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PHYSICAL REVIEW E 82, 030104共R兲 共2010兲
Unifying approach for fluctuation theorems from joint probability distributions
Reinaldo García-García,1 Daniel Domínguez,1 Vivien Lecomte,2,3 and Alejandro B. Kolton1
1
Centro Atómico Bariloche and Instituto Balseiro, 8400 S. C. de Bariloche, Argentina
Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR 7599), Université Pierre et Marie Curie–Paris VI
and Université Paris-Diderot–Paris VII, 2 Place Jussieu, 75005 Paris, France
3
DPMC-MaNEP, Université de Genève, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland
共Received 12 July 2010; published 22 September 2010兲
2
Any decomposition of the total trajectory entropy production for Markovian systems has a joint probability
distribution satisfying a generalized detailed fluctuation theorem, when all the contributing terms are odd with
respect to time reversal. The expression of the result does not bring into play dual probability distributions,
hence easing potential applications. We show that several fluctuation theorems for perturbed nonequilibrium
steady states are unified and arise as particular cases of this general result. In particular, we show that the joint
probability distribution of the system and reservoir trajectory entropies satisfy a detailed fluctuation theorem
valid for all times.
DOI: 10.1103/PhysRevE.82.030104
PACS number共s兲: 05.70.Ln, 05.40.⫺a
The nonequilibrium stochastic thermodynamics of small
systems has attracted a lot of attention. From the experimental side the development of techniques for microscopic manipulation has allowed one to study fluctuations in small systems 关1,2兴. From the theoretical side a group of exact
relations known as fluctuation theorems 共FTs兲 关3–8兴 has shed
light into the principles governing dissipation and fluctuations in nonequilibrium phenomena, as in driven systems in
contact with thermal bath. Formally, the generality of the FTs
can be attributed to the way probability distribution functions
of different observables behave under time-reversal
symmetry-breaking perturbations 共see 关9,10兴 for reviews on
FT兲.
Oono and Paniconi 关11兴 proposed a phenomenological
framework for a “nonequilibrium steady-state 共NESS兲 thermodynamics” aimed at describing fluctuating systems subjected to an external protocol. In this approach, the total exchanged heat during a time interval ␶ by a system initially
prepared in a NESS is expressed as the sum of two contributions: Qtot = Qex + Qhk. The “excess heat” Qex is, in average,
associated with the energy exchange during transitions between steady states while the “housekeeping heat” Qhk corresponds, in average, to the energy we need to supply to
maintain the system in a NESS. Hatano and Sasa 关12兴 introduced a formal framework for these phenomenological ideas
and derived a FT which extends the second law of thermodynamics for transitions between NESSs controlled by external parameters ␴共t兲. The Hatano and Sasa FT is applicable to
trajectories x共t兲 evolved with a Langevin or more generally a
Markovian dynamics starting from an initial condition
sampled from a NESS compatible with the initial values ␴共0兲
of the control parameters.
Under identical conditions different FTs were subsequently proposed for NESS, involving the above decomposition of the exchanged heat. We can distinguish between
the so-called integral and detailed FTs for systems initially
prepared as described above. The integral fluctuation theorems 共IFTs兲 are exact relations for the average over histories
of different stochastic trajectory functionals W关x兴, such as
具e−W典 = 1. Examples are the Jarzynski FT for the total work
关7兴, the Hatano-Sasa FT 关12兴, and the Speck-Seifert
1539-3755/2010/82共3兲/030104共4兲
FT 关13兴. The so-called detailed FTs 共DFTs兲 are, on the
other hand, exact relations for the probability distribution
functions 共PDFs兲 of different observables W, such as
P共W兲 / PR共−W兲 = eW, where PR共W兲 corresponds to the timereversed protocol ␴R共t兲 = ␴共␶ − t兲, and a NESS initial condition compatible with ␴共␶兲. Examples are given by the Crooks
relation 关8兴 or Seifert fluctuation theorem 关14兴. While
observables satisfying a DFT trivially satisfy an IFT, the opposite is not always true. In many recently formulated DFTs,
a modified “dual” PDF P†R共W兲 enters into play 关15,16兴,
which corresponds to trajectories in a system with same
stationary PDF but with reversed steady probability current
关i.e., such that P†R共W兲 = PR共W兲 when detailed balance holds兴.
In general the dual dynamics is different from the
original dynamics of the system. Thus, the presence of dual
distributions is a strong limitation to the experimental use
of such a DFT. A central result of our work is that
generalized DFTs can be established without relying on dual
probabilities by considering joint probability distributions
for different complementary observables, instead of
single PDFs. The joint probability distributions arise
naturally from the above mentioned separation of two contributions to the total heat exchanged in a NESS. From this
joint DFT all the known DFTs and IFTs follow in a straightforward way.
Among the fluctuation theorems formulated for Markovian dynamics, the total trajectory entropy production
S关x ; ␴兴 = ln共P关x ; ␴兴 / P关xR ; ␴R兴兲 plays a fundamental role
关5,6,14兴. Here, P关x ; ␴兴 共P关xR ; ␴R兴兲 denotes the probability
density of trajectory x 共time-reversed trajectory xR兲 in the
forward 共backward兲 protocol. We include in P the initial distribution of x. We omit hereafter the final time ␶ in all trajectory functionals and use calligraphic symbols to denote
functionals and normal symbols to denote their values. The
total trajectory entropy production is odd upon time reversal:
S关xR ; ␴R兴 = −S关x ; ␴兴. We show that any decomposition of S
in M distinct contributions, S关x ; ␴兴 = 兺iM Ai关x ; ␴兴, each of
them being odd Ai关xR ; ␴R兴 = −Ai关x ; ␴兴, has a generating
function satisfying the symmetry
030104-1
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GARCÍA-GARCÍA et al.
冓
冉
M
exp − 兺 ␭iAi关x; ␴兴
=
冓冉
i
PHYSICAL REVIEW E 82, 030104共R兲 共2010兲
冊冔
M
exp − 兺 共1 − ␭i兲Ai关x; ␴R兴
i
冊冔
␤Qtot关x; ␴兴 = − ␤
共1兲
,
R
with S = 兺 Ai .
=
DxP关xR ; ␴R兴O关x; ␴兴e−S关x
DxP关x; ␴R兴O关xR ; ␴兴e
␶
dtẋ兵⳵x␾共x; ␴兲 − ␤关⳵xU共x; ␣兲 − f兴其, 共6兲
for the housekeeping heat, and
␤Qex关x; ␴兴 = −
= 具exp兵−
兺Ni 共1
共3兲
− S关x; ␴ 兴兲典R
R
− ␭i兲Ai关x; ␴ 兴其典R .
R
Equation 共2兲 is proved in a similar way, or also follows from
Eq. 共1兲 since it is a symmetry for the generating function of
the joint distribution P共A1 , . . . , A M 兲. Before considering their
particular application for explicit decompositions of S we
note that symmetries 共1兲 and 共2兲 are valid for all times ␶ for
systems prepared in any initial distribution Pi(x共0兲). In particular, we see that the total trajectory entropy production
FTs 具e−S典 = 1 and P共S兲 / PR共−S兲 = eS hold without further assumption.
We will consider two generic frameworks where our result applies: systems described 共i兲 by Langevin dynamics and
共ii兲 by a Markovian dynamics on discrete configurations, exemplifying their parallel features. First we consider a particle
driven by a constant force f in a potential U,
ẋ = − ⳵xU„x; ␣共t兲… + f + ␰ ,
dtẋ⳵x␾共x; ␴兲,
共7兲
for the excess heat. The Hatano-Sasa functional 关12兴 Y关x ; ␴兴
is then defined as
Y关x; ␴兴 ⬅
R;␴R兴
where we have used S关x ; ␴兴 = −S关xR ; ␴R兴 together with the
change of variable x → xR. Considering S关x ; ␴兴 = 兺iM Ai关x ; ␴兴
the proof comes around:
= 具exp共−
冕
␶
冕
␶
dt␴˙ ⳵␴␾共x; ␴兲 = ␤Qex关x; ␴兴 + ⌬␾共x; ␴兲, 共8兲
0
R
具exp共−
冕
0
−S关x;␴R兴
兺iM ␭iAi关xR ; ␴兴
共5兲
0
i=1
= 具O关xR ; ␴兴e−S关x;␴ 兴典R ,
兺iM ␭iAi关x; ␴兴兲典
␤Qhk关x; ␴兴 =
共2兲
Note that the result involves no use of dual PDFs. To prove
Eq. 共1兲 we start by noting that the average of any observable
O关x兴 over trajectories satisfies the relation
冕
冕
dtẋ关⳵xU共x; ␣兲 − f兴.
The total exchanged heat in a trajectory can be split as
Qtot = Qhk + Qex 关11兴. Defining ␾共x ; ␴兲 = −ln ␳SS共x , ␴兲 from
the steady-state probability density ␳SS共x , ␴兲 at fixed values
of ␴ = 共␣ , f兲 Hatano and Sasa 关12兴 proposed
M
P共A1,A2, . . . ,A M 兲
= eS
R
P 共− A1,− A2, . . . ,− A M 兲
␶
0
where ␭i are arbitrary parameters and 具 . . . 典R denotes average
over trajectories in the reversed protocol 关17兴. This symmetry is equivalent to the following generalized DFT for the
joint probability of Ai关x ; ␴兴’s:
具O关x; ␴兴典 =
冕
共4兲
where ␣共t兲 represents a set of control parameters and
␰共t兲 is the Gaussian white noise, with 具␰共t兲典 = 0 and
具␰共t兲␰共t⬘兲典 = 2T␦共t − t⬘兲 due to a thermal bath at temperature T.
For example, this system can be out of equilibrium when U
has periodic boundary conditions. We consider for simplicity
a single degree of freedom x, but our results are easily generalized, e.g., in larger dimensions and/or with more particles. For a stochastic trajectory generated by Eq. 共4兲 we
define the total exchanged heat as 关18兴
where ⌬␾共x ; ␴兲 = ␾(x共␶兲 ; ␴共␶兲) − ␾(x共0兲 ; ␴共0兲) is a time
boundary term.
We now assume that the system is initially prepared with
a distribution corresponding to a NESS compatible with
␴共0兲, Pi(x共0兲) = ␳SS(x共0兲 , ␴共0兲). With the previous definitions
of Eqs. 共5兲–共8兲 it is known that the total trajectory entropy
production S can be decomposed as the sum of two contributions, in two different ways 关10,11,13,16兴:
S = Y + ␤Qhk = ⌬␾ + ␤Qtot .
共9兲
Similar decompositions also exist for Markovian
dynamics: we consider now discrete configurations 兵C其
with transition rates W共C → C⬘ ; ␴兲 between configurations.
They depend on ␴, an external control parameter which may
vary in time. The probability density at time t obeys the
Markovian dynamics ⳵t P共C , t兲 = 兺C⬘W(C⬘ → C ; ␴共t兲)P共C⬘ , t兲
− r(C ; ␴共t兲)P共C , t兲, where r共C ; ␴兲 = 兺C⬘W共C → C⬘ ; ␴兲 is the escape rate from configuration C. A history of the system is
described by the succession of configuration 共C0 , . . . , CK兲
visited by the system, with Ck being visited between
tk and tk+1. Starting from initial distribution Pi(C共0兲 , ␴共0兲),
the
probability
of
a
history
is
P关C ; ␴兴
K
W共Ck−1 → Ck , ␴tk兲Pi(C共0兲 , ␴共0兲),
= exp关−兰0␶ dtr(C共t兲 ; ␴共t兲)兴兿k=1
meaning that the mean value of an historydependent
observable
O
is
given
by
具O典
We
obtain
= 兺Kⱖ0兺C0¯CK兰t0dtK¯兰t02dt1O关C , ␴兴P关C , ␴兴.
that the total trajectory entropy production S关C ; ␴兴
= ln共P关C ; ␴兴 / P关CR ; ␴R兴兲 has a first decomposition S = ⌬␾
+ ␤Qtot with ⌬␾ = log关Pi(C共0兲 , ␴共0兲) / Pi(C共␶兲 , ␴共␶兲)兴 and
K
␤Qtot = 兺 log
k=1
W共Ck−1 → Ck, ␴tk兲
W共Ck → Ck−1, ␴tk兲
.
共10兲
Although there is no natural definition of ␤ we write ␤Qtot to
exemplify the parallel with Langevin dynamics.
Turning to the first decomposition, let us now assume
that the initial distribution is steady state: Pi = ␳SS = e−␾.
One directly checks that the Hatano-Sasa functional
Y关C , ␴兴 = 兰0␶ dt␴˙ ⳵␴␾ can be written as
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UNIFYING APPROACH FOR FLUCTUATION THEOREMS…
K
Y = 关␾共C, ␴兲兴0␶ − 兺 关␾共Ck, ␴tk兲 − ␾共Ck−1, ␴tk兲兴.
共11兲
k=1
Let us now derive from a unified view the known FTs. As
an intermediate step, it is useful to define a dual trajectory
weight P†关x兴 as 关15,16兴
P†关x; ␴兴 = P关x; ␴兴e−␤Q
Besides, defining the housekeeping work as
K
␤Qhk关C, ␴兴 = 兺 log
k=1
W共Ck−1 → Ck, ␴tk兲
W共Ck → Ck−1, ␴tk兲
K
+ 兺 ␾共Ck, ␴tk兲
k=1
− ␾共Ck−1, ␴tk兲,
共12兲
we check that the decomposition S = Y + ␤Qhk holds 关19兴.
The parallel between Markovian and Langevin frameworks
also appears by specializing to rates W共k → k ⫾ 1兲
= e−共␤/2兲共Vk⫾1−Vk兲 of jumping on a one-dimensional lattice
from site k to k ⫾ 1 in a tilted potential Vk = Uk − kf: in the
continuum limit, one recovers the Langevin observables
关19兴.
In the first decomposition, Y can be identified with the
so-called nonadiabatic contribution 共since it vanishes for
quasistatic protocols兲 to the trajectory entropy Sna ⬅ Y, while
␤Qhk 共which is continuously produced in the steady state兲
can be identified with the so-called adiabatic part Sa
⬅ ␤Qhk 关16兴. On the other hand, in the second decomposition of S, ⌬␾ can be identified with the system contribution
Ss ⬅ ⌬␾, while ␤Qtot can be identified with the reservoir
contribution Sr ⬅ ␤Qtot to the total trajectory entropy production.
The entropy decompositions of Eq. 共9兲 satisfy the conditions for the application of identity 共1兲 or 共2兲 since each term
is odd with respect to time reversal in both decompositions.
We can thus write DFTs 共valid for all times ␶兲 for the joint
probabilities as
P共Y, ␤Qhk兲
Y+␤Qhk
,
R
hk = e
P 共− Y,− ␤Q 兲
共13兲
P共⌬␾, ␤Qtot兲
tot
= e ⌬␾+␤Q .
P 共− ⌬␾,− ␤Qtot兲
共14兲
R
The corresponding IFTs can be written as
hk
hk
具e−␭Y−␬␤Q 典 = 具e−共1−␭兲YR−共1−␬兲␤QR 典R ,
tot
tot
具e−␭⌬␾−␬␤Q 典 = 具e−共1−␭兲⌬␾R−共1−␬兲␤QR 典R .
共15兲
共16兲
Here, XR denotes X关x ; ␴R兴, with X representing Y, Qhk, Qtot,
or ⌬␾. From Eqs. 共13兲 and 共14兲 we have, in terms of Ss, Sr,
Sa, and Sna, that P共Ss , Sr兲 / PR共−Ss , −Sr兲 = eSs+Sr and
P共Sa , Sna兲 / PR共−Sa , −Sna兲 = eSa+Sna. It is worth noting that
these relations do not involve dual PDFs, and thus they can
be tested for a physical system with a given dynamics. We
also note that while one can show that Sa and Sna satisfy each
one separately a DFT by using dual PDFs 关16兴, Ss and Sr
satisfy a joint DFT although they do not satisfy separately a
DFT. 共In the asymptotic long-time limit a DFT for Sr can be
formulated for systems with bounded energy under steadystate conditions 关3–6,14兴, while for systems with unbounded
energies an “extended fluctuation relation” is necessary
关20兴.兲
hk关x;␴兴
共17兲
.
For Markovian dynamics the dual probability P† corresponds
to the dynamics in the so-called dual rates W†共C → C⬘ , ␴兲
⬅ e−关␾共C⬘,␴兲−␾共C,␴兲兴W共C⬘ → C , ␴兲 which share the same steady
state as the original dynamics. In the case of the Langevin
dynamics of Eq. 共4兲, it corresponds to trajectories
generated by the equation ẋ = −⳵xU†(x ; ␣共t兲) + f † + ␰, with
U† = 2␾ / ␤ − U and f † = −f. This equation also has the same
steady state as the original one. Let us stress that the dual
dynamics corresponds to trajectories in a different physical
system. We now follow Eq. 共17兲 to write the dual joint PDF
related to Eq. 共13兲 as
hk
P†共Y, ␤Qhk兲 = P共Y,− ␤Qhk兲e␤Q ,
共18兲
which is normalized. Integrating this relation over Y, we first
hk
obtain the DFT 关16兴 P共␤Qhk兲 = e␤Q P†共−␤Qhk兲, and hence
hk
the IFT 具e−␤Q 典 = 1 关13兴. Using now successively Eqs. 共13兲
and 共18兲,
P共Y兲 = eY
= eY
冕
冕
hk
d共␤Qhk兲e␤Q PR共− Y,− ␤Qhk兲
d共␤Qhk兲P†R共− Y, ␤Qhk兲,
共19兲
we see that the DFT P共Y兲 = eY P†R共−Y兲 关15兴 holds. This implies the corresponding IFT 具e−Y典 = 1 关12兴 关also derived from
setting ␭ = 1 and ␬ = 0 in Eq. 共15兲 and using the Speck-Seifert
IFT兴. Finally, we note that although we have so far assumed
steady-state initial conditions, relations 共14兲 and 共16兲 remain
valid for an arbitrary initial distribution Pi(C共0兲 , ␴共0兲) by
replacing ⌬␾ → log关Pi(C共␶兲 , ␴共␶兲) / Pi(C共0兲 , ␴共0兲)兴. By taking ␭ = ␬ = 1 in Eq. 共16兲 we thus reobtain the IFT noted by
Seifert 关14兴.
As an illustration of our approach, let us show how joint
FTs provide insights on the experimental error in the evaluation of entropy productions. Consider an experiment where
the steady state can be evaluated for different values of the
control parameter ␴, e.g., microspheres optically driven in a
liquid 关21兴. Having in hand an experimental evaluation ␾exp
of ␾, we write
S = Ye + ␦Y + ␤Qhk ,
共20兲
where Ye关x ; ␴兴 = 兰0␶ dt␴˙ ⳵␴␾exp and ␦Y = Y − Ye is the difference between exact and experimental Hatano-Sasa functionals. Starting from the NESS associated with ␴共0兲 and assuming that each of the terms in Eq. 共20兲 is odd upon time
reversal, we can use Eq. 共2兲 for M = 3, obtaining
P共Y e, ␦Y, ␤Qhk兲 = PR共− Y e,− ␦Y,− ␤Qhk兲eY e+␦Y+␤Q .
hk
Using Eq. 共18兲, this gives P共Y e , ␦Y兲 = P†R共−Y e , −␦Y兲eY e+␦Y ,
and hence also the IFT
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具e−Ye典 = 具e−␦YR典R† .
共21兲
From the analysis of 具e−␦YR典R† for specific cases of experimental errors, one can estimate the dispersion of the experimentally obtained 具e−Ye典 around 具e−Y典 = 1 关19兴.
As a second example let us consider a system initially
prepared in a NESS with a variation of its parameters
␴i共t兲 = ␴0i + ␦␴i共t兲 in such a way that 兩␦␴i共t兲 / ␴0i 兩 Ⰶ 1, with
␴0i = ␴i共0兲 and ␦␴i共0兲 = 0. In this case a modified fluctuationdissipation theorem has been derived in 关22兴, which relates
dissipation under small perturbations around a NESS with
fluctuations in the corresponding steady state. Expanding the
exponentials in Eq. 共15兲 up to second order in ␦␴ and then to
second order in ␭ and order zero in ␬, we arrive at 共see 关19兴
for details兲
具Bij共0, ␶兲典0 =
冓
冋
Bij共␶,0兲exp − ␤
冕
␶
0
dtẋ共t兲vs„x共t兲; ␴0…
册冔
,
0
共22兲
where Bij共t , t⬘兲 = 关⳵␾(x共t兲 ; ␴ ) / ⳵␴i兴关⳵␾(x共t⬘兲 ; ␴ ) / ⳵␴ j兴 and
vs = ␤−1⳵x␾ − 共⳵xU − f兲 corresponds to the average velocity in
the NESS associated with ␴. For a Boltzmann-Gibbs steady
state, this result reduces to the symmetry Cij共␶兲 = Cij共−␶兲, with
Cij共␶兲 = 具Bij共0 , ␶兲典0. Equation 共22兲 can also be derived from
Eq. 共19兲 in Ref. 关15兴. With the use of the joint PDF we can
obtain further new results. Let us introduce a weighted correlation function as
0
CW
ij 共␶,0兲
=
冓
冋
0
␤
Bij共0, ␶兲exp −
冓冋
exp −
␤
2
2
冕
␶
0
冕
␶
0
dtẋ共t兲vs„x共t兲; ␴0…
dtẋ共t兲vs„x共t兲; ␴ …
0
册冔
册冔
0
.
0
共23兲
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This correlation function carries explicit information about
the lack of detailed balance and reduces to the usual one
when the system is able to equilibrate. Using Eq. 共15兲 with
␬ = 21 and repeating the same procedure used to obtain Eq.
W
共22兲 we arrive at the symmetry CW
ij 共␶ , 0兲 = Cij 共0 , ␶兲, which
reduces to the known result for equilibrium dynamics when
detailed balance holds.
In conclusion, identities 共1兲 and 共2兲 and their immediate
consequences for Markovian systems are the main results of
this work. Equation 共1兲 or Eq. 共2兲 indeed contains, as particular cases, several known FTs such as the ones previously
derived by Hatano and Sasa 关12兴, Speck and Seifert 关13兴,
Seifert 关14兴, Chernyak et al. 关15兴, and Esposito and Van den
Broeck 关16兴. In addition, an exact DFT, valid for all times ␶,
holds for the joint distribution of the reservoir and system
entropy contributions to the total trajectory entropy production, although each contribution does not do it separately, as
given by Eq. 共14兲. Also a similar DFT holds for the joint
distribution of the adiabatic and nonadiabatic entropy contributions to the total trajectory entropy, as given by Eq. 共13兲.
For the type of NESS discussed here, two-variable joint
PDFs are all that is needed for the corresponding DFTs since
there are M = 2 decompositions of the total trajectory entropy
production, like Eq. 共9兲, for this case. We have shown an
example with M = 3 for handling experimental errors in the
Hatano-Sasa FT. In any case, in the light of Eqs. 共1兲 and 共2兲,
obtaining an adequate minimal M decomposition of the total
trajectory entropy production constitutes the cornerstone toward the derivation of generalized FTs for nonequilibrium
systems.
This work was supported by CNEA, CONICET 共Grant
No. PIP11220090100051兲, and ANPCYT 共Grant No.
PICT2007886兲. V.L. was supported in part by the Swiss
NSF under MaNEP and Division II, and thanks CNEA for
hospitality.
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