution is due to the external driving while
the first is due to the reservoirs and is expressed in terms of
energy and matter currents entering the system,
I
(ν)
E
=
i,j
W
(ν)
ij
p
j
(
i
−
j
), (11)
I
(ν)
N
=
i,j
W
(ν)
ij
p
j
(n
i
− n
j
). (12)
The change in the system entropy,
˙
S =
i
[s
i
− k
b
ln p
i
]
˙
p
i
+
i
˙
s
i
p
i
, (13)
can be decomposed in analogy with irreversible thermody-
namics as [1,3]
˙
S =
˙
S
i
+
˙
S
e
. (14)
The non-negative entropy production is given by
˙
S
i
= k
b
ν,i,j
W
(ν)
ij
p
j
ln
W
(ν)
ij
p
j
W
(ν)
ji
p
i
0. (15)
Non-negativity is proved using ln x x − 1. Entropy pro-
duction only vanishes for reversible transformations, i.e.,
transformations along which the detailed balance condition
(6) is satisfied. The entropy flow in turn is given by
˙
S
e
=
ν,i,j
W
(ν)
ij
p
j
s
i
− k
b
ln
W
(ν)
ij
W
(ν)
ji
+
i
˙
s
i
p
i
=
ν
˙
Q
(ν)
T
(ν)
+
˙
λ∂
λ
S. (16)
The second contribution to the entropy flow is due to the
external driving which reversibly modifies the equilibrium
entropy associated to the internal structure of the states. The
first contribution is due to the reservoirs and is expressed in
term of the heat flowing from reservoir ν to the system,
˙
Q
(ν)
= I
(ν)
E
− μ
(ν)
I
(ν)
M
. (17)
As a result, the work reads
˙
W =
˙
λ∂
λ
E +
ν
μ
(ν)
I
(ν)
M
(18)
and the first law is satisfied,
˙
E =
˙
W +
ν
˙
Q
(ν)
. (19)
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STOCHASTIC THERMODYNAMICS UNDER COARSE GRAINING PHYSICAL REVIEW E 85, 041125 (2012)
We note that some authors do not incorporate the particle
current contribution in the definition of heat and work and
keep it as a distinct contribution to the entropy flow.
B. Single reservoir
We now show the simplifications which take place when the
system interacts with a single reservoir (from now on, we stop
repeating that single reservoir also refers to multiple reservoirs
with identical thermodynamic properties). Using (16) with (4),
the entropy flow becomes
T
˙
S
e
=
i
(
i
− μn
i
)
˙
p
i
+
i
˙
s
i
p
i
=
˙
Q +
˙
λ∂
λ
S, (20)
where heat can be written, using (17), (9), and (10),as
˙
Q = (
˙
E − μ
˙
N) −
˙
λ(∂
λ
E − μ∂
λ
N). (21)
Using (13) with (20), entropy production now reads
T
˙
S
i
=−
i
(ω
i
+ k
b
T ln p
i
)
˙
p
i
0. (22)
Introducing the system nonequilibrium grand potential (or
Landau potential)
= E − μN − TS =
i
(ω
i
+ k
b
T ln p
i
)p
i
, (23)
we find that entropy production can be expressed as
T
˙
S
i
=−(
˙
−
˙
λ∂
λ
) 0. (24)
If we assume that ∂
λ
n
i
= ∂
λ
s
i
= 0, as is often the case, we
find that (24) reduces to
T
˙
S
i
=−(
˙
E −
˙
λ∂
λ
E − μ
˙
N − T
˙
S) 0. (25)
In the absence of external driving, a system prepared in
an arbitrary initial nonequilibrium state will always relax
to equilibrium where all quantities stop evolving (i.e.,
˙
S
i
=
˙
S
e
=
˙
S = 0 and
˙
=
˙
E =
˙
N = 0). Their stationary value
is given by X|
eq
, where X = ,E,N,S, and |
eq
denotes
that p
i
is replaced by p
eq
i
given by (5). Note that using
(23), we verify that |
eq
=
eq
. We now consider a system
initially at equilibrium and subjected to a slow (compared to
the typical relaxation time of the system) external driving.
Its probability distribution p
i
will follow the instantaneous
equilibrium grand-canonical distribution p
eq
i
, and the grand
potential, the average energy and number of particles, and the
entropy all become state functions,
˙
λ(∂
λ
X)|
eq
=
˙
X|
eq
for X = ,E,N,S. (26)
Such quasistatic transformations are called reversible because
entropy production remains zero all along the process
˙
S
i
|
eq
=
0. Consequently, the changes in the system entropy are given
by the entropy flow,
˙
S|
eq
=
˙
S
e
|
eq
=
˙
Q
T
|
eq
+
˙
λ (∂
λ
S)|
eq
. (27)
We note that in this case, we recover the fundamental equation
of equilibrium thermodynamics from (25). This shows how
traditional equilibrium thermodynamics is recovered from
stochastic thermodynamics in the reversible limit.
III. COARSE GRAINING
We now define a set of “mesostates” denoted by k and
assume that each “microstate” i leads to a unique mesostate
k = k(i). This terminology is used for convenience and does
not necessarily refer to a notion of size. We use the compact
notation i
k
to denote microstates which lead to the mesostate
k. The probability to find the system in a mesostate k is given
by
P
k
=
i
k
p
i
k
=
i
δ
Kr
[k − k(i)]p
i
. (28)
The conditional probability to be in the microstate i
k
being in
the mesostate k is denoted
P
i
k
= p
i
k
/P
k
. (29)
We verify that
i
k
P
i
k
= 1.
A. Dynamics
Writing the master equation (2) in terms of (29), we find
˙
P
k
P
i
k
+ P
k
˙
P
i
k
=
k
P
k
ν,j
k
W
(ν)
i
k
j
k
P
j
k
. (30)
Summing this equation over i
k
, we find a master equation
ruling the dynamics of the mesostates,
˙
P
k
=
ν,k
V
(ν)
kk
P
k
. (31)
This equation is not closed because the mesoscopic rate matrix,
V
(ν)
kk
=
i
k
,j
k
W
(ν)
i
k
j
k
P
j
k
, (32)
depends on the dynamics of the microstates through P
j
k
.
We verify that
k
V
kk
= 0. In general, even for a time-
independent microscopic rate matrix, as long as the distribution
of the microlevels evolves (i.e., P
i
k
is time-dependent), V
kk
will be time-dependent.
B. Time-scale separation
We temporarily consider systems which accommodate a
coarse-graining procedure such that the dynamics between mi-
crostates belonging to the same mesostate is much faster than
that between microstates belonging to different mesostates.
In other words, the rate matrix is such that W
i
k
j
k
W
i
k
j
k
for k = k
, and the results presented in this subsection can
be proved using first-order perturbation theory as shown in
Appendix A. In such situations, the conditional probabilities
P
i
k
evolve much faster than the mesostate probabilities P
k
.On
short time scales, denoted τ
mic
, the mesostate probabilities P
k
barely change while the P
i
k
’s obey an almost isolated dynamics
inside the mesostates k, eventually relaxing to the stationary
distribution P
st
j
k
defined by
j
k
W
i
k
j
k
P
st
j
k
= 0. (33)
If the transitions between microstates belonging to a given
mesostate k are due to a single reservoir, due to the local
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MASSIMILIANO ESPOSITO PHYSICAL REVIEW E 85, 041125 (2012)
detailed balance property (4), P
st
i
k
will be given by the
equilibrium distribution
P
eq
i
k
= exp
−
ω
i
k
−
eq
(k)
k
b
T
, (34)
and all currents within the mesostate vanish (i.e., detailed
balance is satisfied within k),
W
i
k
j
k
P
eq
j
k
= W
j
k
i
k
P
eq
i
k
. (35)
Turning back to the general case of multiple reservoirs,
for times much longer than τ
mic
,theP
k
’s will start evolving
following the approximate mesostate dynamics (31),
˙
P
k
=
ν,k
V
(ν)st
kk
P
k
. (36)
This equation is closed because, thanks to the time-scale sepa-
ration, the exact mesoscopic rate matrix can be approximated
by
V
(ν)st
kk
=
i
k
,j
k
W
(ν)
i
k
j
k
P
st
j
k
. (37)
Over a characteristic time τ
mes
,theP
k
’s will also reach a
stationary distribution P
st
k
defined by
ν,k
V
(ν)st
kk
P
st
k
= 0. (38)
If the entire system is in contact with a single reservoir, the
mesoscopic rate matrix V
(ν)st
kk
becomes V
eq
kk
, meaning that P
st
i
k
is replaced by P
eq
i
k
in Eq. (37). In this case, using (4) and (32),
we recover the property of local detailed balance at the level
of the mesoscopic rates,
V
eq
kk
V
eq
k
k
= exp
−
eq
(k) −
eq
(k
)
k
b
T
. (39)
As a result, the stationary distribution of the mesoscopic states
P
st
k
is given by the equilibrium distribution
P
eq
k
= exp
−
eq
(k) −
eq
k
b
T
. (40)
Using (5), (34), and (40), we verify that
p
eq
i
k
= P
eq
k
P
eq
i
k
. (41)
This means that over times larger than τ
mes
, the full system
reaches equilibrium and the detailed balance condition (6) is
satisfied.
C. Energy and entropy
We now turn back to an arbitrary coarse-graining and
consider its effect on the system energy, number of particles,
and entropy. The system energy and particle number (7) can
be expressed as
E =
k
E(k)P
k
,N=
k
N(k)P
k
, (42)
where the average energy and number of particles conditional
on being on a mesostate k are given by
E(k) =
i
k
i
k
P
i
k
, N(k) =
i
k
n
i
k
P
i
k
. (43)
Their evolution can be expressed as
˙
E =
k
E(k)
˙
P
k
+
k
˙
E(k)P
k
, (44)
˙
N =
k
N(k)
˙
P
k
+
k
˙
N(k)P
k
. (45)
The system entropy (8) can be rewritten as
S =
k
[S(k) − k
b
ln P
k
]P
k
, (46)
where the entropy conditional on being on a mesostate k is
given by
S(k) =
i
k
s
i
k
− k
b
ln P
i
k
P
i
k
. (47)
The entropy evolution reads
˙
S =
k
[S(k) − k
b
ln P
k
]
˙
P
k
+
k
˙
S(k)P
k
, (48)
where
˙
S(k) =
i
k
s
i
k
− k
b
ln P
i
k
˙
P
i
k
+
i
k
˙
s
i
k
P
i
k
. (49)
We note that the evolution of energy, number of particles, and
entropy, expressed in terms of the mesostates [(44), (45), and
(48)], has the same form as the original evolution expressed in
terms of microstates [(9), (10), and (13)]. The key difference
is that the evolution of the quantities defined on the mesostates
is no longer exclusively due to the external driving, but also
contains the internal dynamics of the mesostates. Remarkably,
the form of this internal dynamics expressed in term of
conditional probabilities, P
i
k
, is also the same as the original
evolution expressed in terms of microstates.
IV. ENTROPY BALANCE
A. Single reservoir
We now formulate the entropy balance for an arbitrary
coarse-graining in the case of a single reservoir. Using (29) and
(30) in Eqs. (20) and (21), the entropy flow can be rewritten as
T
˙
S
e
=
˙
Q +
˙
λT
k
∂
λ
S(k)P
k
, (50)
where heat is given by
˙
Q =
k
[E(k) − μN(k)]
˙
P
k
+
k
(
˙
E(k)
− μ
˙
N(k) −
˙
λ[∂
λ
E(k) − μ∂
λ
N(k)])P
k
. (51)
It is worth noting that
˙
E(k) − μ
˙
N(k) −
˙
λ[∂
λ
E(k) − μ∂
λ
N(k)]
=
i
k
i
k
+ μn
i
k
˙
P
i
k
. (52)
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STOCHASTIC THERMODYNAMICS UNDER COARSE GRAINING PHYSICAL REVIEW E 85, 041125 (2012)
The form taken by the entropy flow at the mesostate level
contains two types of contributions. The first is made of the
second term of (50) and the first term of (51). It has the exact
same form as the entropy flow at the microstates level (20) with
(21). The second contribution consists in an ensemble average
over the mesostates probabilities, P
k
, of the heat flow within
each mesostate. We now turn to entropy production. Using the
system entropy (48), we find that entropy production (22) can
be written as
T
˙
S
i
=−
k
[(k) + k
b
T ln P
k
]
˙
P
k
−
k
[
˙
(k) −
˙
λ∂
λ
(k)]P
k
. (53)
We introduced the mesostate grand potential
(k) = E(k) − μN(k) − T S (k)
=
i
k
ω
i
k
+ k
b
T ln P
i
k
P
i
k
, (54)
which is connected to the grand potential (23) by
=
k
[(k) + k
b
T ln P
k
]P
k
. (55)
It is worth noting that
˙
(k) −
˙
λ∂
λ
(k) =
i
k
i
k
+ k
b
T ln P
i
k
˙
P
i
k
. (56)
As for the entropy flow, the form taken by the entropy
production at the mesostate level contains two types of
contributions. The first term in Eq. (53) has the exact same
form as the entropy production at the microstates level (22),
and the second term is an ensemble average over the mesostates
probabilities, P
k
, of the entropy production arising from within
each mesostate.
We now turn to the situation described in Sec. III B where
microvariables evolve faster than mesovariables. We consider
the system evolution over time scales longer than τ
mic
.We
assume that the external driving is sufficiently slow to keep
the microstates within mesostates at equilibrium, i.e., P
i
k
is
replaced by P
eq
i
k
given by (34). It can, however, be fast enough
to keep the mesostate probabilities, P
k
, far from equilibrium.
As a result, using (54), we verify that
(k)|
eq
=
eq
(k), (57)
where |
eq
means that P
i
k
in the expression has to be replaced
by P
eq
i
k
. Defining
X
eq
(k) ≡ X(k)|
eq
for X = ,E,N,S, (58)
we find the important property
˙
λ∂
λ
X
eq
(k) =
˙
X(k)|
eq
for X = ,E,N,S, (59)
which translates the fact that microstates within mesostates
evolve reversibly. As a result, using (48), the system entropy
evolves as
˙
S =
k
[S
eq
(k) − k
b
ln P
k
]
˙
P
k
+
˙
λ
k
∂
λ
S
eq
(k)P
k
. (60)
The entropy flow, using (59) with (50) and (51), becomes
T
˙
S
e
=
˙
Q +
˙
λ
k
∂
λ
S
eq
(k)P
k
, (61)
where heat is given by
˙
Q =
k
[E
eq
(k) − μN
eq
(k)]
˙
P
k
. (62)
Entropy production, using (59) with (53) and (59), reads
T
˙
S
i
=−
k
[
eq
(k) + k
b
T ln P
k
]
˙
P
k
. (63)
We notice that the second term in Eq. (51) as well as in Eq. (53),
which both arise from the dynamics within the mesostates,
have vanished due to (59). Using the local detailed balance
property of the mesoscopic rates (39), the entropy flow (61)
can finally be rewritten as
˙
S
e
=
k,k
V
eq
kk
P
k
S
eq
(k) − k
b
ln
V
eq
kk
V
eq
k
k
+
˙
λ
k
∂
λ
S
eq
(k)P
k
(64)
and the entropy production (63) as
˙
S
i
= k
b
k,k
V
eq
kk
P
k
ln
V
eq
kk
P
k
V
eq
k
k
P
k
0. (65)
This clearly shows that when microstates within mesostates are
at equilibrium, stochastic thermodynamics assumes the same
form at the mesostate level as at the microstate level. This can
be clearly seen when comparing (63) with (22) or (65) with (15)
as well as when comparing (61) with (20) or (64) with (16).
This important result demonstrates a posteriori that the theory
of stochastic thermodynamics makes a key assumption at its
most fundamental level of description described in Sec. II:
the internal structure of the states entering the stochastic
description may evolve due to external driving but always
does so reversibly, i.e., by remaining at equilibrium. In the
next section, we will generalize this result to mesostates in
contact with multiple reservoirs.
All terms containing
˙
λ vanish in the absence of driving.
However, the terms containing
˙
λ at the level of the internal
dynamics of the mesostates [e.g., in Eqs. (60) and (61)]may
be omitted even in the presence of driving if this latter can be
assumed to act similarly on all the microstates belonging to
the same mesostate.
B. Multiple reservoirs
We now consider the most general case of an arbitrary
coarse-graining and different reservoirs. We start by separating
the evolution of the system entropy (46) in three contributions,
˙
S =
˙
S
(1)
+
˙
S
(2)
+
˙
S
(3)
. (66)
The first is the evolution of the (Shannon) information
entropy expressed in terms of the mesostate probabilities. It
corresponds to the entropy evolution of a system made of
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