Thermodynamic formalism and linear response theory for non-equilibrium steady states

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by  Thomas Speck

Thermodynamic formalism and linear response theory
for non-equilibrium steady states
Thomas Speck
Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
We study the linear response in systems driven away from thermal equilibrium into a non-
equilibrium steady state with non-vanishing entropy production rate. A simple derivation of a
general response formula is presented under the condition that the generating function describes a
transformationthat(tolowestorder)preservesnormalizationandthusdescribesaphysicalstochas-
ticprocess. ForMarkovprocessesweexplicitlyconstructtheconjugatequantitiesanddiscusstheir
relationwithknownresponseformulas. Emphasisisputontheformalanalogywiththermodynamic
potentials and some consequences are discussed.
6 I. INTRODUCTION ganizedasfollows: First,webrieflyoutlinethecanonical
1 structure of intensive affinities and extensive generalized
0 distances for NESS. We then derive a general response
One of the objectives of computational sciences is the
2 formula and show that it contains previously derived re-
accurate prediction of material properties. The determi-

g sults, in particular the response formula by Warren and
nation of transport coefficients (e.g., conductivities and
u Allen [11] and the path weight representation [6, 14, 18].
mobilities) remains a challenge since, in general, it im-
A Before concluding we discuss our results in the light of a
plies currents and thus non-equilibrium conditions. Il-
possible thermodynamic formalism for NESS.
4 lustrative as well as technologically important examples
includetheefficienttransportofchargesinorganicsemi-

] conductors [1, 2] and across thin membranes in reverse
h
c osmosis [3]. While many sophisticated numerical meth- II. THERMODYNAMIC FORMALISM
e ods have been developed based on thermal equilibrium,
m for driven systems one typically has to resort to brute-
A. Conjugate variables
force computer experiments.
-
at Sufficiently close to equilibrium transport coefficients
The mathematical structure of equilibrium statistical
t can be determined from equilibrium fluctuations via the
s mechanics is based on pairs of an extensive quantity
. fluctuation-dissipationtheorem[4]. Therehavebeencon-
t (volume, particle number) and the conjugate intensive
a siderableeffortstofindgeneralprinciplesalsoforthelin-
quantity(pressure,chemicalpotential),whicharerelated
m ear response of non-equilibrium states [5–11] (for more
throughthermodynamicpotentials(freeenergies). What
- complete reviews we refer to Refs. 12–14 and references
makes the formalism so powerful is that these potentials
d therein), which find application in “field-free” numeri-
n are also generating functions encoding the full statistics
cal algorithms [15–17]. There is now a “zoo” of dif-
o of the non-conserved extensive quantities. As a corol-
ferent approaches and derivations yielding (sometimes
c lary, fluctuations encode the response of thermodynamic
[ unrecognized) equivalent results. One reason might be
that actually several conjugate observables (and their observables to a small external perturbation.
2
linear combinations) are equivalent in determining the
v
response [18].
0
4 Extendingthenotionofstatisticalensemblestotrajec- (a) (b)
5 tories(time-orderedsequencesofdynamicevents)iscur-
3 rentlyreceivingconsiderableattention[19–21]. Acanoni-
0 calstructureforthejointprobabilityofmicroscopicprob-
.
1 abilitiesandtheircurrentsasbeenformulatedinRef.22. φ
0 Incontrast,hereweareconcernedwithmacroscopiccur-
f
6 rents without information about microscopic probabili- enzyme
1
ties (or densities). Another concept is that of “canoni-
:
v cal” path ensembles (also appearing under the names s-
FIG. 1: Examples for affinities and distances. (a) Colloidal
i ensemble [23, 24], tilted ensemble, or Esscher transform)
X particleinaringtrapwithradiusR. Theparticleisdrivenby
in which trajectories are biased by a time-integrated ob-
a constant force f (the affinity) while X = Rφ is the total
r τ
a servable. Under certain conditions typical trajectories distance travelled during time τ. (b) Sketch of an enzyme
in the canonical path ensemble are equivalent to trajec- drivingthereaction◦→(cid:46). Thegeneralized“distance”X =
τ
tories in the original processes with fixed value of the N(cid:46) = −N◦ now corresponds to the number of (cid:46) molecules
τ τ
observable [25–28]. The purpose of this paper is to fol- producedduringtimeτ. Theaffinityf =−(µ(cid:46)−µ◦)isgiven
low these ideas and apply them to the linear response by the difference of chemical potential, which we assume to
around a non-equilibrium steady state (NESS). It is or- be fixed by chemiostats.
2
Pairs of apparently conjugate quantities (fi,Xi) also with symmetric Onsager coefficients Lij = Lji following
arise for non-equilibrium steady states (NESS), where from the Green-Kubo relations
non-zero intensive affinities fi (the generalized forces)
∂ 1
give rise to transport and thus extensive (generalized) (cid:104)XiXj(cid:105)(cid:16)Lij. (5)
distances Xi ∼ τ (measured over time τ), see Fig. 1 ∂τ 2 τ τ
τ
for two examples. Their product determines the entropy
production Σ = (cid:80) fiXi. Truly conjugate quantities,
τ i τ C. Canonical path ensembles
however,wouldrequiretheexistenceofanon-equilibrium
“potential” Φ(f;τ) so that (cid:104)Xi(cid:105)= ∂Φ, which more gen-
τ ∂fi Away from the linear response regime for NESS char-
erally would determine state functions and justify vari-
acterized by the affinities f we can still define the gener-
ation principles [29, 30]. Since this also implies strict
ating function
convexity, it would preclude established phenomena like
a negative differential mobility [31]. (cid:90)
Zf(s;τ)≡ dX esXPf(X;τ)(cid:16)eτφ∗f(s), (6)
B. Linear response regime wheresatthispointisjusttheargumentofthisfunction.
Moments and cumulants are obtained through differen-
A thermodynamic description does, however, apply to tiation with respect to s around s = 0. The function
the linear response regime. To this end, consider a gen- φ∗f(s) is the large deviation function, which by construc-
eralized distance X ∼ τ measured in thermal equilib- tionisaconvexfunction. Itisrelatedtotheratefunction
τ
rium(i.e., f =0)withprobabilitydistributionP0(X;τ). Pf(X;τ) (cid:16) e−τφf(x) for the current x ≡ Xτ/τ through
Clearly, the average (cid:104)X (cid:105)=0 vanishes. The time τ now the Legendre-Fenchel transform [33]
τ
plays a role similar to system size N in conventional sta-
φ∗(s)=sup[xs−φ (x)]. (7)
tistical mechanics. We define the generating function f f
x
(cid:90)
Z0(f;τ)≡ dX e21fXP0(X;τ)(cid:16)eτ12σ(f) (1) Oconnedcitainonnothweapsaktthheenfsoellmowbliengtoqucoensttiaoinn:oAnslysutmraejethctaotrwiees
with large deviation function σ(f), where (cid:16) denotes the
asymptotic limit of τ becoming larger than the longest ��
dctioeorsnrcaerlliabttehioetnrhmetoimsdayemn.eamFpoihclysloswwiceianlagsskyt:hsteDemoaensbaluZotg0(ynfo;wwτi)thwfoitcrhofnnv(cid:54)=oenn0-- ����������� �� ��� ���������� ���� ���
zeroaffinity(i.e.,drivenintoaNESS)?Apositiveanswer ��� �� ��� ��
would imply that � ��� ���� ��� ����
�� �� �� �� �� ��
∂σ � ���������
(cid:104)X (cid:105)=τ (2)
τ ∂f
��
holds, which, however, is not the case for arbitrary f. ����� �� � ������������������
Onlyforsmall|f|(cid:28)1inthelinearresponseregimedoes ���
�
suchaninterpretationyieldthecorrectresultwithmean �� ���
1 (cid:90) �� �� ��
(cid:104)Xτ(cid:105)= Z dX Xe12fXP0(X;τ) �������
0 (3)
= 1(cid:104)(X )2(cid:105)f +O(f2). FIG.2: Illustrationofthedualityofsandcurrentx=Xτ/τ
2 τ fortheasymmetricrandomwalk(fordetailsseeAppendixA,
with f = 1 and k+ =1): (a) Large deviation function φ∗(s)
2 f
This result is the well-known fluctuation-dissipation the- (thickline). Theslopeats=0(steadystateI)correspondsto
orem [4] through which fluctuations in thermal equilib- the mean current x0. (b) Rate function φf(x) of the current
(thick line) with minimum at x . Conditioning currents to
riumdeterminehowthesystemreactstoasmallapplied 0
a value x > x leads to the steady state (II) with s > 0
force f. Indeed, Onsager’s seminal insight has been that s 0
determined by the slope. (c) The unbiased average current
in the linear response regime (half) “the rate of increase
as a function of driving force f (thick line) and the current
of the entropy plays the role of a potential” [32], namely x = ∂ φ∗ (thin line) from the generating function, where
s s f
the large deviation function s=(f −f)/2 quantifies a perturbation of the steady state
eff
with effective force f . Both currents agree for s = 0 but
eff
σ(f)= 1(cid:88)Lijfifj (4) deviate for increasing values of the perturbation s. For fixed
2 k+ the unbiased current is bounded (dashed black line).
ij
3
with a fixed value x for the current. As discussed in nowexploretheconsequencesofdemanding thatφˆ (ω,t)
s s
detail by Chetrite and Touchette [25, 26], in the limit of remains a normalized function for s (cid:54)= 0. Since for non-
large τ this “microcanonical” ensemble becomes equiva- negative φ(ω,R,t) Eq. (9) implies that also φˆ (ω,t) is
s
lent (under mild assumptions) to a “canonical” ensem- non-negative,itcanbeinterpretedastheprobabilitydis-
ble in which the current fluctuates but its mean equals tribution of state ω for a system parametrized by s. At
xs. This canonical path ensemble is described through this point the physical meaning of s is not obvious but
the generating function Eq. (6) for a value of s deter- in the next section we will construct explicitly conjugate
minedthroughtheconditions=∂xφf|xs. Whilesandx pairs (s,R).
are thus conjugate quantities (see Fig. 2 for an illustra- Now consider a system described by the probability
tion for a specific system), changing s does not trace a distribution φˆ (ω,t). The expectation value for an arbi-
s
changeoftheaffinitiesbutinvolvesarathercomplicated, trary observable A(ω) becomes
non-local transformation (Doob’s transform) also of the
interactions [25, 26, 34–36]. It is exactly this behavior (cid:104)A(t)(cid:105) =(cid:88)A(ω)φˆ (ω,t), (10)
s s
that complicates a general thermodynamic description ω
of NESS. However, in the following we demonstrate that
whichreducesto(cid:104)A(cid:105)forφˆ (ω,t)=ψ(ω)ats=0. Hence,
smallscanbeinterpretedasaperturbationofthesteady 0
inthefollowingweinterprettheconjugatevariablesap-
state,leadingtriviallytolinearresponserelations. More-
pearinginthegeneratingfunctiontodescribeanexternal
over,basedonthisresultwecanconstructadifferentset
perturbation applied to the system at t = 0 and driving
of conjugate quantities which extend Onsager’s result to
it towards a neighboring steady state. The response to
NESS driven beyond the linear response regime.
this perturbation is
III. A GENERAL RESPONSE FORMULA ∂(cid:104)A∂(sτ)(cid:105)s(cid:12)(cid:12)(cid:12)(cid:12)s=0 =(cid:88)ω A(ω) ∂φˆs∂(ωs,τ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s=0
We consider a steady state maintained by at least one (cid:88)(cid:90) (11)
= dRA(ω)Rφ(ω,R,τ)
non-vanishing affinity and thus having a non-vanishing
entropy production rate (cid:104)Σ (cid:105) > 0. For clarity, in this ω
τ
section we consider a single perturbation but the gener- =(cid:104)A(τ)Rτ(cid:105),
alizationtomorethanoneisstraightforward. Tobesuffi-
whichfollowsinsertingEq.(9). Thisisourcentralresult.
cientlygeneral,wedefineourquantityofinterestthrough
It relates the response (sometimes called sensitivity) of
the stochastic (Riemann-Stieltjes) integral of the form
an observable to the correlations of this observable with
(cid:90) τ (cid:90) τ theamountofRaccumulatedsincetheperturbationwas
R [ω ]≡ dr(ω ) = dtr˙(ω ,ω˙ ) (8)
τ t t t t applied. The correlations are to be determined in the
0 0
unperturbed steady state corresponding to s = 0. The
over a process ω representing the state ω of the system
t result Eq. (11) is quite general and does not require any
at time t. The integral maps a single trajectory {ω }τ
t 0 assumptions on the dynamics.
of length τ (cid:62) 0 onto a real number. Clearly, R ∼ τ is
τ
a time-extensive quantity. It will be convenient later to
also introduce the generalized velocity r˙(ω,ω˙), which for IV. CONSTRUCTING CONJUGATE PAIRS
stochasticprocessesistobeunderstoodsymbolicallyand
followsthenotationalconventionthattypicallyisusedin The response Eq. (11) follows for functions φˆ (ω,t)
s
physics.
that, at least for small s, are normalized. This places
Thejointprobabilityφ(ω,R,t)describestheprobabil-
some restrictions onto what integrals R [ω ] are actually
τ t
ity to observe the system in state ω at time t having
admissible. One property follows immediately by choos-
accumulated an amount R = R up to time t (cid:54) τ start-
t ing A(ω) = 1, which implies that the average (cid:104)R (cid:105) = 0
t
ing with R = 0. Hence, the initial condition factorizes
0 vanishes for any t>0.
toφ(ω,R,0)=ψ(ω)δ(R), whereψ(ω)isthesteadystate
probability of state ω and δ(R) is the Dirac δ-function.
It is often more convenient to work with the (Laplace)
A. Stochastic dynamics
transform
(cid:90)
φˆ (ω,t)≡ dResRφ(ω,R,t) (9) To be more specific, we consider a continuous Markov
s
process
with initial condition φˆ (ω,0)=ψ(ω) following from the dω =F(ω )dt+dξ(ω ) (12)
s t t t
factorization of the joint probability, where the integral
with effective drift vector F(ω) and random increments
runs over all possible values of R. Moreover, for s = 0
we have that φˆ (ω,t) = ψ(ω) is the steady state prob- (cid:88)
0 dξ(ω)= σ (ω)◦dW (t), (13)
ability. While in general φˆ (ω,t) is not normalized, we α α
s α
4
where W (t) are independent Wiener processes and the C. Coupling to state changes
α
symbol ◦ denotes the Stratonovich rule for stochastic in-
tegrals. With the symmetric diffusion matrix A more general form of time-extensive observables is
1(cid:88) given by
Dij(ω)≡ σi(ω)σj(ω) (14)
2 α α
α dr(ω)=h(ω)dt+g(ω)◦dω, (22)
the Markov generator reads
wherethevectorg(ω)nowcouplestotheevolutionofthe
L =F ·D·∇+∇·D·∇=(F +∇)·D·∇. (15) state ω. The generator follows as
0
Its adjoint L† generates the time evolution of the proba- L =F·D·(∇+sg)+(∇+sg)·D·(∇+sg)+sh. (23)
0 s
bilitydistribution,∂ ψ =L†ψ. Here,F(ω)isthephysical
t 0
Expanding to lowest order we again find Eq. (21) for the
force such that the effective drift becomes
coefficient g(ω) and the condition to preserve normaliza-
1(cid:88)
F =D·F + (∇·σ )σ . (16) tion now becomes
2 α α
α
F ·D·g+∇·(D·g)+h=0. (24)
Throughout we set Boltzmann’s constant and temper-
ature to unity so that entropies are dimensionless and It is straightforward to check that the time-integrated
the mobility matrix coincides with the diffusion matrix observable R following from Eq. (22) can be written as
τ
Eq. (14). the derivative
(cid:12)
B. The response formula of Warren and Allen Rτ =− ∂∂Asτ(cid:12)(cid:12)(cid:12) (25)
s=0
with stochastic action
We first consider
(cid:90) τ
dr(ω)=h(ω)dt+g(ω)◦dξ(ω), (17) A [ω ]≡ dtL (ω ,ω˙ ), (26)
τ t s t t
0
where the vector g(ω) couples to the same noise as in
Eq. (12). Following Chetrite and Touchette [25, 26], the where (still employing the Stratonovich rule)
tilted (or deformed) generator for the evolution ∂ φˆ =
t s 1
L†φˆ becomes L (ω,ω˙)= (ω˙ −D·F )·D−1·(ω˙ −D·F )
s s s 4 s s
L =F ·D·∇+(∇+sg)·D·(∇+sg)+sh, (18) 1
s + ∇·(D·F ). (27)
2 s
which for s=0 reduces to the generator L in Eq. (15).
0
Expanding the second term to linear order of s, we can Hence, employing Eq. (22), the conjugate observable R
τ
recast this generator into the form nowcorrespondstothe“pathweightrepresentation”dis-
L =(F+2sg+∇)·D·∇+s[∇·(D·g)+h]+O(s2), (19) cussed in Refs. 6, 14, 18.
s
which manifestly preserves normalization if ∇·(D·g)+
h = 0, which thus determines h(ω). Note that changing V. DISCUSSION
from Stratonovich to Itˆo calculus, this condition implies
that dr = g · dξ. Clearly, since then state and noise A. Thermodynamic formalism
are independent, the expectation value of R vanishes as
required.
It is straightforward to extend Eq. (11) to multiple
We now assume that the perturbed steady state is de-
affinities s = {si}. We restrict our considerations to the
scribed by the forces F (ω) depending on s. Expanding
s set of observables {Ri} with vanishing mean, for which
the forces to linear order, τ
we can derive a local potential. To this end, from the
(cid:12)
Fs(ω)=F(ω)+s ∂F∂ss(ω)(cid:12)(cid:12)(cid:12) +O(s2), (20) terraantisnfogrfmunedctjioonint probability Eq. (9) we define the gen-
s=0
we read off the coefficient Zf(s;τ)=(cid:88)φˆs(ω,τ)(cid:16)eτϕ∗f(s), (28)
(cid:12)
g(ω)= 1 ∂Fs(ω)(cid:12)(cid:12) (21) ω
2 ∂s (cid:12)
s=0 where we make explicit the dependency on the affinities
by comparing with Eq. (19). This is the result found f driving the system. In the limit τ → ∞ the large de-
in Ref. 11 following quite a different approach. Pro- viationfunctionagainfollowsfromtheLegendre-Fenchel
vided we know how the forces depend on s, we have transform
thus constructed one possible observable R to be used
τ ϕ∗(s)=sup[r·s−ϕ (r)], (29)
in Eq. (11). f f
r
5
where we have assumed that a large deviation principle afterinsertingthespeedv(f+s)=(cid:104)x˙(cid:105) =D [−(cid:104)∂ U(cid:105) +
s 0 x s
Pf(R;τ) (cid:16) e−τϕf(r) holds with ri ≡ Ri/τ. As a conse- f+s]. Duetotheadditivityoftheperturbation,thecon-
quence,ϕ∗(s)isalways aconvexfunctionandconstitutes jugate variable (cid:104)r˙(cid:105) is a simple linear function of s inde-
f
our local potential around a steady state determined by pendentoff implyingthepotentialsϕ∗(s)= 1D s2 and
4 0
theaffinitiesf. Forapotentialϕ∗(s)thatisdifferentiable ϕ(r) = 1 r2. Hence, while v(f) is a non-linear function
at s = 0, the correlations are mfanifestly symmetric and ofthedrDiv0ingforcef,thelocalpotentialdescribestrivial,
follow as equilibrium-like fluctuations [7]. Close to equilibrium in
(cid:104)RτiRτj(cid:105)= ∂s∂i∂2sj lnZf(cid:12)(cid:12)(cid:12)(cid:12)s=0 (cid:16) τ∂∂s2iϕ∂s∗fj(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s=0 =τLifj tahseelxipneeactrerde.sponseregimeonerecovers(cid:104)x˙(cid:105)=2(cid:104)r˙(cid:105)=D0f
(30)
with steady state susceptibilities C. Fluctuation theorem
1∂(cid:104)Ri(cid:105) ∂(cid:104)r˙i(cid:105)
Lij ≡ lim τ s = s. (31) Whatisthephysicalmeaningofs? Togetsomeinsight
f τ→∞τ ∂sj ∂sj let us assume that s shifts the steady state to f+s with
ThisresultextendstheOnsagerpotentialEq.(4)tonon- probability
zero affinities and emphasizes the canonical structure.
P (R;τ)∼esRP (R;τ), (36)
An interesting consequence is that, employing Legendre f+s f
transforms as in conventional thermodynamics, we can which is the expression that appears in the generating
now switch between affinity si and current (cid:104)r˙i(cid:105) depend- function. For a large class of observables (including
ing on what is the more convenient variable for a spe- the example from the previous subsection) we can write
cific situation. Moreover, susceptibilities are related by R = 1(X −S )asacurrentX minusanothertermS ,
Maxwell and further relations (similar to, e.g., the rela- boτth o2f whτich hτave the same avτerage (cid:104)X (cid:105)=(cid:104)S (cid:105) in thτe
τ τ
tion between the heat capacities at constant volume and unperturbed NESS. While the current is antisymmetric
constant pressure). with respect to time reversal, X† = −X , the second
τ τ
term S† =S is invariant. The fluctuation theorem [13]
τ τ
then becomes
B. Illustration: Single particle in a ring trap
P (R;τ) P (X,S;τ)
f+s = f esX =e(f+s)X, (37)
Tobrieflyillustrateourresultsweconsidertheparadig- Pf+s(R†;τ) Pf(−X,S;τ)
matic single colloidal particle moving in a ring trap [36–
where the final expression involves the entropy Σ =
τ
39], see Fig. 1(a). The state of the system is given by
(f +s)X produced in the perturbed NESS. This shows
τ
the position x with force F(x) = −∂ U(x)+f, where
x that the parameter s of the generating function, for
U(x) is an external, periodic potential energy and f is
the pair (s,R), indeed corresponds to a change of the
the constant driving force. The diffusion coefficient D
0 affinity f determining the unperturbed NESS. The im-
isindependentofx. Theparticleisdrivenintotheunper-
portance of the time-symmetric contribution S for the
τ
turbed NESS through the force f with non-zero average
non-equilibriumlinearresponsehasbeendiscussedbyC.
speed v(f) = (cid:104)x˙(cid:105) = D [f −(cid:104)∂ U(cid:105)]. As perturbation we
0 x Maes and coworkers [6, 40].
consider a change of the driving force, f →f +s, with
F (x)=−∂ U(x)+f +s. (32)
s x
D. Linear response regime
From Eq. (20) we immediately find
g = 1∂Fs(cid:12)(cid:12)(cid:12) = 1. (33) peAarsinegluidneEdqt.o(i1n1t)hiesninottroudnuiqcutieo.nT,thhisebobecsoemrveasblaepRpaτraenpt-
2 ∂s (cid:12) 2
s=0 inthelinearresponseregimeperturbingthermalequilib-
From Eq. (24) one then obtains h(x)=−1D F (x) and rium when choosing the current Rτ → Xτ, which also
2 0 f has vanishing mean. Again appealing to the fluctuation
thus from Eq. (22) the generalized velocity
theorem we have for small |f|(cid:28)1
1
r˙(x,x˙)= 2[x˙ −D0Ff(x)]. (34) Pf(X;τ) = e12f·XP0(X;τ) =efX. (38)
Thisisindeedoneoftheadmissiblechoicesfordetermin- Pf(−X;τ) e−12f·XP0(−X;τ)
ing the response with respect to a change of the driving
Here we have used that the currents change sign un-
force [18].
der time reversal, whereby in equilibrium P (−X;τ) =
0
The average of Eq. (34) for a perturbed NESS with
P (X;τ) holds. Following Eq. (36) one sees that now we
0
f +s becomes have to use s → 1f leading to the definition of the gen-
2
1 1 erating function Eq. (6) given in Sec. IIB, which in turn
(cid:104)r˙(cid:105)s = 2[v(f +s)+D0(cid:104)∂xU(cid:105)s−D0f]= 2D0s (35) leads to the famous Onsager result.
6
VI. CONCLUSIONS ward with rates k+ and k−, respectively. The affinity
is simply the force f = ln(k+/k−). For N+ steps for-
τ
InthispaperwehavestudiedthetiltedMarkovgener- ward and Nτ− steps backward, the distance traveled is
ator under the condition that for small tilt s it preserves Xτ =Nτ+−Nτ− with average
normalization and thus describes a physical stochastic
process. Identifyingthisprocessasashiftedsteadystate (cid:104)Xτ(cid:105)=τ(k+−k−)=τk+(1−e−f). (A1)
has allowed us to interpret the abstract tilt parameter
s of the generating function as a perturbation of the Note that here the distance takes only discrete integer
original steady state. For Markov processes we have ex- values. Its probability is known analytically [44]
plicitly constructed two types of conjugate observables
√
R that encode the system’s response and thus allow to P (X;τ)=I (2 k+k−τ)(k+/k−)X/2e−(k++k−)τ,
f X
determine transport coefficients from correlations in the (A2)
unperturbedsteadystate. Onlyforcesarerequiredasin- where I (z) is the modified Bessel function of the first
n
put, no explicit knowledge of the stationary distribution kind of order n. The generating function
or entropy production is necessary.
Whatisperhapsmostinterestingisthenotionofdiffer- ∞
(cid:88)
ent ensembles analogous to conventional thermodynam- Zf(s;τ)= esXPf(X;τ) (A3)
ics. Consider for example the situation that we require X=−∞
can be calculated exactly using [45]
a transversal transport coefficient for fixed longitudinal
field (affinity) although the simulations (or experiments)
∞
have to be performed at fixed current. Transport coef- (cid:88) I (z)cX =exp(cid:2)(z/2)(c+c−1)(cid:3). (A4)
ficients in one ensemble could then be calculated from X
X=−∞
those in another ensemble much in the same way the
heat capacity at constant pressure is calculated from the
The result is Z (s;τ)=exp[τφ∗(s)] with
heat capacity at constant volume. The approach pre- f f
sented here might pave the way for a systematic theory
(cid:16) (cid:17)
althoughthesimpleexampleofatrappedBrownianpar- φ∗(s)=k+ es+e−(f+s)−e−f −1 . (A5)
f
ticle demonstrates that not all informations about the
steady state are encoded in the corresponding local po- Clearly, the derivative
tential.
∂φ∗ (cid:16) (cid:17)
x = f =k+ es−e−(f+s) (A6)
s ∂s
Appendix A: Asymmetric random walk
only agrees with the current Eq. (A1) for s = 0. Note
Asaspecificexampleweconsidertheasymmetricran- that for this simple example the same current can be
dom walk (ARW) [41–43], for which we can perform the achieved through the effective force f = f +2s while
eff
transformations analytically. The ARW describes the simultaneously rescaling time. The slightly more com-
motion of a walker on an infinite lattice [cf. Fig. 1(a)] plex example of a bias random walker with two internal
with discrete sites. The walker jumps forward and back- states has been treated in Ref. 35.
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