Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: curvature, gauge field, and generalized fluctuation-dissipation theorem.

Abstract

The driving force of the dynamical system can be decomposed into the gradient of a potential landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to near equilibrium systems with detailed balance. The response due to a small perturbation can be expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT that the response function is composed of two parts: (1) a spontaneous correlation representing the relaxation which is present in the near equilibrium systems with detailed balance and (2) a correlation related to the persistence of the curl flux in steady state, which is also in part linked to a internal curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where the entropy production rate can be decomposed into spontaneous relaxation driven by gradient force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium environment and breaks the detailed balance. On any particular path, the medium heat dissipation due to the non-zero curl flux is analogous to the Wilson lines of an Abelian gauge theory.

Full text

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arXiv:1108.5680v1 [cond-mat.stat-mech] 29 Aug 2011
Potential and Flux Decomposition for Dynamical Systems and Non-Equilibrium
Thermodynamics: Curvature, Gauge Field and Generalized Fluctuation-Dissipation
Theorem
Haidong Feng1and Jin Wang2, ∗
1Department of Chemistry, Physics, and Applied Mathematics, State University of New York at Stony Brook
2State Key Laboratory of Electroanalytical Chemistry,
Changchun Institute of Applied Chemistry, Chinese Academy of Sciences
(Dated: August 30, 2011)
The driving force of the dynamical system can be decomposed into the gradient of a potential
landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to
near equilibrium systems with detailed balance. The response due to a small perturbation can be
expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT
that the response function is composed of two parts: (1) a spontaneous correlation representing the
relaxation which is present in the near equilibrium systems with detailed balance; (2) a correlation
related to the persistence of the curl flux in steady state, which is also in part linked to a internal
curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the
equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where
the entropy production rate can be decomposed into spontaneous relaxation driven by gradient
force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium
environment and breaks the detailed balance.
The global stability is essential in understanding the
dynamical non-equilibrium systems. The driving force
of the dynamical system often is not integrable and can
not be written in terms of the gradient of a potential.
The driving force however can be decomposed into the
gradient of a potential and a curl flux (current) [1]. The
potential is related to the steady state probability and
the gradient force gives the normal dynamics analogous
to equilibrium system, while the curl flux force is di-
rectly linked to the non-equilibrium contribution from
detailed balance breaking. For non-equilibrium dynam-
ics, the dual description with both potential and flux is
necessary.
In addition, the fluctuation-dissipation theorem (FDT)
plays a central role for systems in near equilibrium sys-
tems with detail balance [2, 3]. It links the fluctuations of
the system quantified by the correlation function with the
response of the system quantified by the response func-
tion. Many efforts have been made to extend the FDT
to non-equilibrium systems [4–16]. It was found that the
FDT involves the correlation function of a variable that
is conjugate with entropy [17]. Furthermore, by choosing
proper observables, the FDT for non-equilibrium systems
can be uncovered [18].
In this letter, we found another way to generalize FDT
for non-equilibrium processes, specifically for direct ob-
servables such as xi, under Markov dynamics in contin-
uous space described by Langevin dynamics or Fokker-
Planck equations. Particularly, the response function can
be split into two parts. One is from the correlation of
the observable itself representing the spontaneous relax-
ations, which also exists in systems with detailed bal-
ance. The other one relates to the heat dissipation in the
medium, representing the detailed balance breaking con-
tribution, which directly links to the curl flux part of the
force. On a closed loop, the medium heat dissipation can
be described by the internal curvature introduced by the
non-gradient force or curl flux part, which is analogous to
Abelian Gauge Theory [19]. On any particular path, the
medium heat dissipation is analogous to the Wilson lines
of Abelian gauge theory [19]. In the equal time limit, the
generalized FDT naturally leads to non-equilibrium ther-
modynamics [20–22]. In addition, this generalized FDT
is also related to Fluctuation Theorem [23–28].
Markov dynamics in continuous space can be charac-
terized by Langevin equations:
˙ xi= Fi(x) + Bij(x)ξj(t)(1)
where Fi(x) is the driving force and ξi(t) is the Gaussian
distributed white noise: ?ξi(t)ξ′
the Einstein notation is used: when an index i appears
twice in a single term, it implies that we are summing
over all of its possible values. The probability obeys the
Fokker-Planck equation:
j(t′)? = δ(t − t′). Here
˙P(x,t) =ˆL(x)P(x,t) (2)
with the operatorˆL(x) =
the diffusion coefficient Dij(x) =1
venience, we use ∂i ≡
the time dependent probability distribution and PSS(x)
to indicate the time independent steady state probability
distribution. The flux can be defined as:
?
− ∂iFi(x) +∂i∂jDij(x)
?
and
2(BBT)ij(x). For con-
∂xi, P(x) ≡ P(x,t) to represent
∂
−˜Fi(x)P(x) + Dij(x)∂jP(x) = ji(x)(3)
where˜Fi= Fi−∂jDij. Then Fokker-Planck equation can
be rewritten as
dt
= ∂ · j. The system is considered
dP(x,t)

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to be in detailed balance if the steady state flux:
−˜Fi(x)PSS(x) + Dij(x)∂jPSS(x) = jSS
i
(x) (4)
is zero: jSS= 0. For general non-equilibrium systems
without detailed balance: jSS?= 0, the steady state flux is
a divergence free vector with ∂ · jSS= 0. The force term
˜Fj(x) can be decomposed into two parts: a potential gra-
dient term −Dij(x)
and flux term −jSS
bilistic velocity: vi(x). Alternatively, the gradient of po-
tential −lnPSS(x) can also be decomposed into a force
term and a curl flux term :
∂
∂xiU(x) where U(x) = −lnPss(x)
j (x)/PSS(x) ≡ −vj(x), with a proba-
− ∂iln[PSS(x)] = D−1
ij(x)[−˜Fj(x) − vSS
j (x)] (5)
Using perturbation theories, FDT for equilibrium sys-
tems with detailed balance was investigated [3]. Here
we will extend it to non-equilibrium systems. Consider
a linear perturbation on the force: Fi(x) → F′
Fi(x) + h(t)δFi(x), we haveˆL →ˆL′=ˆL − h(t)δˆL, with
δˆL = δFi(x)∂i+ ∂iδFi(x). The probability evolves as
i(x) =
P(x,t) = exp
??t
t′dt(ˆL − hi(t)δˆL)
?
P(x,t′) (6)
δ?Ω(t)? = ?Ω(t)? − ?Ω? =?dxΩ(x)[P(x,t) − PSS(x)]
Therefore, for t ≥ t′, the response function reads as
RΩ
i(t − t′) = δ?Ω(t)?/δh(t′)

δF=0
=
?
dxΩ(x)eˆL(t−t′)(−δˆL)PSS(x)(7)
Using the decomposition in equ. (5), we have
RΩ
?
i(t − t′)
dxΩeˆL(t−t′){δFi[−˜Fk− vSS
(8)
=
k]D−1
ik− ∂iδFi}PSS
= −?Ω(t)∂iδFi(t′)? −
?
?Ω(t)δFi(t′)˜Fk(t′)D−1
ik(t′)?
+?Ω(t)δFi(t′)vSS
k(t′)D−1
ik(t′)?
?
This is the general relation between response functions
and correlation functions.
tween two observables Ω1and Ω2is CΩ1Ω2(t′,t) =
?Ω1(t′)Ω2(t)? − ?Ω1(t′)??Ω2(t)? with ?Ω1(t′)Ω2(t)? =
PSS(xi)Ω1
sition probability from initial state xiat time t′to final
state xjat time t. For the perturbation independent on
x: δFi= 1, we obtain
Here, the correlation be-
xiΩ2
xjP(xi,t′|xj,t). P(xi,t′|xi,t) is the tran-
RΩ
i(t − t′) = −?Ω(t)∂iln[PSS(x)]?
?
which is a generalized FDT for non-equilibrium systems
[21]. With the force decomposition in equ. (5), the re-
sponse of the system is composed of two terms. The first
term, just as equilibrium cases, is related to the usual
(9)
= −?Ω(t)˜Fk(t′)D−1
ik(t′)? + ?Ω(t)vSS
k(t′)D−1
ik(t′)?
?
correlation of the variable with the driving force. This
term exists even for FDT of equilibrium systems obeying
the detailed balance (this is the case where the gradient
of the logarithm of probability is equal to the driving
force). The second term however is directly related to
the non-zero flux which violates the detailed balance and
measures the degree of the non-equilibrium-ness (how far
away the system is from equilibrium).
FDT in equ. (9) can also be generalized to the case
that the system is not prepared in steady state but an
arbitrary distribution P(x). For t ≥ t′, we have
RΩ
i(t − t′) =
?
dxΩ(x)eˆL(t−t′)(−δˆL)P(x) (10)
= −
?
?Ω(t)˜Fk(t′)D−1
ik(t′)? + ?Ω(t)vk(t′)D−1
ik(t′)?
?
Choose the observable Ω = viand sum over i from equ.
(10), the response function in equal time limit t = t′is:
Rvi
i(t) =
?
dxvi(x)[−∂iP(x)] =
?
dx[∂ · j(x)]lnP(x)
=d
dt
?
dxP(x)lnP(x) = −˙S
= −
?
?vi(t)˜Fk(t)D−1
ik(t)? + ?vi(t)vk(t)D−1
ik(t)?
?
(11)
Then, the Gibbs entropy S = −?d(x)P(x)lnP(x) has
two parts:
˙S = ?vi∂iln[P(x)]?
= ?viD−1
ijvj? + ?viD−1
ij˜Fj? = ep−˙Sm
(12)
ep≥ 0 is the average entropy production rate of the sys-
tem and T˙Sm= ?T ˙ sm? is the average heat dissipation in
the medium. The rate of heat dissipation in the medium
is ˙ q = Fi˙ xjD−1
ij
= T ˙ sm, where the exchanged heat q is
identified with the increase of entropy smin the medium
of temperature T [21].
˙S links with the gradient of the
time dependent probability distribution: ∇P, which is
composed of two terms. One is from the bulk entropy
production of the system which links with the flux v and
the other is from the heat dissipation into the medium
(surface) which links with the driving force˜F [21]. We
see that the driving force for entropy production is the
flux. With the detailed balance, only the time dependent
flux contributes to entropy production. While without
detailed balance, entropy production has both time de-
pendent and steady state flux contributions. We would
like to separate the contribution of time dependent and
independent entropy production of the system and re-
late that to the relaxation of time dependent probability
and steady state flux explicitly. Therefore, if we take the
observable Ω = vi− vSS
i
and sum over i, the response
function in equ. (10) with equal time limit t = t′gives:
?vi∂iln
?
PSS(x)/P(x)
?
? =˙Ffree/T
= ?vSS
i
D−1
ijvj? − ?viD−1
ijvj? = Qhk/T − ep
(13)

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It leads to Tep= Qhk−˙Ffreewith free energy Ffree=
T?ln?
T?vSS
ij(t)?=
Tep + ˙Ffree
= T˙Sm +˙U and total energy U
−T?dxP(x)ln[PSS(x)], which was given in previous lit-
erature [20–22]. The change of the total internal energy
is˙U = T?vi(t)∂iln[PSS(x)]?.
origins of the total entropy production ep. ˙Ffreeis from
spontaneous non-stationary relaxation which links with
the gradient of relative potential −∂iln
the driving force necessary to sustain the non-equilibrium
environment, which links with the steady state flux
vSS(x). For the non-equilibrium steady state,˙Ffree= 0.
Qhk equals the medium dissipated heat for maintain-
ing the violation of detailed balance: Qhk = T˙Sm =
−T?vSS
total entropy production of the system equals the spon-
taneous relaxation of free energy Tep = −˙Ffree. Here
we found that the generalized FDT in the equal time
limit t′= t naturally leads to non-equilibrium thermo-
dynamics with total entropy production from both non-
stationary spontaneous relaxation and stationary house
keeping part. This is our first main result.
In addition, we can relate the non-equilibrium Fokker-
Planck equation with Abelian Gauge Theory and in-
ternal curved space, as in Quantum Electrodynamics
(QED)[19].With the covariant derivative ∇i = ∂i−
D−1
ten as: Dij(x)∇jP(x) = ji(x). The curvature of internal
charge space due to the Abelian gauge field Aiis:
P(x)
PSS(x)
i (t)vj(t)D−1
?? = U −TS, the house keeping heat Qhk=
T?vSS
i (t)vSS
j (t)D−1
ij(t)?=
=
There are two different
?
P(x)
PSS(x)
?
. Qhkis
i D−1
ij˜Fj?. For detail balanced cases, Qhk= 0 and
ij˜Fj= ∂i+Ai, Fokker-Planck equation can be rewrit-
Rij= ∂iAj− ∂jAi= [∇i,∇j].(14)
where [·] indicates a commutator of two operators. Ac-
cording to equ. (4), for the detailed balance case: jSS=
0, Ai= ∂iln(PSS) is a pure gradient and the curvature
is zero: Rij= 0 which corresponds to a flat space. While
for non-equilibrium cases, A can’t be written as a gradi-
ent and Rij?= 0 which corresponds to a curved internal
space. Rijis gauge invariant tensor: for a gauge transfor-
mation Ai→ Ai+ ∂iφ, Rij→ R′
the probabilistic velocity v(x,t) and the flux j(x,t) are
also related to this internal curvature as:
ij= Rij. Furthermore,
∂i(D−1
jkvk) − ∂j(D−1
ikvk) = Rij
(15)
or in the case of constant diffusion coefficient Dij= Dδij:
∂ivj−∂jvi= Rij. We noticed that equ. (15) is also gauge
invariant. It means if we change Ai→ Ai+∂φ, although
P(x,t), v(x,t) and j(x,t) are all changed, equ. (15) is
always satisfied with a same curvature Rij. Moreover,
even v(x,t) and j(x,t) depend on the solution of P(x,t),
they always satisfy equ. (15), either for steady state so-
lutions or time dependent solutions. So Rij represents a
measurement of internal geometry of the non-equilibrium
dynamics. This curvature of internal space relates to the
the heat dissipation in the medium along closed loop.
Along any specific path x(t), T∆smis the heat dissipa-
tion in the medium:
T∆sm(x′(t′),x(t)) = T
?t
t′
˙ smdt(16)
=
?t
t′D−1
ij(x(t))˜Fj(x(t))˙ xidt = −
?t
t′Ai(x(t))dxi(t)
Using the Stokes’s theorem and the current definition in
equ. (3), the entropy increase of the medium ∆smalong
a close loop C can be written as:
T∆sC
m= −
?
C
Ai(x)dxi= −
?
C
D−1
ij(x)vSS
j (x)dxi
= −1
2
?
Σ
dσijRij
(17)
where Σ is the surface of the closed loop C, dσijis the an
area element on this surface, and Rijis the curvature due
to the gauge field A. Both the curvature Rijand the close
loop heat dissipation in the medium T∆sC
invariant under gauge transformation Ai → Ai+ ∂iφ.
Thus, we related non-equilibrium dynamics to an internal
curved space. The presence of the non-zero flux destroys
the detailed balance, leads to non-zero internal curvature
and a global topological non-trivial phase analogous to
quantum mechanical Berry phase [1]. This is our second
main point.
The medium heat dissipation ∆smin equ. (16) plays
an important role in the time irreversibility for non-
equilibrium systems [21, 23, 24]. We will see it also gives
an important contribution in generalized FDT for non-
equilibrium dynamics and such contribution links with
this gauge field and internal curvature .
In the following, we will focus on cases of constant dif-
fusion coefficients Dij for simplicity. If j = 0, it is the
equilibrium system with detailed balance, which has time
reversal invariant:?Ω(t)Fj(x(t′))? = ?Fj(x(t))Ω(t′)?.
Using the Langevin equation (1), ?Fi(x(t))Ω(t′)? =
?[˙ xi(t) − ξi(t)]Ω(t′)? = ?˙ xi(t)Ω(t′)?, since random force
will not correlate with Ω of previous time (t > t′):
?ξi(t)Ω(t′)? = 0. Then, we arrive at:
mare gauge
RΩ
i(t − t′) = −D−1
ik
?d
dt?xk(t)Ω(t′)?
?
(18)
In particular, for the operator Ω(x) = xj, we see
Rxj
i(t − t′) = −D−1
ik
?d
dt?xk(t)xj(t′)?
?
(19)
which is the FDT near equilibrium [3].
However, if the system is in non-equilibrium state,
there is no detailed balance: j ?= 0. We are often more
interested in the direct observable xi and a FDT as
the form of equilibrium case as in equ. (19), in which
we can split out the correlation ?xk(t)xi(t′)?.
out detailed balance, the system is time irreversible:
With-

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?Ω(t)Fj(x(t′))? ?= ?Fj(x(t))Ω(t′)?.
Fluctuation theorem [21, 23, 24], we have
According to the
lnPSS(x′)˜P(x,t|x′,t′)
PSS(x)˜P(x′,t|x,t′)
= ∆sm+ lnPSS(x′)
PSS(x)
(20)
with ˜P(x,t|x′,t′) (˜P(x′,t|x,t′)) the probabilities of a
forward (backward) path. We define ?Ω(t)Fi(x(t′))? −
?Fi(x(t))Ω(t′)? =?dxdx′Ω(x)Fi(x′)A(x,x′,t − t′) with
A(x,x′,t − t′)
= PSS(x′)P(x,t|x′,t′) − PSS(x)P(x′,t|x,t′)
(21)
= PSS(x′)
?
D[x]˜P(x,t|x′,t′)
?
1 −PSS(x)
PSS(x′)e−∆sm?
D[x] is the path integral from x′(t′) to x(t). Then, we
get
RΩ
i(t − t′) = −D−1
ik
?d
?
dt?xk(t)Ω(t′)?
?
−D−1
ik
dxdx′Ω(x)Fk(x′)A(x,x′,t − t′)
−D−1
ik?Ω(t)vSS
k(t′)?(22)
For the operator Ω(x) = xj, the response function reads
Rj
i(t − t′) = −D−1
ik
?d
?
dt?xk(t)xj(t′)?
?
−D−1
ik
dxdx′xjFk(x′)A(x,x′,t − t′)
−D−1
ik?xj(t)vSS
k(t′)?(23)
The first term is similar to the equilibrium case in equ.
(19). The last two terms in equ. (23) are zero for de-
tailed balance case. These two terms are related to the
internal curvature due to the gauge field in space, as
shown in equ. (15) and (17). In equ. (21), the fac-
tor U(x,y) = e−∆sm= e
T
the Wilson loop or Wilson line in Abelian gauge theory,
with
?
[19]. It describes the irreversibility determined by the
heat dissipation in the medium. The function inside the
path integral of equ. (21) is U(x,y)PSS(x)
where ∆qhkis the housekeeping heat along a trajectory.
It was proved ?e−∆qhk/T? = 1 [21]. Along a closed loop,
e−∆qC
U(x,y) transforms as: U(x,y) → eφ(x)U(x,y)e−φ(y). It
also satisfies the differential equation:
1
?
PAi(x)dxiis very similar to
Pindicating the integral for a path from x to y
PSS(x′)= e−∆qhk/T,
hk/T= U(x,x). Under the gauge transformation,
˙ xi∇iU(x,y) = 0(24)
It means that the gradient of phase factor (Wilson lines)
contribution from the heat dissipation or house keeping
part for non-equilibrium systems is perpendicular to the
dynamics just as the case in the circular motion. The
origin of the non-zero curvature is the non-zero flux which
breaks the detailed balance for non-equilibrium systems.
This is the third and last main result of the paper.
∗Corresponding author:jin.wang.1@stonybrook.edu
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