A meaningful expansion around detailed balance

0.16 MB·English

by  Matteo Colangeli

A meaningful expansion around detailed balance
Matteo Colangeli,1 Christian Maes,2 and Bram Wynants3
1Dipartimento di Matematica, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy∗
2Instituut voor Theoretische Fysica, K. U. Leuven, 3001 Leuven, Belgium
1 3Institut de Physique The´orique, CEA-Saclay,
1
0 F-91191 Gif-sur-Yvette Cedex, France
2

n Abstract
a
J
We consider Markovian dynamics modeling open mesoscopic systems which are driven away from de-
8
1
tailedbalancebyanonconservative force. Asystematicexpansion isobtained ofthestationary distribution

]
h around an equilibrium reference, in orders of the nonequilibrium forcing. The first order around equilib-
p
- rium has been known since the work of McLennan (1959), and involves the transient irreversible entropy
h
t
a flux. The expansion generalizes the McLennan formula to higher orders, complementing the entropy flux
m
[ with the dynamical activity. The latter is more kinetic than thermodynamic and is a possible realization

1
ofLandauer’s insight (1975) that, fornonequilibrium, the relative occupation ofstates also depends onthe
v
7
noisealongpossibleescaperoutes. Inthatwaynonlinearresponsearoundequilibriumcanbemeaningfully
8
4
3 discussed intermsoftwomainquantities only, theentropy fluxand thedynamical activity. Theexpansion
.
1
makesmathematical senseasshowninthesimplestcasesfromexponential ergodicity.
0
1
1
: PACSnumbers:05.40.-a,05.70.Ln,05.10.Gg
v
i
X
r
a
∗Electronicaddress:[email protected]
1
I. INTRODUCTION
Recent years haveseen an intensivesearch fora theoreticalframework underlyingnonequilib-
rium fluctuations. The nature of nonequilibriumis diverseand rich but we hope to uncover some
unifying structures that, ideally, would give an extension of the Gibbs formalism. Still, today,
the mathematical models are very often simplified to Markov dynamics for jump or diffusion
processesthatrepresentopensystemsinweakcontactwithoneormorereservoirs. Thesesystems
(suchasmolecularconfigurations)canbeverysmallbutneverthelesswehavelearnthowtoattach
thermodynamic meaning to various path-dependent quantities, see e.g. [27, 33], as they relate to
what happens in the large environment. Most of that has concentrated around the concepts of
energy and entropy, so that much of standard irreversible thermodynamics also got a formulation
for mesoscopic systems. Another question has concerned response theory for these Markov
dynamics and some systematics have been obtained there also, e.g. in [1, 2, 22, 29, 31, 35].
A related issue concerns the characterization of the stationary distribution for a dynamics that
breaks the condition of detailed balance. Of course we need a physically meaningful violation of
detailed balance, and a useful interpretation is given by the condition of local (sometimes called,
generalized) detailed balance, [5, 11, 18, 19, 34]. The latter requires that the power dissipated to
the environment during a transition equals the logarithmic ratio between forward and backward
transitionrates. Fromthereonecanhopetofindthecorrespondingstationarydistributioninterms
of these irreversible entropy fluxes. That is exactly what was achieved by McLennan in 1959,
[30]. Inthatway,thestationarydistributionpicksupthermodynamicinformationandisnolonger
“justastationarysolution”oftheMasterequation. However,thatMcLennanproposalonlyworks
close-to-equilibrium[12–14,26]. Theredoesnotappeartobeareadyextensionbeyondthelinear
regime in terms of entropy considerations only. The present paper takes a next step but we need
to go beyond the purely entropic concepts we are used to from heterogeneous equilibrium. That
is, to go to second and higher order in an expansion around detailed balance we need another
concept to complement the entropy fluxes. That novel quantity is called the dynamical activity
and is much related to the notion of escape rate: it measures the reactivity and instability of a
trajectory. Dynamicalactivityisthusmuchmoreconcernedwithkineticsthanitisembeddedinto
thermodynamics but by introducing it, we can complete the expansion beyond linear order in the
nonconservative forces around equilibrium. In that sense, we add to the spirit of the McLennan
proposal the insights of Landauer and others that for nonequilibria, the noise behavior along in-
2
and outgoingtrajectoriesenters criticallyintothedeterminationofstateplausibilities,[15–17].
In the expansion of the stationary distribution, every term at any order in the nonequilib-
rium forcing, just contains the same (dynamical) observables in various combinations of time-
correlationfunctionsunderthereferenceequilibriumprocess. Forexample,insomeprecisesense
correct up tothird orderweget
1 1
ρ(x) = ρ (x)[1 S o + S o S(S2 +3 2 12 ) o]
o −h ix 2 h T1ix − 24h T1 − T2 ix
for the stationary distribution ρ on states x in terms of the equilibrium distribution ρ . The
o
averages o are over the detailed balance process started from state x while S is the irreversible
h·ix
entropy flux and , denote the first and second order to the dynamical activity. These
1 2
T T
path observables S and depend of course on the nonequilibrium dynamics: S is basically
1,2
T
determined by the work done by the nonconservative force in a particular trajectory, and
1,2
T
measures the expected dynamical activity long the followed trajectory. Specific details, rewriting
and mathematicalprecision followbelow.
In the next section, we specify the Markovian set-up and we define the various objects such
as ρ,S and in the above. We concentrate on overdamped diffusions and jump processes for
T
introducing(inaparticularway)thenonequilibriumdriving. SectionsIII–IV–Vgivethemainidea
and the structure of the expansion with some writing out for specific models to lowest order. The
lastsectionaboutnonlinearresponsesuggestssomeimmediateapplicationbeforeweconcludethe
paper. AppendixBdefinestheentropyfluxandthedynamicalactivityforunderdampeddiffusions.
II. SET-UPOFMARKOVSTOCHASTICDYNAMICS
We restrict our analysis to Markovian stochastic models for mesoscopic systems driven by a
nonconservative force, i.e., a force that cannot be derived from a potential. We imagine such
systems to be immersed in an environment in thermal equilibrium at some inverse temperature
β. All conditions are time-independent and we want to characterize the statistical distribution of
statesforouropen systemwhen reaching stationarity.
A first standardchoiceisto consideroverdampeddiffusionsforstatex Rn, according towhich
∈
χ
x˙ = χ F(x )+ 2 ξ , ξ = standardwhitenoise (1)
t t t t
· β
r
3
where the mobilityχ is a positivedefiniten n-matrix not depending on x (for simplicityonly).
×
Thetotalforceequals
F = ǫf U
−∇
for the energy U of the system, and with f the nonconservativeforce with amplitudeǫ (our small
number). When ǫ = 0, thedynamicssatisfies the conditionofdetailed balance, and givenenough
time the distribution of states converges to the equilibrium distribution ρ (x) e−βU(x), so we
o
∝
assume.
Whenever ǫ = 0 the system is not in equilibrium. The Fokker-Planck equation for the evolution
6
ofdistributionsµ is
t
∂µ χ
t
(x)+ [χF(x)µ (x) µ (x)] = 0
t t
∂t ∇· − β∇
The stationary solution ρ thus satisfies β F(x)ρ(x) = ∆ρ(x), but we would be much helped
∇ ·
by further more explicit, physical or systematic understanding of that ρ. One option is to find an
expansion for that stationary distribution ρ(x) of the system in orders of ǫ, assuming uniformly
and exponentially fast relaxation behavior. That is the programme of the present paper and we
will find that the main quantities in such an expansion are directly related to two specific path
observables,thatwenowintroduce.
Wefix alarge time-interval[0,T]and weconsiderpaths ω = (x ,t [0,T]),on whichwedefine
t
∈
ǫ2β T T T
(ω) = dtf χf ǫβ dtf χ U +ǫ dtχ f (2)
T 2 · − · ∇ ∇·
Z0 Z0 Z0
T
S(ω) = ǫβ dx f(x )
t t
◦
Z0
The last stochasticintegral (with the ) is in the sense of Stratonovich; in that way S is identified
◦
withβ timestheworkdonebythenonconservativeforce—forshort,wespeakabouttheentropy
fluxS. Thequantity islessfamiliar,anditcontainsbothorderǫandorderǫ2. Correspondingly,
T
we write = + for the first and second order. Its meaning is best understood before the
1 2
T T T
continuumlimit,in termsofjumpprocessesto whichweturnnext.
We consider a Markov jump process with discrete states x and as a reference we take jump rates
k (x,y)forthetransitionx y whichsatisfydetailedbalance, i.e.,
o
→
k (x,y)e−βU(x) = e−βU(y)k (y,x)
o o
In this case the stationary Master equation is solved by ρ (x) exp( βU(x)). To this reference
o
∝ −
weadd an extraflux f(x,y) = f(y,x) ofenergy inthetransitionx y,and write
− →
kǫ(x,y) = ko(x,y)eβ2ǫf(x,y) (3)
4
Toestablishnonequilibriumweaskthatthefluxesf(x,y)cannotallberewrittenasthedifference
V(x) V(y) of a unique potential V, which means that there are loops x x ...x = x
1 2 n 1
− → →
over which the sum f(x ,x ) + f(x ,x ) + ...f(x ,x ) = 0 does not vanish. In terms of
1 2 2 3 n−1 n
6
the condition of local detailed balance, these fluxes f(x,y) should be interpreted as the product
of a displacement of a certain quantity and a nonconservative force, see [11], but here we do not
need this formulation. The ǫ is the magnitude of the nonequilibrium forcing. Finally, in (3) we
havechosentoomitanextrasymmetricprefactorψ (x,y)becausethemoregeneralchoicefor(3)
ǫ
wouldbe k (x,y)/k (x,y) = ψ (x,y) exp[βǫf(x,y)/2]. Wetakehoweverψ (x,y) = ψ (y,x) =
ǫ o ǫ ǫ ǫ
1 formuchgreatersimplicity.
Again,forǫ = 0,thestationaryprobabilitylawρisonlyknownindirectlyassolutionoftheMaster
6
equation [ρ(y)k(y,x) ρ(x)k(x,y)] = 0 forallx.
y −
ThistimePtheentropyflux is
S(ω) = ǫβ f(xt−,xt) (4)
t≤T
X
as asumoverthejumptimestin thetrajectory ω = (x ,s [0,T]),and thedynamicalactivityis
s
∈
T
(ω) = 2 dt ko(xt,y)[eβ2ǫf(xt,y) 1] (5)
T −
Z0 y
X
Nowweseebetterwherethenameactivitycomesfrom: (ω)isthedifferenceintheescaperates,
T
integrated over the trajectory. The escape rate k(x,y) measures the frequency by which the
y
system exits state x, and in that way it counts Pthe expected number of transitions away from x.
In other words, (ω) sees how the escape rate away from the trajectory ω changes when adding
T
the forcing f. In the appropriate rescaling the expression (2) is simply the continuum limit of (5)
from Markov jump to (overdamped)diffusions. We do that computationin Appendix A; the case
of underdamped or inertial diffusions is shortly discussed in B. We repeat that we use here (3),
i.e., with prefactor ψ = 1; otherwisetheexpressionfortheactivitygets morecomplicated — the
ǫ
majorpart oftheanalysiswouldhoweverremain unchanged.
III. EXPANSION:MAINIDEA
The expansion of the stationary distribution starts from a simple idea which was applied al-
ready in [7, 12, 20, 26]: the single-time distributions on states, and in particular the stationary
distribution, can be obtained from its embedding in the path space distribution. The latter is the
5
distribution on the level of trajectories or paths ω and gives the weight P(ω) for path-integrals.
P is much more directly obtained and is much better-behaved than its projections on single time
layers. In fact, wecan giveexplicitexpressionsforthe“action”A(ω)in
P(ω) = e−A(ω)Po(ω) (6)
that connectsthedistributionP on paths startingfromρ but withdrivingf, withthefullequilib-
o
riumreference distributionPo, seee.g. [28]forsomeusefultechniques. TheactionAis typically
localinspace-timeandthusissimilartoHamiltoniansorLagrangiansthatwemeetin(equilibrium
statistical)mechanics, seee.g. [23, 27]. Wecan verify that
A = ( S)/2
T −
as defined above for overdamped and jump processes. Furthermore, the action A in (6) is left
unchangedwhen bothprocesses startfrom thesamestatex,
P (ω) = e−A(ω)Po(ω) (7)
x x
Nowcomestheembedding. Asdefinedbefore,bothnonequilibriumandequilibriumprocesses
in (6) start at time t = 0 from data distributed with the equilibrium ρ . At time T later, the
o
probabilityto find thedrivensystemin statex is
p(x,T) = δ(x x) ǫ
h T − iρ0
averaging over the trajectories for the nonequilibrium dynamics. We assume that for T +
↑ ∞
thereisexponentiallyfastconvergencep(x,T) ρ(x)(inthesenseofdensities)tothestationary
→
distributionofthenonequilibriumprocess, uniformlyin ǫ. By (6)wecan rewrite
p(x,T) = δ(x x)e−A(ω) o (8)
T
−
(cid:10) (cid:11)
whichisnowanexpectationvalueforthefullequilibrium(detailedbalance)process(andthenwe
omitthesubscriptρ ). By time-reversalinvariance,(8) equals
o
p(x,T) = δ(x πx)e−A(θω) o = ρ (x) e−A(θω) o (9)
0 − o x0=πx
(cid:10) (cid:11) (cid:10) (cid:11)
wherethetime-reversaloperatorθ acts as
θω = ((πx) ,0 t T)
T−t
≤ ≤
6
with πx equal to x except for flipping the velocities (if they are part of the state-description) or
othervariables withnegativeparity undertime-reversal. In equilibriumρ (πx) = ρ (x). Equality
o o
(9)can stillberewrittenintermsofS and getting
T
p(x,T)ρ (x) e−(S+T)/2 o (10)
o x0=πx
(cid:10) (cid:11)
Indeed, thedecompositionA = ( S)/2followsthesymmetryundertime-reversal,
T −
S(ω) = A(θω) A(ω)
−
(ω) = A(θω)+A(ω)
T
The fact that the quantity A(θω) A(ω) is the excess entropy flux from the system into the
−
environment during the process ω is one of the main discoveries for the construction of nonequi-
librium statistical mechanics of the last decade, see [4, 25] and the references in e.g. [8, 23, 29].
Excess means the difference between the nonequilibrium process and the reference equilibrium
process. Specifically, this excess is here equal to the work done by the nonequilibrium force ǫf,
multiplied by β. The dynamical activity has been introduced and used before [21, 24] but has
T
no thermodynamictradition. Its role is kinetic and that it influences the relativestability of states
was somehowemphasizedlongbefore, cf. [15–17].
Theleft-handsideof(10)isassumedtoconvergeexponentiallyfasttothestationarylawρ(x),
but there is a problem with its right-hand side because both S and are time-extensive of order
T
T, being sent to infinity. We will therefore need to control the limit T + and to worry about
↑ ∞
the exchange with the sum of the perturbation series. An important point here that follows from
(7), isthenormalization
e(S−T)/2 o = 1 (11)
x
(cid:10) (cid:11)
valid under the equilibrium process but started from an arbitrary state x. That itself can be ex-
panded in orders of ǫ, and takes care of many cancellations. Finally, one must use that S is
anti-symmetric,and is symmetricundertime-reversal,so thatoverall time-intervals[0,T]
T
Sn m o = 0, nodd (12)
h T i
for all m and for all odd powers n. With these ingredients (10)–(11)–(12), combined with fast
relaxationfortheequilibriumprocess,all isinplacetostartasystematicexpansion. In thepartic-
ular cases we have in mind, see Section II, the path function S is simply first order in ǫ and is
T
7
either = + secondorderinǫfordiffusions,see(2),orofarbitraryorder = + +...
1 2 1 2
T T T T T T
for Markov jump processes. For practical matters our expansion including second or third order
around equilibriumis already newand relevant.
IV. GENERALEXPANSION
Formally,expanding(10)justgives
p(x,T) 1 1 (S + )2 o
= 1 S + o + + T1 +O(ǫ3)
ρ (x) − 2 h T1ix0=πx 2 −T2 4
o (cid:28) (cid:29)x0=πx
On theotherhand,(11)gives
1 (S )2
0 = S o , 0 = −T1 T2 o ,...
2h −T1ix0=πx h 8 − 2 ix0=πx
Adding or subtracting these relations from the corresponding orders of the expansion of the sta-
tionary distribution simplifies matters. In the end, in every order of the expansion we can choose
thatonlythosetermssurvivewhichare averages ofquantitiesantisymmetricin time,
p(x,T) 1
= 1 S o + S o (13)
ρ (x) −h ix0=πx 2 h T1ix0=πx
o
1
S(S2 +3 2 12 ) o +O(ǫ4)
−24 T1 − T2 x0=πx
(cid:10) (cid:11)
Theaboveconsiderationscanbesystematized. Theformalexpansionthatresults,afteralsotaking
intoaccount thenormalizationin(11), is
∞
S + S o
p(x,T) = ρ (x) 1+ B T B −T (14)
o m m
" mX=1D (cid:18)− 2 (cid:19)− (cid:18) 2 (cid:19)Ex0=πx#
where we introduced a shorthand notation for the path-dependent functionals B (G) acting on
m
pathobservablesG(ω,ǫ),defined via
B (G) = B (G ,...,G )
m m 1 m−k+1
m 1 G b1 G bm−k+1
= 1 ... m−k+1 (15)
b !...b ! 1! (m k +1)!
k=1 σ 1 m−k+1 (cid:18) (cid:19) (cid:18) − (cid:19)
XX
for G (ω) = dℓG/dǫℓ(ω,0). The sum in (15) extends over all sequences σ of non-negative
ℓ
coefficients σ = (b ,...,b ), such that m−k+1b = k and m−k+1jb = m. Note that the
1 m−k+1 j=1 j j=1 j
Bm are versions of the so called completePBell polynomials. TPhere exist alternative, although
8
equivalent, expressions for (14), see e.g. [9]. On the other hand, the expression we use has the
advantageofbeingcompact and suitablefornumericalimplementation.
From (14) and (15), we can write the explicit expression for the m-th order in ǫ. Remember
thatwewrite = + +...and thatS isoforderǫwhile isoforderǫn. Theresultis
1 2 n
T T T T
∞
p(x,T) 1 1 o
= 1+2 (− )Pbj Sb0 b1 b2... bm (16)
ρo(x) 2 b0!b1!...bm! T1 T2 Tm x0=πx
mX=1Xσm D E
The sum in (16) extends over all sequences σ of non-negativeintegers (b ,b ,...,b ), such that
m 0 1 m
b is odd and b + m jb = m. For illustration to construct (13), m = 1 requires b = 1 and
0 0 j=1 j 0
all other bj = 0; mP= 2 requires b0 = b1 = 1 with all the other bj = 0; m = 3 allows three cases
b = b = 1, b = 1,b = 2and b = 3 each timewithall otherb = 0.
0 2 0 1 0 j
In the case of diffusions (where p(x,T) must be understood as a probability density with respect
to dx), see (2), we have = 0 for n > 2 so that we mustthen also require b = 0,j 3 in each
n j
T ≥
σ .
m
Thefirst importantthingtoobserveabout theexpansion(16)isthat itconvergesforfixed state
xand uniformlyintimeT ifthereis c = c(x) < suchthat each term isboundedlike
↑ ∞ ∞
o
Sb0 b1 b2... bm ǫmcPbj (17)
| T1 T2 Tm x0=πx| ≤
D E
Thereason isthat
∞ c 1 ∞ m ck(m+1)k
ǫm ( )Pbj ǫm
2 b !b !...b ! ≤ 2kk!
0 1 m
mX=1 Xσm mX=1 Xk=1
converges forsmallenough ǫ.
Weonlyarguefor(17)explicitlyforthefirstandthesecondorder. ThefirstorderistheMcLennan
formula(seeimmediatelybelow)andhasbeentreatedbeforein[26]. Thesecondorderaddsanew
complication(tobetreatedbelow)andthatcomplicationisrepeatedforthehigherordertermsand
canbesolvedinthesameway. Atanyratewecannotquiteleaveitwith(13)orwith(16)because
weareinterestedin thelimitT and S, do notmakeanysenseinthatlimit. We thusneed a
↑ ∞ T
furtherrewritingforwhichweneedsomemoremodel-dependentinputandtowhichweturnnext.
9
V. EXPANSIONDETAILS
A. McLennanformula: thefirstorder
The first order in the expansion of the stationary distribution (13) has been known for a long
time [30] and has been reconsidered more recently in [12, 26]. In fact, the idea of obtaining the
McLennan-formulaviatheembedding described under Section III originates from [12]. Theway
howtodealwiththelimitingbehaviorT ,ǫ 0wastreatedin[26]. Webrieflyrepeatthisand
↑ ∞ ↓
weconcentrateon theMarkovprocesses ofSection II.
Themain pointis that
T
S(ω) o = ǫβ dt w(x ) o (18)
h ix h t ix
Z0
where, for jump processes w(x) = k (x,y)f(x,y), and for overdamped diffusions
y o
w = χ f/β χf U, see [26]. ThePexpression (18) allows the limit T uniformly in ǫ
∇· − ·∇ ↑ ∞
onceweassumetheequilibriumprocessto beirreducibleand exponentiallyergodic.
Plugging(18)into(16)thusgivesthelinearorderexpression
ρ(x)/ρ (x) = 1 ǫβh(x)+O(ǫ2) (19)
o
−
with
∞
h(x) = dt w(x ) o
h t ix
Z0
inwhichtheintegralisexponentiallyconvergent. Therelationwithlocalequilibriumdistributions
isalso discussedin[26].
B. Secondorder
Welookat them = 2 termin(16). Themainobject toconsiderforjumpprocesses is
o
T T
ds f(xt−,xt) ko(xs,y)f(xs,y)
Z0 * t y +
X X x
where the sum is over the jump times. On the other hand, for overdamped diffusions we can
introduce σ(x ,dx ) = ǫβdx f(x ) and (x) = βǫf(x) χ U + ǫχ f, so that we must
t t t t 1
◦ T − · ∇ ∇ ·
10