Entropy fluctuation theorems in driven open systems: Application to electron counting statistics
Massimiliano Esposito,*Upendra Harbola, and Shaul MukamelDepartment of Chemistry, University of California, Irvine, California 92697, USA 共Received 30 April 2007; revised manuscript received 4 July 2007; published 26 September 2007兲
The total entropy production generated by the dynamics of an externally driven systems exchanging energy and matter with multiple reservoirs and described by a master equation is expressed as the sum of three contributions, each corresponding to a distinct mechanism for bringing the system out of equilibrium: Non-equilibrium initial conditions, external driving, and breaking of detailed balance. We derive three integral fluctuation theorems共FTs兲 for these contributions and show that they lead to the following universal inequality: An arbitrary nonequilibrium transformation always produces a change in the total entropy production greater than or equal to the one produced if the transformation is done very slowly共adiabatically兲. Previously derived fluctuation theorems can be recovered as special cases. We show how these FTs can be experimentally tested by performing the counting statistics of the electrons crossing a single level quantum dot coupled to two reservoirs with externally varying chemical potentials. The entropy probability distributions are simulated for driving protocols ranging from the adiabatic to the sudden switching limit.
DOI:10.1103/PhysRevE.76.031132 PACS number共s兲: 05.70.Ln, 05.40.⫺a, 73.63.⫺b
I. INTRODUCTION
In statistical mechanics, thermodynamic laws are recov-ered at the level of ensemble averages. The past decade has brought new insights into nonequilibrium statistical mechan-ics due to the discovery of various types of fluctuation rela-tions valid arbitrarily far from equilibrium关1–17兴. These re-lations identify, at the level of the single realization of a statistical ensemble, the “trajectory entropy” which upon en-semble averaging reproduces the thermodynamic entropy. They therefore quantify the statistical significance of non-thermodynamic behaviors which can become significant in small systems关18,19兴. Various experimental verifications of these FTs have been reported关20–26兴.
In this paper, we consider an open system, described by a master equation 共ME兲, exchanging matter and energy with multiple reservoirs. The system can be externally driven by varying its energies or the different temperature or chemical potentials of the reservoirs. There are three mechanisms for bringing such a system out of equilibrium: preparing it in a nonequilibrium state, externally driving it, or putting it in contact with multiple reservoirs at different temperatures or chemical potentials thus breaking the detailed balance con-dition共DBC兲. We show that each of these mechanisms make a distinct contribution to the total entropy production共EP兲 generated by the nonequilibrium dynamics of the system. The two first contributions are nonzero only if the system is not in its steady state and are therefore called nonadiabatic. The third contribution is equal to the EP for slow transfor-mation during which the system remains in a steady-state and is therefore called adiabatic. We derive three FTs, for the total EP and its nonadiabatic and adiabatic contribution and show that they lead to exact inequalities valid arbitrary far from equilibrium. Previously derived FTs are recovered by
considering specific types of nonequilibrium transforma-tions. Steady state FTs 关6,12,13兴 are obtained for systems maintained in a nonequilibrium steady state共NESS兲 between reservoirs with different thermodynamic properties. The Jarzynski or Crooks type FTs关4,8,9兴 are derived for systems initially at equilibrium with a single reservoir which are ex-ternally driven out of equilibrium by an external force. The Hatano-Sasa FT 关10,11兴 is recovered for externally driven systems initially in a NESS with multiple reservoirs.
To calculate the statistical properties of the various con-tributions to the total EP and to demonstrate the FTs, we extend the generating function 共GF兲 method 关6兴 to driven open systems. Apart from providing clear proofs of the vari-ous FTs, this method is useful for simulations because it does not require to explicitly generate the stochastic trajectories. Some additional insight is provided by using an alternative derivation of the FTs similar to the Crooks derivation关8,9兴, where the total EP and its nonadiabatic part can be identified in terms of forward-backward trajectory probabilities. By do-ing so, we connect the trajectory approach previously used for driven closed systems 关8,9,14兴 with the GF approach used for steady state systems关6,12兴.
We propose to experimentally test these new FTs in a driven single orbital quantum dot where the various entropy probability distributions can be measured by the full electron counting statistics which keeps track of the four possible types of electron transfer共in and out of the dot through either lead兲. Such measurements of the bidirectional counting sta-tistics have become feasible recently关27兴. We calculate the entropy probability distributions, analyze their behavior as the driving is varied between the sudden and the adiabatic limits, and verify the validity of the FTs.
In Sec. II we present our stochastic model and in Sec. III we describe the various contributions to the total EP gener-ated during a nonequilibrium transformation and the in-equalities that these contributions satisfy. In Sec. IV, we de-fine the various trajectory entropies which upon ensemble averaging give the various contributions to the EP. We then present the GF formalism used to calculate the statistical properties of these trajectory entropies. In Sec. V, we derive *Also at Center for Nonlinear Phenomena and Complex Systems,
the various FTs and the implied inequalities. Alternative deri-vations of FTs in terms of forward-backward trajectories are given in the Appendix A. By considering specific nonequi-librium transformations, we recover most of the previously derived FTs. Finally in Sec. VI, we apply our results to the full counting statistics of electrons in a driven quantum dot. Conclusions are drawn in Sec. VII.
II. MASTER EQUATION
We consider an externally driven open system exchanging particles and energy with multiple reservoirs. Each state m of the system has a given energy⑀mand Nmparticles. The total number of states m is finite and equal to M. The probability to find the system in a state m at time t is denoted by pm共t兲. The evolution of this probability is described by the ME
p˙m共t兲 =
兺
m⬘Wm,m⬘共t兲pm⬘共t兲, 共1兲 where the rate matrix satisfies
兺
m Wm,m⬘共t兲 = 0. 共2兲 We assume that if a transition from m to m⬘
can occur, the reversed transition from m⬘
to m can also occur. Various parameters, such as the energies ⑀m of the system or the chemical potential and the temperature−1 of the res-ervoir can be varied in time externally according to a known protocol. This is described by the dependence of the rate matrix on several time-dependent parameterst. If the tran-sition rates are kept constant, the system will eventually reach the unique steady state solution pmst共兲 which satisfies
p˙m
st共兲=0 关
28兴.
The transition rates will be expressed as sums of contri-butions from different reservoirs
Wm,m⬘共兲 =
兺
Wm,m共兲 ⬘共兲, 共3兲 each satisfying Wm共兲⬘,m共兲 Wm,m共兲 ⬘共兲= exp兵共兲关„⑀m共兲 −⑀m⬘共兲… −共兲共Nm− Nm⬘兲兴其. 共4兲 If all reservoirs have the same thermodynamic properties 共temperature−1 and chemical potential兲, the steady statedistribution coincides with the equilibrium distribution
pmst共兲=pmeq共兲 which satisfies the detailed balanced condition
共DBC兲
Wm,m共兲⬘共兲pmeq⬘共兲 = Wm共兲⬘,m共兲pm
eq共兲. 共5兲
As a consequence of Eqs.共4兲 and 共5兲, the equilibrium distri-bution then assumes the grand canonical form
pmeq共兲 =
exp关−共兲„⑀m共兲 −共兲Nm…兴
⌶共兲 , 共6兲
where⌶共兲 is the grand canonical partition function. How-ever, in the general case where the reservoirs have different 共兲and共兲, the DBC does not hold and p
m
st共兲 is a NESS.
III. ENTROPIES
The Gibbs entropy of the system is a state function de-fined as
S共t兲 ⬅ −
兺
mpm共t兲ln pm共t兲. 共7兲 Using Eqs.共1兲 and 共2兲, the system EP reads
S˙共t兲 = −
兺
m p˙m共t兲ln pm共t兲 = −兺
m,m⬘ Wm,m⬘共t兲pm⬘共t兲ln pm共t兲 pm⬘共t兲 . 共8兲 This can be partitioned as关12,14,32–34兴S˙共t兲 = S˙tot共t兲 − S˙r共t兲, 共9兲 with the total EP
S˙tot共t兲 ⬅ −
兺
m,m⬘, Wm,m共兲 ⬘共t兲pm⬘共t兲ln Wm共兲⬘,m共t兲pm共t兲 Wm,m共兲 ⬘共t兲pm⬘共t兲 艌 0 共10兲 and the reservoir EP共also called medium entropy or entropy flow兲 S˙r共t兲 ⬅ −兺
m,m⬘, Wm,m共兲 ⬘共t兲pm⬘共t兲ln Wm共兲⬘,m共t兲 Wm,m共兲 ⬘共t兲 . 共11兲 We note that S˙tot共t兲艌0 follows from Wm,m共兲 ⬘共t兲pm⬘共t兲⬎0 if m⬘
⫽m and ln x艋x−1 for x⬎0 共if m⬘
= m the log in zero兲,by using the fact that 兺m,Wm⬘,m
共兲 共
t兲pmst共t兲=0 and 兺m,Wm,m共兲⬘共t兲=0. S˙共t兲 is the contribution to S˙tot共t兲 coming
from the changes in the system probability distribution and
S˙r共t兲 is the contribution coming from matter and energy ex-change processes between the system and its reservoirs.
We further separate the reservoir EP into two components 关11,15兴
S˙r共t兲 ⬅ S˙ex共t兲 + S˙a共t兲, 共12兲 with the excess EP
S˙a共t兲 ⬅ −
兺
m,m⬘, Wm,m共兲⬘共t兲pm⬘共t兲 ⫻ln Wm⬘,m 共兲 共 t兲pm st共 t兲 Wm,m共兲 ⬘共t兲pm⬘ st 共 t兲 艌 0. 共14兲The positivity of Eq. 共14兲 follows from the same reason as Eq.共10兲. If a transformation is done very slowly, the system remains at all times in the steady state distribution pm共t兲 = pmst共t兲. Such a transformation is called adiabatic. We then have S˙tot共t兲=S˙a共t兲 and S˙ex共t兲=−S˙共t兲. We also notice that
S˙a共t兲=0 when the DBC is satisfied.
We next define the state function quantity
Sb共t兲 ⬅ −
兺
m pm共t兲ln pm共t兲 pmst共t兲 共15兲 which is obviously zero when the system is at steady state. When considering a transformation between steady states, ⌬Sb共T,0兲=兰0T
dS˙b共兲=Sb共T兲−Sb共0兲=0. We call S˙b共t兲 the boundary EP共the terminology will be explained shortly兲 and
separate it in two parts
S˙b共t兲 = S˙na共t兲 − S˙d共t兲, 共16兲 where the nonadiabatic EP is
S˙na共t兲 ⬅ −
兺
m p˙m共t兲ln pm共t兲 pm st共 t兲 = −兺
m,m⬘ Wm,m⬘共t兲pm⬘共t兲ln pm共t兲pm⬘ st共 t兲 pm st共 t兲pm⬘共t兲 艌 0, 共17兲and the driving EP is
S˙d共t兲 ⬅
兺
mpm共t兲˙m共t兲, 共18兲 with
m共t兲 ⬅ − ln pmst共t兲. 共19兲 The positivity of Eq.共17兲 is again shown in the same way as for Eqs. 共10兲 and 共14兲. If no external driving acts on the system, is time independent and from Eq. 共18兲, S˙d共t兲=0. For an adiabatic transformation, since pm共t兲=pmst共t兲, from Eqs.共15兲 and 共17兲, we see that S˙na共t兲=0 as well as S˙b共t兲=0. From Eq. 共16兲, this also means that S˙d共t兲=0. Therefore, S˙d共t兲⫽0 only for nonadiabatic driving. Using Eq. 共17兲 with Eqs.共8兲 and 共13兲, we find
S˙na共t兲 = S˙ex共t兲 + S˙共t兲 = S˙tot共t兲 − S˙a共t兲. 共20兲 It is clear from the last equality why we call S˙na共t兲 the
nona-diabatic EP. The inequality S˙ex共t兲艌−S˙共t兲 which follows from
the first line is a generalization of the “second law of steady state thermodynamics”关11,35,36兴 derived for transitions be-tween steady states.
We next summarize our results
S˙tot共t兲 = S˙na共t兲 + S˙a共t兲 艌 0, 共21兲 S˙na共t兲 = S˙d共t兲 + S˙b共t兲 艌 0, 共22兲
S˙a共t兲 艌 0. 共23兲
The total EP is always positive and can be separated into two positive contributions, adiabatic 共which are nonzero only when the DBC is violated兲 and nonadiabatic effects. The latter can be due to a nonadiabatic external driving acting on the system or to the fact that one considers transformation during which the system is initially or finally not in a steady state. We therefore have a minimum EP principle stating that the total EP for arbitrary nonequilibrium transformations takes its minimal value if the transformation is done adiabati-cally共very slowly兲. The equality sign in Eq. 共21兲 is satisfied for adiabatic transformations which occur at equilibrium. The equality sign in Eq.共22兲 holds for adiabatic transforma-tions. The equality sign in Eq. 共23兲 only occurs when the DBC is satisfied.
IV. TRAJECTORY ENTROPIES
The evolution described by the ME can be represented by an ensemble of stochastic trajectories involving instanta-neous jumps between states. This will allow us to define fluctuating共trajectory兲 entropies.
A. Definitions
We denote a trajectory taken by the system between t = 0 and t = T by m共兲=兵0 − m0→ 1 m1→ 2 ¯ mj−1→ j mj→ j+1 ¯ mN−1→ N mN− T其. At t = 0 the system is in m0, and stays there until it jumps at 1to m1, etc., jumps atNfrom mN−1to mNand stays in mN until t = T共see Fig.1兲. N is the total number of jumps during this trajectory.
We next introduce various types of “trajectory entropy production”共TEP兲. We will see at the end of this section that when ensemble averaged, these correspond to the various EP defined in Sec. III.
The trajectory Gibbs entropy is defined as
s关m共兲,t兴 ⬅ − ln pm共兲共t兲, 共24兲 where pm共兲共t兲 represents the value of pm共t兲 along the trajec-tory m共兲. The system TEP is given by
s˙关m共兲,t兴 = −
冏
p˙m共t兲 pm共t兲冏m共兲 +兺
j=1 N ␦共t −j兲ln pmj−1共j兲 pmj共j兲 . 共25兲 The first term represents the smooth changes of s关m共兲, t兴in the expression changes depending on which horizontal segment along the trajectory one considers. The second term represents the discrete changes of s关m共兲, t兴 along the vertical
segments of the trajectory. These changes are singular and only due to the change in the system state.
Separating the trajectory system TEP similarly as the sys-tem EP in Sec. III, we get
s˙tot关m共兲,t兴 = s˙关m共兲,t兴 + s˙r关m共兲,t兴, 共26兲 where the total TEP is
s˙tot关m共兲,t兴 ⬅ −
冏
p˙m共t兲 pm共t兲冏m共兲 +兺
j=1 N ␦共t −j兲 ⫻lnWmj,mj−1 共j兲 共 j兲pmj−1共j兲 Wm共j−1j兲 ,mj共j兲pmj共j兲 , 共27兲and the reservoir TEP
s˙r关m共兲,t兴 ⬅
兺
j=1 N ␦共t −j兲ln Wm j,mj−1 共j兲 共 j兲 Wm j−1,mj 共j兲 共 j兲 . 共28兲We further separate the reservoir TEP into
s˙r关m共兲,t兴 = s˙a关m共兲,t兴 + s˙ex关m共兲,t兴, 共29兲
with the adiabatic TEP
s˙a关m共兲,t兴 ⬅
兺
j=1 N ␦共t −j兲ln Wm j,mj−1 共j兲 共 j兲pmj−1 st 共 j兲 Wm共j−1j兲 ,mj共j兲pmj st共 j兲 , 共30兲 and the excess TEPs˙ex关m共兲,t兴 ⬅
兺
j=1 N ␦共t −j兲ln pmstj共j兲 pmj−1 st 共 j兲 . 共31兲The nonadiabatic TEP
s˙na关m共兲,t兴 ⬅ −
冏
p˙m共t兲 pm共t兲冏m共兲 +兺
j=1 N ␦共t −j兲ln pmj−1共j兲pmj st共 j兲 pmstj−1共j兲pmj共j兲 共32兲 is made of the sum of two termss˙na关m共兲,t兴 ⬅ s˙d关m共兲,t兴 + s˙b关m共兲,t兴, 共33兲 the boundary TEP
s˙b关m共兲,t兴 ⬅
冏
− p˙m共t兲 pm共t兲冏m共兲 −˙t m共兲共t兲 t +兺
j=1 N ␦共t −j兲ln pmj−1共j兲pmj st共 j兲 pmstj−1共j兲pmj共j兲 , 共34兲 and the driving TEPs˙d关m共兲,t兴 ⬅ ˙t
m共兲共t兲 t
.
As in Sec. III, since
s˙na关m共兲,t兴 = s˙关m共兲,t兴 + s˙ex关m共兲,t兴, 共35兲
we get
s˙tot关m共兲,t兴 = s˙na关m共兲,t兴 + s˙a关m共兲,t兴. 共36兲 We generically denote these TEP by a关m共兲, t兴. The change
of a关m共兲, t兴 along a trajectory m共兲 of length T is given by
⌬a关m共兲,T兴 =
冕
0T
dt a˙关m共兲,t兴. 共37兲 Notice that ⌬s关m共兲, T兴 and ⌬sb关m共兲, T兴 are state function TEP τ1 τ2 τj-1 τj τN T 0 m0 m1 mj-2 m2 mj-1 mj mN-1 mN m(τ)
⌬s关m共兲,T兴 = ln pm0共0兲 pmN共T兲 = smN共T兲 − sm0共0兲, 共38兲 where sm共t兲⬅−ln pm共t兲, and ⌬sb关m共兲,T兴 = 关smN共T兲 −mN共T兲兴 − 关sm0共0兲 −m0共0兲兴. 共39兲 The other TEP are path functions.
B. Deriving trajectory entropies from measured currents Using Eq.共4兲, the reservoir TEP can be expressed as s˙r关m共兲,t兴 = −
兺
共t兲共Iheat 共兲关m
共兲,t兴 −共t兲Imat共兲关m共兲,t兴兲,
共40兲 where the heat current between the reservoir and the sys-tem is
Iheat共兲关m共兲,t兴 =
兺
j=1 N␦,j共t −j兲关⑀mj共j兲 −⑀mj−1共j兲兴 共41兲
and the matter current between the reservoir and the sys-tem
Imat共兲关m共兲,t兴 =
兺
j=1 N␦,j共t −j兲共Nmj− Nmj−1兲. 共42兲
The currents are positive if the system energy 共matter兲 in-creases.␦,
j共t−j兲 is a Dirac distribution centered at timej
only if the transition is due to the reservoir=i. Otherwise, it is zero. We thus confirm that the reservoir EP is the en-tropy associated to system-reservoir exchange processes.
We assume that the parametric time dependence of the energies, temperatures, and chemical potentials is known. Except in degenerate cases for which two different transi-tions between states have the same energy difference and number of particle difference, the trajectory of the system can be uniquely determined by measuring the heat and mat-ter currents between the system and the reservoirs. The sys-tem steady state probability distribution can be calculated by recording the steady state currents for sufficiently long times for different values of the energies, temperatures, or chemi-cal potentials. The driven system probability distribution can in principle be calculated by reproducing the measurement of the currents multiple times. All trajectory entropies contain-ing the logarithm of the transition rates can be expressed in terms of a combination of the reservoir EP共directly measur-able via current兲 and other trajectory entropies which can be expressed in terms of the system probability distribution 共ac-tual or steady state兲. Therefore, provided the current mea-surements can be repeated often enough to get good statis-tics, all the trajectory entropies are in principle measurable.
C. Statistical properties using generating functions The GF formalism allows to compute the probability dis-tributions and all statistical properties of the TEP without
having to generate the trajectories themselves. It further pro-vides a direct means for proving the FTs.
The GF associated with the changes of a关m共兲, t兴 along a
trajectory is given by
G共␥,t兲 ⬅ 具exp兵␥⌬a关m共兲,t兴其典, 共43兲
where 具·典 denotes an average over all possible trajectories. The probability that the system follows a trajectory with the constraint⌬A=⌬a关m共兲, t兴 at time t can be obtained from the GF using
P共⌬A,t兲 ⬅ 具␦共⌬A − ⌬a关m共兲,t兴兲典 = 1 2
冕
−⬁⬁
d␥e−i␥⌬AG共i␥,t兲. 共44兲 By inverting Eq.共44兲, we get
G共i␥,t兲 =
冕
−⬁ ⬁
d⌬A ei␥⌬AP共⌬A,t兲. 共45兲
The moments of the distribution are given by derivatives of the GF 具⌬ak关m 共兲,t兴典 =
冏
k G共␥,t兲 ␥k冏
␥=0 , k = 1,2, . . . . 共46兲In order to compute the GF, we recast it in the form
G共␥,t兲 =
兺
mgm共␥,t兲, 共47兲
where
gm共␥,t兲 = pm共t兲具exp兵␥⌬a关m共兲,t兴其典m 共48兲 is the product of the probability to find the system in state m at time t multiplied by the expectation value of exp兵␥⌬a关m共兲, t兴其 conditional on the system being in state m
at time t. Since⌬a关m共兲, t兴=0 for a trajectory of length t=0, we have gm共␥, 0兲=pm共0兲. We also have G共0,t兲=1 and gm共0,t兲=pm共t兲.
The time derivative of Eq.共47兲 gives G˙ 共␥,t兲 =
兺
m
g˙m共␥,t兲, 共49兲 where g˙m共␥, t兲 depends on the TEP of interest. Below, we will derive equations of motion for the gm共␥, t兲’s associated to the various TEP.
1. State function trajectory entropy production
The generating function associated with a state function TEP⌬a关m共兲, t兴=am共t兲−an共0兲 like ⌬s关m共兲, t兴 or ⌬sb关m共兲, t兴 may be straightforwardly obtained using Eq. 共47兲 with Eq. 共48兲. We get
G共␥,t兲 =
兺
m,nexp兵␥关am共t兲 − an共0兲兴其pm共t兲pn共0兲. 共50兲
2. Excess trajectory entropy production
and it remains constant along a given state m of the system. This means that
g˙m共ex兲共␥,t兲 =
兺
m⬘Wm,m⬘共t兲pm⬘共t兲 + exp兵␥sex共m,m
⬘
兲其⫻具exp兵␥⌬sex关m共兲,t兴其典m⬘, 共51兲 which using Eq.共48兲 can be rewritten
g˙m共ex兲共␥,t兲 =
兺
m⬘冉
pmst共t兲 pmst⬘共t兲冊
␥ Wm,m⬘共t兲gm⬘ 共ex兲共␥,t兲. 共52兲3. Reservoir trajectory entropy production and currents
Each time a transition along the system trajectory occurs,
sr关m共兲, t兴 acquires an amount sr共m,m⬘兲=ln共Wm,m⬘ 共兲 共 t兲/ Wm ⬘,m 共兲 共
t兲兲. Similarly to the excess entropy, we get g˙m共r兲共␥,t兲 =
兺
,m⬘冉
Wm,m共兲⬘共t兲 Wm⬘,m 共兲 共 t兲冊
␥ Wm,m共兲⬘共t兲gm⬘ 共r兲共␥,t兲. 共53兲This equation has been used in the study of steady state FTs 关6,12,37兴.
In Eq.共40兲, we expressed the reservoir TEP in terms of currents. The time integrated individual currents give the heat and matter transfer between the reservoir and the system qheat共兲关m共兲, t兴=兰0tdt
⬘
Iheat共兲关m共兲, t⬘
兴 and qmat共兲关m共兲, t兴 =兰0tdt⬘
Imat共兲关m共兲, t⬘兴. Their statistics can be calculated using
g˙m共␥ជ,t兲 =
兺
,m⬘ exp兵␥heat共兲关⑀m共t兲 −⑀m⬘共t兲兴其 ⫻exp兵␥mat共兲共Nm− Nm⬘兲其Wm,m⬘ 共兲 共 t兲gm⬘共␥ជ,t兲, 共54兲 where␥ជ is a vector whose elements are the different␥heat共兲’s and␥mat共兲’s. The GF calculated from Eq.共54兲 is therefore as-sociated with the joint probability distribution for having a certain heat and matter transfer with each reservoir.4. Adiabatic trajectory entropy production
Each time a transition along the system traject-ory occurs, sa关m共兲, t兴 acquires an amount sa共m,m
⬘
兲 = ln关W m,m⬘ 共兲 共 t兲pm⬘ st 共 t兲/Wm⬘,m 共兲 共 t兲pm st共 t兲兴. We therefore get g˙m共a兲共␥,t兲 =兺
,m⬘冉
Wm,m共兲 ⬘共t兲pm⬘ st 共 t兲 Wm共兲⬘,m共t兲pm st共 t兲冊
␥ Wm,m共兲⬘共t兲gm⬘ 共a兲共␥,t兲. 共55兲5. Total trajectory entropy production
Each time a transition along the system trajectory occurs,
stot关m共兲, t兴 acquires an amount ln关Wm,m
⬘ 共兲 共 t兲pm⬘共t兲/ Wm ⬘,m 共兲 共
t兲pm共t兲兴. In addition it also changes by an amount −d关ln pm共t兲兴/dt during an infinitesimally small time on a given state m of the system. Combining the two, we have
g˙m共tot兲共␥,t兲 = −␥
冉
p˙m共t兲 pm共t兲冊
gm共tot兲共␥,t兲 +兺
,m⬘冉
Wm,m共兲 ⬘共t兲pm⬘共t兲 Wm共兲⬘,m共t兲pm共t兲冊
␥ Wm,m共兲 ⬘共t兲gm⬘ 共tot兲共␥,t兲. 共56兲6. Nonadiabatic trajectory entropy production
Like the total TEP, sna关m共兲, t兴 acquires an amount
ln关pmst共t兲pm⬘共t兲/pm⬘
st共
t兲pm共t兲兴 each time a transition from a states m
⬘
to m occurs, and also changes by an amount −d关ln pm共t兲兴/dt during an infinitesimally small time on a given state m. This givesg˙m共na兲共␥,t兲 = −␥
冉
p˙m共t兲 pm共t兲冊
gm共na兲共␥,t兲 +兺
m⬘冉
pmst共t兲pm⬘共t兲 pmst⬘共t兲pm共t兲冊
␥ Wm,m⬘共t兲gm⬘ 共na兲共␥,t兲. 共57兲7. Driving trajectory entropy production
Since⌬sd关m共兲, t兴 exclusively accumulates along the seg-ments of the system trajectory, we get
g˙m共d兲共␥,t兲 =␥˙m共t兲gm共d兲共␥,t兲 +
兺
m⬘Wm,m⬘共t兲gm⬘
共d兲共␥,t兲. 共58兲
It follows from Eq.共46兲 that the average change of a TEP is obtained from its GF by differentiation with respect to␥at ␥= 0. By differentiating the GF evolution equations of this section one recovers the evolution equation for the EPs of Sec. III. The EPs are therefore the ensemble average of the TEPs introduced in this section A˙ 共t兲=具a˙关m共兲, t兴典 and ⌬A共T,0兲=具⌬a关m共兲, T兴典.
V. FLUCTUATION THEOREMS A. General integral fluctuation theorems
We can easily verify that gm共␥= −1 , t兲=pm共t兲 is the solu-tion of the evolusolu-tion equasolu-tions共56兲 and 共57兲. It immediately follows from probability conservation and Eq. 共49兲 that G˙tot共−1,t兲=G˙na共−1,t兲=0. Summing both sides of Eq. 共55兲
over m, we also verify that G˙a共−1,t兲=0. Because gm共z兲共−1,0兲=pm共0兲, Gz共−1,0兲=1 where z=tot,na,a. There-fore, we find that Gz共−1,t兲=1. Using Eq. 共43兲, this results in the three FTs
具exp兵− ⌬stot关m共兲,t兴其典 = 1, 共59兲
具exp兵− ⌬sna关m共兲,t兴其典 = 1, 共60兲
Equation共59兲 is the generalization to open systems of the integral FT for the TEP obtained earlier for closed systems 关14兴. The TEP 共27兲 needs to specify which reservoir is re-sponsible for the transitions occurring along the trajectory 共by labeling the rates with reservoir index兲. This point, made earlier for an open system at steady state关12兴, is generalized here for driven systems with an arbitrary initial condition. Equation共60兲 will be shown in the next section to reduce to the integral Hatano-Sasa FT 关11兴 for systems initially in a steady state. Equation共61兲 generalizes the integral FT for the adiabatic entropy关15兴 previously derived for closed system initially in a steady state.
Some additional insights can be gained by an alternative proof of the FTs共59兲 and 共60兲 which use a forward-backward trajectory picture of the dynamics. This is given for com-pleteness in the Appendix.
B. Transitions between steady states
We consider a system initially共t=0兲 at steady state and subjected to an external driving force between t = tdi and t = tdf. For tdi⬎t⬎0, the system remains in the steady state
corresponding to=tdi. The time protocol oftduring the driving tdf⬎t⬎tdiis arbitrary. If ttris the characteristic
tran-sient time needed for the system to reach a steady state from an arbitrary distribution, for t⬎tdf+ ttr, the system is in the
new steady state corresponding to tdf. The system is mea-sured between t = 0 and t = T.
1. Fluctuation theorem for the reservoir entropy production
We restrict our analysis to cases where the system is at steady state at t = 0 and the driving starts at least a time ttr
after the measurement started: tdi⬎ttr.
We use the bra-ket notation where兩p共t兲典 is the probability vector with components pm共t兲 and Wˆt denotes the rate ma-trix.兩I典 denotes a vector with all components equal to one. The ME共1兲 now reads
兩p˙共t兲典 = Wˆ t兩p共t兲典. 共62兲
The generating function for the reservoir EP共53兲 for␥= −1 evolves according to the adjoint equation of Eq.共62兲
兩g˙共r兲共− 1,t兲典 = Wˆ t
†兩g共r兲共− 1,t兲典. 共63兲
The initial condition of Eqs. 共62兲 and 共63兲 is 兩p共0兲典 =兩g共r兲共−1,0兲典=兩pst共tdi兲典. The formal solution of Eq. 共63兲 for t⬍tdibefore the driving starts reads
Gr共− 1,t兲 = 具I兩exp兵Wˆ†t其兩pmst典. 共64兲 We now insert a closure relation in terms of right and left eigenvectors of the adjoint rate matrix between the evolution operator and the initial condition. Because the rate matrix and its adjoint have the same eigenvalues共all negative and one zero兲, for ttr⬍t⬍tdi, only the right and left eigenvector associated with the zero eigenvalue survive. Since the right 共left兲 eigenvector of Wˆ † is 兩I典 关具p
m
st共
tdi兲兩兴, we get for ttr⬍t
⬍tdi
Gr共− 1,t兲 = 具I兩exp兵Wˆ†t其兩I典具pst兩pst典. 共65兲 For longer times, even when the system starts to be driven, 兩I典 remains invariant under the time evolution operator as can be seen using Eq.共2兲 in Eq. 共63兲. We get
Gr共− 1,T兲 = M具pst兩pst典 for T 艌 ttr, 共66兲
where M =具I兩I典 is the total number of states. This implies the following integral FT for the reservoir TEP
M艌 具exp兵− ⌬sr关m共兲,T兴其典 = M
兺
m=1 M 共pm st共 tdi兲兲2艌 1. 共67兲 The equality on the left-hand side关right-hand side兴 is satis-fied if pmst共tdi兲=␦n,m关pmst共tdi兲=1/M兴. Jensen’s inequality im-plies ⌬Sr共T,0兲艌0. Note that since ⌬s关m共兲, T兴 is a state function, it is easily verified thatM
兺
m=1 M pm 2共0兲 = 具exp兵⌬s关m 共兲,T兴其典. 共68兲2. Hatano-Sasa fluctuation theorem
We assume that the driving starts at the same time or later as the measurement共tdi艌0兲. We define T
⬘
⬎tdf+ ttr.We have pointed out at the end of Sec. III that for transi-tions between steady states⌬Sb共T⬘, 0兲=0, so that
⌬Sna共T⬘,0兲 = ⌬Sd共T⬘,0兲 = ⌬Sd共tdf,tdi兲 艌 0. 共69兲
We used the fact that⌬Sd共T⬘, 0兲 starts 共stops兲 evolving at tdi
共tdf兲. The same is true at the trajectory level, since from Eq.
共39兲 we have ⌬sb关m共兲, T
⬘
兴=0 and therefore⌬sna关m共兲,T
⬘兴 = ⌬s
d关m共兲,T⬘兴,
共70兲 where ⌬sd关m共兲,T⬘兴 ⬅
兺
j=0 N ln pmj st共 j兲 pmstj共j+1兲 =兺
j=0 N 关mj共j+1兲 −mj共j兲兴 =冕
0 T⬘ dt˙t m共兲共t兲 t =冕
tdi tdf dt˙t m共兲共t兲 t . 共71兲 The integrand in the third line contributes only during the time intervals between jumps provided the driving is chang-ing. Therefore, if without loss of generality we choose the measurement time such that T艌tdf共if T⬍tdf one canrede-fine tdfas equal to T兲, ⌬sd关m共兲, T兴=⌬sd关m共兲, T
⬘兴. This means
that for a transition between steady states, the FT 共60兲 re-duces to the Hatano-Sasa FT关11兴具exp兵− sd关m共兲,T兴其典 = 1. 共72兲 Alternatively, Eq.共72兲 can be proved from Eq. 共58兲 by show-ing that when␥= −1, gm共d兲共−1,t兲=exp兵−m共t兲其=pm
st共
t兲 is the solution of Eq.共58兲.
slow兲 driving, the inequality in Eq. 共69兲 becomes an equality. In the other extreme of a sudden driving, where t=tdi +⌰共t−tdi兲共tdf−tdi兲 and ˙t=␦共tdi兲共tdf−tdi兲,
⌬sd关m共兲,T兴 =m0共tdf兲 −m0共tdi兲 共73兲 becomes a state function and its average takes the simple form
⌬Sd共T,0兲 =
兺
mpmst共tdi兲关m共tdf兲 −m共tdi兲兴. 共74兲 Using Eq.共35兲 with Eq. 共70兲, and since
⌬s关m共兲,T兴 =mT共tdf兲 −m0共tdi兲, 共75兲 we find that
⌬sex关m共兲,T兴 =m0共tdf兲 −mT共tdf兲 共76兲 also becomes a state function and its average becomes
⌬Sex共T,0兲 =
兺
m0,mT
pmst0共tdi兲pmstT共tdf兲关m0共tdf兲 −mT共tdf兲兴. 共77兲 C. Transitions between equilibrium states
For a system coupled to a single reservoir 共or multiple reservoirs with identical thermodynamical properties兲, the DBC共5兲 is satisfied. A nondriven system in an arbitrary state will reach after some transient time ttrthe equilibrium grand
canonical distribution 共6兲. We again choose T
⬘
⬎tdf+ ttr andT艌tdf. From the TEP of Sec. IV, we find in this case ⌬sa关m共兲,T
⬘兴 = 0, ⌬s
r关m共兲,T⬘兴 = ⌬s
ex关m共兲,T⬘兴,
⌬stot关m共兲,T
⬘
兴 = ⌬sna关m共兲,T⬘
兴. 共78兲The two FTs共59兲 and 共60兲 become identical and the FT 共61兲 becomes trivial. Using Eq. 共6兲, we also find that Eq. 共19兲 becomes
m共兲 =共兲兵⑀m共兲 −共兲Nm−⍀共兲其, 共79兲 where⍀共兲=−ln ⌶共兲 is the thermodynamic grand canoni-cal potential.
We next consider transitions between equilibrium states, so that the procedure is the same as in Sec. V B 2 but with the DBC 共5兲 now satisfied. We therefore have ⌬sna关m共兲, T
⬘兴=⌬s
d关m共兲, T⬘兴. The driving implies externally
modulating the system energies, the chemical potential or the temperature of the reservoir.When driving the system energy, using Eqs.共71兲 and 共79兲, we find
⌬sd关m共兲,T兴 = ⌬sd关m共兲,T
⬘兴 =
w关m共兲兴 −⌬⍀, 共80兲 where the work is given by w关m共兲兴=兰tditdf
dt⑀˙m共兲共t兲 and ⌬⍀ =⍀关⑀共tdf兲兴−⍀关⑀共tdi兲兴. Both FTs, Eqs. 共59兲 and 共60兲, lead to
the same Jarzynski relation关4兴
具exp兵−w关m共兲兴其典 = exp兵−⌬⍀其. 共81兲
When driving the reservoir chemical potential, Eqs.共71兲 and共79兲 give ⌬sd关m共兲,T兴 = ⌬sd关m共兲,T
⬘兴 =
w˜关m共兲兴 −⌬⍀, 共82兲 where w˜关m共兲兴=−兰t di tdf dt˙ Nm共兲and⌬⍀=⍀关共tdf兲兴−⍀关共tdi兲兴.Both FTs, Eqs.共59兲 and 共60兲, now lead to
具exp兵−w˜关m共兲兴其典 = exp兵−⌬⍀其. 共83兲
The case where reservoir temperature is driven can be calculated similarly.
D. No driving: Steady state fluctuation theorem In a NESS, the relations of Sec. IV give
⌬sna关m共兲,t兴 = ⌬sd关m共兲,t兴 = 0,
⌬s关m共兲,t兴 = − ⌬sex关m共兲,t兴, ⌬stot关m共兲,t兴 = ⌬sa关m共兲,t兴. 共84兲 Furthermore, since⌬S共t,0兲=0, we get
⌬Stot共t,0兲 = ⌬Sr共t,0兲 艌 0. 共85兲 We shall rewrite the GF evolution equation for the reservoir TEP共53兲 in the bracket notation
兩g˙共r兲共␥,t兲典 = Vˆ共␥兲兩g共r兲共␥,t兲典, 共86兲
so that
Gr共␥,t兲 = 具I兩exp兵Vˆ共␥兲t其兩pst典, 共87兲 where兩I典 is a vector with all elements equal to one. Since from共53兲 the generator has the property Vˆ共␥兲=Vˆ†共−␥− 1兲, its eigenvalues have the symmetry s共␥兲=s共−␥− 1兲. Further-more, since exp兵Vˆ共␥兲t其 is a positive matrix, the Frobenious-Perron theorem 关29–31兴 ensures that all eigenvalues are negative or zero and that the left and the right eigenvectors, 兩0共␥兲典 and 兩˜0共␥兲典, associated with the largest eigenvalue
s
0共␥兲 exist. Adopting the normalization 具
˜
0共␥兲兩0共␥兲典=1, we
find for long times
Gr共␥,t兲 = t→⬁ exp兵s0共␥兲t其具I兩˜ 共␥兲典具0 0共␥兲兩p st典, 共88兲 and that Gr共−␥− 1,t兲 = t→⬁ exp兵s0共␥兲t其具I兩˜ 共−0 ␥− 1兲具0共−␥− 1兲兩p st典. 共89兲 This means that the cumulant generating function
Fr共␥兲 ⬅ lim t→⬁ 1
t ln Gr共␥,t兲 共90兲
satisfies the symmetry
P共⌬Sr兲 P共− ⌬Sr兲
= t→⬁
e⌬Sr, 共92兲
where P共⌬Sr兲 is the probability for a trajectory of the system to produce a reservoir TEP equal to⌬Sr.
⌬sr关m共兲, t兴 grows in average with time because it depends on the number of jumps along the trajectory. However, ⌬s关m共兲, t兴 is bounded. The FT 共92兲 can therefore be viewed
as a consequence of the detailed FT for⌬Stotderived in Eq. 共A13兲. The long time limit is needed in order to neglect the contribution from⌬s关m共兲, t兴 to ⌬stot关m共兲, t兴.
The FT共67兲 remains valid at steady state. FTs for currents can also be derived关19,37,42兴.
VI. ENTROPY FLUCTUATIONS FOR ELECTRON TRANSPORT THROUGH A SINGLE LEVEL
QUANTUM DOT
We have seen in Sec. IV B that the various entropies can be calculated by measuring the different currents between the system and the reservoirs. The counting statistics of electrons through quantum dots has recently raised considerable theo-retical 关38–42兴 as well as experimental 关27,43–46兴 interest. The single electrons entering and exiting a quantum dot con-nected to two leads can be measured. One can therefore cal-culate all currents, deduce the system trajectories, and calcu-late the various trajectory entropies presented earlier.
We will analyze the probability distribution for the vari-ous trajectory entropies in a single level quantum dot of en-ergy⑀connected to two leads with different chemical poten-tials共t兲, where= l , r. We neglect spin so that the dot can either be empty 0 or filled 1. The ME is of the form 共1兲 关40,47–52兴
冉
p˙1共t兲 p˙0共t兲冊
=冉
−vt wt vt − wt冊冉
p1共t兲 p0共t兲冊
, 共93兲 where vt=兺
vt共兲=兺
a关1 − f共t兲兴, wt=兺
wt 共兲=兺
af共t兲. 共94兲The coefficients acharacterize the coupling between the dot and the lead with Fermi distribution f共t兲⬅1/(exp{兵⑀ −共t兲其}+1). If ប=1, 关a兴=关energy兴=关time−1兴. By renormal-izing energies by ⑀, all parameters of our model become dimensionless. The steady state distribution of the system is
p1共st兲= wt
vt+ wt
, p0共st兲= vt
vt+ wt
, 共95兲
and the steady state currents are given by
具I典1,3共st兲= vt共l,r兲wt vt+ wt , 具I典2,4共st兲=wt 共l,r兲v t vt+ wt . 共96兲
We switch the chemical potential of the left leadl共t兲=0
+ V共t兲 using the protocol
V共t兲 =Vf− Vi
2 tanh关c共t − tm兲 + 1兴, 共97兲 while holding the right lead chemical potential fixed r共t兲 =0. We can therefore calculate all the trajectory entropies’ probability distributions using the GF method described in Sec. IV C. We solve numerically the evolution equations for the gm共i␥, t兲’s associated with the different entropies for dif-ferent values of ␥ with the initial condition gm共i␥, 0兲 = pm共0兲. After calculating the G共i␥, t兲’s using Eq. 共47兲, the probability distribution is obtained by a numerical inverse Fourier transform Eq. 共44兲. In all calculations we used  = 5,⑀= 1, al= 0.2, and ar= 0.1.
We start by analyzing the different contributions to the EP as defined in Sec. III for the l共t兲 protocol shown in Fig. 2共a兲.
The system is initially in a nonequilibrium distribution different from the steady state. The solution of the ME共93兲 as well as its steady state solution are displayed in Fig.2共b兲. Between t = 0 and t = 20,l共t兲 is essentially constant and the system undergoes an exponential relaxation to the steady state. Between t = 20 and t = 50,l共t兲 changes from0+ Vito 0+ Vffast enough for the system distribution to start differ-ing again from the instantaneous steady state distribution 共adiabatic solution兲. After t=50, l共t兲 remains constant and the system again undergoes a transient relaxation to the new steady state corresponding to0+ Vf. Figure 2共c兲shows the time dependent EP S˙totand its adiabatic S˙aand nonadiabatic contribution S˙na. As predicted, these three quantities are
al-ways positive关see Eqs. 共21兲–共23兲兴. We also demonstrate that S˙na only contributes when nonadiabatic effects are
signifi-cant, i.e., when the actual probability distribution is different from the steady state one 关pm共t兲⫽pm
共st兲(共t兲)兴. S˙
a= 0 only once at t⬇33, whenl共t兲=r共t兲=0.5 and the DBC is satis-fied. Otherwise the DBC is broken and S˙a⬎0. In Fig. 2共d兲, we present the two contributions to the nonadiabatic EP S˙na,
the driving EP S˙d, and the boundary EP S˙b关see Eq. 共22兲兴. The driving EP S˙d only contributes whenl共t兲 changes in time. One can also guess that⌬Sb=兰2060dt S˙b= 0 due to the fact that the change of boundary EP during an interval between two steady states is zero.⌬Sb=兰0
20
dt S˙b⫽0 because the system is initially not in a steady state. Figure2共e兲shows the alterna-tive partitioning of the nonadiabatic EP into the system EP S˙ and the excess EP S˙ex关see Eq. 共20兲兴. Finally the splitting of the total EP in the reservoir EP and the system EP关see Eq. 共9兲兴 is shown in Fig.2共f兲. We see that at steady state S˙ = 0 so that S˙tot= S˙r.
when the system reaches its new steady state at0+ Vf. Dif-ferent measurement times are represented by a , b , c , d.
In Fig.4, we display P共⌬S兲. Since ⌬s is a state function, P共⌬S兲 is the same for the various protocol. Because we
con-sider a two level system, ⌬s can only take four possible values which correspond to the four possible changes in the
system state between its initial and final condition. The tran-sitions 0→0 and 0→1 are much more probable because the probability to initially find the system in the empty state 0 is much higher共0.96兲 than finding it in the filled state 1 共0.04兲. The transition 0→0 is more probable than 0→1 because the system has a final probability 0.64 to be in its empty state and 0.36 to be in its filled state.
In the left column of Fig. 5, we depict P共⌬Sd兲. Here ⌬sb= 0 so that ⌬sd=⌬sna 关see Eq. 共16兲兴. All curves 共i兲–共v兲
satisfy the FT共60兲. For the sudden switch 共i兲, ⌬sdbecomes a state function which only depends on the initial state of the system关see Eq. 共73兲兴. ⌬sd can therefore take two possible values corresponding to the empty 0 or filled 1 orbital with a
0 200 400 600 800 1000 t 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Μl t
ii iii iv v i
a b c d
FIG. 3. 共Color online兲 Five driving protocols for the left lead chemical potentiall共t兲=0+ V共t兲 where V共t兲 follows Eq. 共97兲 with Vi= −0.25, Vf= 0.5 and共ii兲 c=0.2, tm= 75,共iii兲 c=0.05, tm= 200,共iv兲 c = 0.02, tm= 300, and共v兲 c=0.01, tm= 500.r共t兲=0= 0.5.共i兲 The sudden switch limitl共t兲=0+⌰共t兲Vf 关limit c→⬁ with tm= 0 in Eq.共97兲兴. The system is initially at steady state where p0st= 0.96 and p1st= 0.04. In the final steady state p0st= 0.64 and p1st= 0.36. a , b , c , d correspond to different measurement times. In all calculations  = 5,⑀=1, al= 0.2, and ar= 0.1. 10 20 30 40 50 60 t 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Μl t 10 20 30 40 50 60
t
0 0.05 0.1 0.15 0.2 0.25 Sna Sd Sb 10 20 30 40 50 60t
0.2 0.4 0.6 0.8 p1t p0t p1stΛt p0stΛt 10 20 30 40 50 60t
0 0.1 0.2 0.3 Sex Sna S 10 20 30 40 50 60t
0 0.05 0.1 0.15 0.2 0.25 0.3 Sa Stot Sna 10 20 30 40 50 60t
0 0.1 0.2 0.3 Sr Stot S (a) (b) (c) (d) (e) (f)FIG. 2. 共Color online兲 共a兲 Driving protocol of the left lead chemical potential l共t兲=0 + V共t兲 where V共t兲 follows Eq. 共97兲
with Vi= −0.25, Vf= 0.5, c = 0.2, tm= 0.2, and 0= 0.5. The right lead chemical potential remains constant atr=0.共b兲 Solid: The probability distribution of the dot obtained by solving the ME共93兲
with initial condition p0共0兲=0.4 and p1共0兲=0.6. Dotted: The adia-batic probability distribution. 共c兲 Decomposition of S˙tot共t兲 accord-ing to Eq.共21兲. 共d兲 Decomposition
of S˙na共t兲 according to Eq. 共22兲. 共e兲 Decomposition of S˙na共t兲 according to Eq.共20兲. 共f兲 Decomposition of
S˙tot共t兲 according to Eq. 共9兲.
3 2 1 0 1 2 3 0.1 0.2 0.3 0.4 0.5 0.6 PS 01 00 11 10
respective probability 0.96 or 0.04. When the driving speed slows down in共ii兲 the peaks are broadened. In the adiabatic limit 共V兲, P共⌬Sd兲 becomes a broad distribution with zero average. In the right column of Fig.5, we depict P共⌬Sex兲.
For sudden switch 共i兲, ⌬sex turns to a state function which
only depends on the final steady state distribution 关see Eq. 共76兲兴. It is clear from Eq. 共76兲 that the transitions 0→0 and 1→1 lead to ⌬sex= 0 and 1→0 and 0→1 to the same ⌬sex with opposite sign. The probabilities to observe these transi-tions follow from the fact that the system is initially more
2 1.5 1 0.5 0 0.5 0.2 0.4 0.6 0.8
i PS
d 3 2 1 0 1 2 3 0.1 0.2 0.3 0.4 0.5 0.6i PS
ex 01 00 11 10 2 1.5 1 0.5 0 0.5 0 10 20 30 40 50ii PS
d 3 2 1 0 1 2 3 0 2 4 6 8 10 12ii PS
ex 2 1.5 1 0.5 0 0.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75iii PS
d 3 2 1 0 1 2 3 0 0.2 0.4 0.6 0.8 1iii PS
ex 2 1.5 1 0.5 0 0.5 0 0.5 1 1.5 2iv PS
d 3 2 1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4iv PS
ex 2 1.5 1 0.5 0 0.5 0 0.5 1 1.5 2 2.5 3v PS
d 3 2 1 0 1 2 3 0 0.25 0.5 0.75 1 1.25 1.5 1.75v PS
exlikely to be in 0共prob 0.96兲 than in 1. The probability for the final state 0 共1兲 is 0.64 共0.36兲. Therefore, the most likely transition is 0→0 followed from 0→1. As the driving speed slows down like in 共ii兲, the peaks get broadened. Since ⌬sex=⌬sd−⌬s 关see Eq. 共20兲兴 and since in the adiabatic switch limit共v兲 P共⌬Sd兲 is centered around zero, P共⌬Sex兲 共v兲
has the same peak structure as P共−⌬S兲 共see Fig. 4兲, but broadened by⌬sd.
In Fig.6, we display P共⌬Sr兲 for different measurement times and protocols. Plots with the same driving but different measurement times关共i兲a and 共ii兲d or 共ii兲a and 共ii兲d or 共iii兲b and共iii兲d兴 show the evolution of P共⌬Sr兲 in the final steady state. The plots 共ii兲–共v兲 satisfy the FT 共67兲 which for our parameters imply具exp兵−⌬sr其典=1.846. The FT is not satisfied for 共i兲 because the driving starts at the same time as the measurement关see Sec. V B 1兴. To understand the structure of
P共⌬Sr兲, we time integrate Eq. 共40兲 and use the fact that in our model the heat current is proportional to the matter
cur-rent between the reservoir and the system I共兲= Imat共兲 = Iheat共兲 /⑀, where I共兲共t兲 =
兺
j=1 N ␦,j共t −j兲共Nmj− Nmj−1兲. 共98兲 We get ⌬sr共t兲 = −兺
冕
0 t d„⑀−共兲…I共兲共兲. 共99兲 In the sudden switch limit共i兲, we get⌬sr共t兲 = −
兺
„⑀
−共T兲…N共兲共t兲, 共100兲 whereN共兲=兰0tdI共兲共兲 is the net number of electron
trans-ferred from the reservoir to the system between 0 and t. This explains why ⌬sr in 共i兲 only takes a discrete value
0 5 10 15 20 25 30 0 2.5 5 7.5 10 12.5 15 17.5
ia PS
r 40 50 60 70 80 90 100 110 0 0.2 0.4 0.6 0.8 1id PS
r 5 0 5 10 15 20 0 1 2 3 4iia PS
r 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3iid PS
r0 10 20 30 40 0 0.02 0.04 0.06 0.08 0.1
iiib PS
r 20 40 60 80 100 0 0.01 0.02 0.03 0.04 0.05iiid PS
r 0 10 20 30 40 50 0 0.01 0.02 0.03 0.04 0.05ivc PS
r 0 10 20 30 40 50 60 70 0 0.01 0.02 0.03 0.04vd PS
rwhich are multiples of each other. The distance between the peaks of 2.5 observed in 共i兲 is due to the right lead only 关共⑀−r兲=2.5 because our parameters are such that (⑀ −l共T兲)=0兴. The new peaks which appear in 共ii兲 with a spac-ing 0.125 are due to the fact that the drivspac-ing starts some time after the measurement so that(⑀−l共0兲)=3.75 also contrib-utes. As the driving speed slows down共iii兲–共v兲, the discrete structure broadens and⌬srcan take continuous values.
In Fig. 7, we display P共⌬Sa兲. All curves satisfy the FT 共61兲. The verification 共not shown兲 is best done on the GF 关Ga共−1,t兲=1兴 because the numerical accuracy of the tail of the distribution is not sufficient. The peak structure of
P共⌬Sa兲 can be understood from P共⌬Sr兲 and P共⌬Sex兲 because
⌬sr=⌬sa+⌬sex. This is particularly clear for the sudden
switch 共i兲 where the possible values of the entropies are strongly restricted. Indeed, in共i兲 each peak of P共⌬Sr兲 is split in three smaller peaks which have the same structure as
P共⌬Sex兲. As the speed of the driving decreases 共ii兲–共v兲, the
peak structure disappears.
In Fig.8, we show P共⌬Stot兲. All curves satisfy the FT 共59兲
关verification was done on the GF 共not shown兲兴. The structure of P共⌬Stot兲 can be understood using P共⌬Sr兲 and P共⌬S兲 be-cause⌬stot=⌬sr+⌬s. This is clear for the sudden switch 共i兲 where the peaks of P共⌬Sr兲 are split in smaller peaks which have the structure of P共⌬S兲.
VII. CONCLUSIONS
For a driven open system in contact with multiple reser-voirs and described by a master equation, we have proposed a partitioning of the trajectory entropy production into two parts. One contributes when the system is not in its steady state and contains two contributions due to the external driv-ing and the deviation from steady state in the initial and final probability distribution of the system. The second part comes from breaking of the detailed balance condition by the mul-tiple reservoirs and becomes equal to the total entropy
pro-0 5 10 15 20 25 30 0 2 4 6 8 10
ia PS
a 40 50 60 70 80 90 100 110 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7id PS
a5 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3
iia PS
a 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2iid PS
a 0 10 20 30 40 0 0.02 0.04 0.06 0.08 0.1 0.12iiib PS
a 20 40 60 80 100 0 0.01 0.02 0.03 0.04 0.05 0.06iiid PS
a 0 10 20 30 40 50 0 0.01 0.02 0.03 0.04 0.05ivc PS
a 0 10 20 30 40 50 60 70 0 0.01 0.02 0.03 0.04vd PS
aduction when the system remains in its steady state all throughout the nonequilibrium process. Both parts as well as the total entropy production satisfy a general integral fluc-tuation theorem which imposes positivity on their ensemble average. This partitioning also provides a simple way to identify which part of the entropy production contributes during a specific type of nonequilibrium process共see Fig.9兲. Previously derived integral fluctuation theorems can be re-covered from our three general fluctuation theorems and in addition we derived a new integral fluctuation theorem for the part of the entropy production due to exchange processes between the system and its reservoirs共reservoir entropy pro-duction兲. Our results strictly apply to systems described by a master equation共1兲. However, as has often been the case for previous fluctuation relations, one could expect similar re-sults to hold for other types of dynamics. For electron trans-port through a single level quantum dot between two reser-voirs with time dependent chemical potentials, we have simulated and analyzed in detail the probability distributions
of the various trajectory entropies and showed how they can be measured in electron counting statistics experiments.
ACKNOWLEDGMENTS
The support of the National Science Foundation 共Grant No. CHE-0446555兲 and NIRT 共Grant No. EEC 0303389兲 is gratefully acknowledged. M.E. is partially supported by the FNRS Belgium共collaborateur scientifique兲.
APPENDIX: FLUCTUATION THEOREMS IN TERMS OF FORWARD BACKWARD TRAJECTORY
PROBABILITIES
We show that the FTs共59兲 and 共60兲 have an interesting interpretation in terms of the ratio of the probability of a forward dynamics generating a given trajectory and the prob-ability of the time-reversed trajectory during some backward dynamics. This is an alternative to the GF approach which
0 5 10 15 20 25 30 0 0.25 0.5 0.75 1 1.25 1.5
ia PS
tot 40 50 60 70 80 90 100 110 0 0.2 0.4 0.6 0.8 1 1.2id PS
tot 5 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7iia PS
tot 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3iid PS
tot 0 10 20 30 40 0 0.02 0.04 0.06 0.08iiib PS
tot 20 40 60 80 100 0 0.01 0.02 0.03 0.04iiid PS
tot 0 10 20 30 40 50 0 0.01 0.02 0.03 0.04 0.05ivc PS
tot 0 10 20 30 40 50 60 70 0 0.01 0.02 0.03 0.04vd PS
totconnects the detailed form to the integral form of the FTs. The forward dynamics is described by the ME 共1兲. We introduce the probability共in trajectory space兲 P关m共兲兴 that the system follows a trajectory m共兲
P关m共兲兴 = pm0共0兲
冋
兿
j=1 N exp冉
冕
j−1 j d⬘Wmj−1,mj−1共⬘兲冊
⫻Wm共jj,m兲 j−1共j兲册
exp冉
冕
N T d⬘WmN,mN共⬘兲冊
. 共A1兲 The Wm j,mj−1 共j兲 共j兲 factors in this expression represent the
probability that the system undergoes a given transition whereas the exponentials describe the probability for the sys-tem to remain in a given state between two successive jumps. Summation over all possible trajectories will be denoted by 兺m共兲. It consists of time-ordered integrations over the N time variablesjfrom 0 to T共this gives the probability of having a path with N transitions兲 and then summing over all possible
N from 0 to⬁. Normalization in trajectory-space implies that
兺m共兲P关m共兲兴=1.
The backward dynamics is described on the time interval
t =关0,T兴 by the ME p
˜˙m共t兲 =
兺
m⬘W˜m,m⬘共T−t兲p˜m⬘共t兲, 共A2兲 where the new rate matrix W˜m,m⬘共t兲 satisfies 兺mW˜m,m⬘共t兲 = 0. We require that the parametric time dependence共via the
driving protocol t兲 of the rate matrix in Eq. 共A2兲 is time-reversed compared that of Eq.共1兲 and that the diagonal part of the rate matrix in Eqs.共1兲 and 共A2兲 is the same
W˜m,m共t兲 = Wm,m共t兲. 共A3兲 This still leaves room for different choices of W˜m,m⬘共t兲. We will later specify two choice of W˜m,m⬘共T−t兲 关Eqs. 共A11兲 and 共A14兲兴 that will result in two FTs.
We define the time-reversed trajectory of m共兲by
m¯共兲=兵0 − mN——→ T−N mN−1——→ T−N−1 ¯ mj——→ T−j mj−1 ——→ T−j−1 ¯ m1——→ T−1 m0− T其.
The probability P˜ 关m¯共兲兴 that the system described by Eq. 共A2兲 follows the time-reversed trajectory m¯共兲is given by
P˜ 关m¯共兲兴 = p˜mN共0兲
冋
兿
j=1 N exp冉
冕
T−N−j+2 T−N−j+1 d⬘ ⫻W˜ mN−j+1,mN−j+1共T−⬘兲冊
⫻W˜ mN−j,mN−j+1 共N−j+1兲 共 T−N−j+1兲册
⫻exp冉
冕
T−1 T d⬘W˜m0,m0共T−⬘兲冊
, 共A4兲 whereN+1= T. Normalization in the reverse path ensemble implies兺m¯共兲P˜ 关m¯共兲兴=1.
We consider the ratio of the two probabilities 共A1兲 and 共A4兲,
r关m共兲兴 ⬅ lnP关m共兲兴 P˜ 关m¯共兲兴
. 共A5兲
Due to Eq. 共A3兲, the contributions from the exponentials 共which represent the probabilities to remain on a given state兲 in Eq.共A5兲 cancel, so that
r关m共兲兴 = ln pm0共0兲 p ˜mN共0兲 +
兺
j=1 N ln Wm j,mj−1 共j兲 共 j兲 W˜m共j−1j兲 ,mj共j兲 . 共A6兲We can partition Eq.共A6兲 in the form r关m共兲兴 = ln pm0共0兲 p ˜mN共0兲 +
兺
j=1 N ln pmj st共 j兲 pmj−1 st 共 j兲 +冉
兺
j=1 N ln pmstj−1共j兲Wmj,mj−1 共j兲 共 j兲 pmstj共j兲W˜mj−1,mj 共j兲 共 j兲冊
. 共A7兲 We assume for the moment that r关m共兲兴 can be expressedexclusively in terms of quantities of the dynamics共1兲, i.e., a recipe has to be provided to express the tilde quantities in
s ∆ tot
