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Fluctuation-dissipation relations for steady state systems

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a r X i v : 0 9 0 8 . 4 5 4 1 v 2 [ c o n d - m a t . s t a t - m e c h ] 1 7 M a r 2 0 1 0
Fluctuation-dissipation relationsfar from equilibrium
Jianhua XingDepartment of biological sciences, Virginia Polytechnic Institute and StateUniversity, Blacksburg, VA 24061
Abstract
The ﬂuctuation-dissipation (F-D) theorem is a fundamental result forsystems near thermodynamic equilibrium, and justiﬁes studies betweenmicroscopic and macroscopic properties. It states that the nonequilibriumrelaxation dynamics is related to the spontaneous ﬂuctuation at equilib-rium. Most processes in Nature are out of equilibrium, for which we havelimited theory. Common wisdom believes the F-D theorem is violated ingeneral for systems far from equilibrium. Recently we show that dynam-ics of a dissipative system described by stochastic diﬀerential equationscan be mapped to that of a thermostated Hamiltonian system, with anonequilibrium steady state of the former corresponding to the equilib-rium state of the latter. Her we derived the corresponding F-D theorem,and tested with several examples. We suggest further studies exploitingthe analogy between a general dissipative system appearing in variousscience branches and a Hamiltonian system. Especially we discussed theimplications of this work on biological network studies.
It is ubiquitous to observe a system at a state invariant with time (with theapproximation that the relevant constraining parameters changes much slowerthan the time scale under interest). It can be a thermodynamic equilibriumstate, or more likely a nonequilibrium steady-state. Some examples are homeo-static states of living organisms, a stable eco- or ﬁnancial system. Quite often itis important to know how a system initially subject to a perturbation relaxes toa steady state (including the equilibrium state) after removal of the perturba-tion. The ﬂuctuation-dissipation theorem states that the relaxation dynamicsfor a process close to equilibrium is related to the spontaneous ﬂuctuation atequilibrium. Originally formulated by Nyquist in 1928[1], and ﬁrst proved byCallen and Welton in 1951[2], the F-D theorem is related to many importantresults in statistical physics. Examples are the Einstein-Smoluchowski relationon the diﬀusion constant and drag coeﬃcient [3, 4], Onsager’s regression hy-pothesis[5,6], and the linear response theory[7]. The F-D relation also has
practical importance. It allows deducing nonequilibrium dynamics from equi-librium measurements, and justiﬁes the relation between macroscopic dynamicsand microscopic level simulations, e.g., calculating the diﬀusion constant. Inrecent years, ﬂuctuation theories of nonquilibrium processes, especially for sys-tems far from equilibrium received great attention[8, 9,10,11,12].
On studying problems from physics, chemistry, cellular biology, ecology, en-gineering, ﬁnance, and many other ﬁelds, the following form of equations arewidely used[14, 15, 16],
dx
i
/dt
=
G
i
(
x
) +
m
j
=1
g
ij
(
x
)
ζ
j
(
t
)
,i
= 1
,
···
,n.
(1)In general
m
and
n
may be diﬀerent,
ζ
j
(
t
) are temporally uncorrelated, statisti-cally independent Gaussianwhite noise with the averagessatisfying
< ζ
j
(
t
)
ζ
′
j
(
t
)
>
=
δ
jj
′
δ
(
t
−
t
),
g
(
x
) is related to the
n
×
n
diﬀusion matrix
gg
T
= 2
D
/β
, where thesuperscript
T
refers to transpose. For a physical system
β
is the inverse of theBoltzmann’s constant multiplying temperature. For a non-physical system, onecan deﬁne an eﬀective temperature relating to
β
. Recently we proved that thereexists a mapping between a stochastic dissipative system described by Eqn.1and a thermostated Hamiltonian system[17]. The mapping allows many re-sults from equilibrium statistical physics directly applicable to nonequilibriumprocesses. Speciﬁcally in this work, we will derive the F-D relation applica-ble to processes far from equilibrium. An F-D relation, if exists, would allowpredicting the relaxation dynamics to a steady-state through measurements of steady-state ﬂuctuations. The latter are in general easier to measure.1
1 Theory
1.1 Existence of mapping between linear stochastic dissi-pative systems and Hamiltonian systems
First we brieﬂy summarize the main result of [17]. The mapping is based aseminal work of Ao, which shows that one can always construct a symmetricmatrix
S
and an anti-symmetric one
T
, and transform Eqn.1into, [18]
(
S
+
T
)
d
x
dt
= (
S
+
T
)(
G
(
x
) +
g
(
x
)
ζ
(
t
))=
−∇
x
φ
(
x
) +
g
′
(
x
)
ζ
(
t
) (2)where
φ
is a scalar function corresponding to the potential function in a Hamil-tonian system satisfying (
∂
×
∂φ
)
ij
≡
(
∂
i
∂
j
−
∂
j
∂
i
)
φ
= 0, and
g
′
g
′
T
= 2
S
/β
.Then
S
and
T
are uniquely determined by
∂
×
[(
MG
(
x
)] = 0
,
(
M
)
−
1
+ (
M
)
−
T
= 2
gg
T
,
(3)where
M
=
S
+
T
, with proper choice of the boundary conditions. In [17], weﬁrst demonstrated that starting with the corresponding Fokker-Planck equa-tions, one can derive the transformation matrix more transparently following astandard procedure used by Graham and by Eyink
et al.
previously [19, 20].Then we showed that one can map the dynamics described by Eqn.2to aHamiltonian system in the zero mass limit,
H
=(
˜p
−
A
(
x
))
2
2
m
+
φ
(
x
)+
N
α
α
=1
N
j
=1
12
p
2
αj
+12
ω
2
αj
(
q
αj
−
a
α
(
x
)
/
(
√
Nω
2
αj
))
2
(4)The term
A
is a vector potential satisfying
T
ij
=
j
∂A
i
∂x
j
−
∂A
j
∂x
i
. The lastterm is a type of bath Hamiltonian discussed by Zwanzig[21]. The Hamilto-nian mathematically corresponds to Dirac’s constrained Hamiltonian [22]. Itdescribes a massless particle, coupled to a set of harmonic oscillators, mov-ing in a hypothetical
n
-dimensional conservative scalar potential and magnetic(the vector potential) ﬁeld. The steady state distribution is thus given by theBoltzmann distribution of the Hamiltonian system,
ρ
ss
(
x
) =
d
p
d
Y
exp(
−
βH
)
d
x
d
p
d
Y
exp(
−
βH
)=exp(
−
βφ
)
d
x
exp(
−
βφ
)(5)where
Y
represents all the bath variables. Eqn.5is also conjectured by Ao[18], and a general proof is given in[23].
2
1.2 General ﬂuctuation-dissipation theorem
Let’s denote the steady-state ensemble average of a generic dynamic quantity
O
as,
<
O
>
=
d
xO
ρ
ss
(
x
) (6)Consider a system initially at a steady state deﬁned by Eqn.1with an ex-tra inﬁnitesimal perturbation
δ
G
(
λ
), where
λ
refers to the parameters be-ing perturbed. The corresponding perturbation Hamiltonian term is given by
∇
(
δH
) =
−
(
M
′
G
′
−
MG
)
≈−
M
δ
G
−
δ
MG
, where
δ
M
is the variation of
M
due to the perturbation. At time 0
δ
G
is removed, and the system relaxes tothe steady-state of
G
. Or the Hamiltonian of the corresponding mapped systemchanges from
H
′
=
H
+
δH
=
H
+
∂φ∂λ
·
δλ
. Then the nonequilibrium relaxationdynamics of
O
follows¯
O
(
t
)
−
<
O
>
=
d
x
d
pO
(
t
)exp(
−
βH
′
)
d
x
d
p
exp(
−
βH
′
)
−
<
O
>
≈−
β
[
<
O
(
t
)
∂φ
(
x
(0))
∂λ
)
>
−
<
O
><∂φ∂λ>
]
·
δλ
(0)(7)In the above derivation, we used the stationary property of equilibrium ensembleaverage, so
<
O
(
t
)
>
=
<
O
(0)
>
=
<
O
>
. This is the generalized F-D relationfor systems obeying or violating detailed balance, which states that nonequilib-rium relaxation dynamics can be predicted from steady-state ﬂuctuations. If one relates the relaxation function to the linear response function,¯
O
i
(
t
)
−
< O
i
>
≈
tt
0
dt
′
χ
ij
(
t
−
t
′
)
δλ
j
(
t
′
) (8)one obtains the diﬀerential form of the F-D relations,
χ
ij
(
t
−
t
′
) =
−
βddt
< O
i
(
t
)
∂φ
(
x
(
t
′
))
∂λ
j
)
>
(9)Eqns7and9are the central results of this work. These results are actually
mathematically trivial with the replacement
φ
=
−
ln
ρ
ss
[12]. The mapping,however, provides direct connection between
ρ
ss
and the Langevin equations(see especially case 4 in the next section), and allows uniﬁed treatment forsystems obeying or violating detailed balance. One can generalize the resultsdiscussed in this work to higher orders of
δ
G
.One type of perturbation of special interest is a system coordinate cou-ples to some constant external force linearly,
δH
=
f
·
x
, which corresponds to
δ
G
=
−
M
−
1
f
with
δ
M
= 0. Notice that
δ
G
is in general a nonlinear func-tion of
x
. This situation has been previously discussed by Graham, and byEyink
et al.
[19,20]. Under this special type of perturbation, all the famil-
iar results obtained on studying relaxations near an equilibrium state follow3
[24]. One can deﬁne a response function
χ
(
t,t
′
) so¯
x
(
t
)
−
<
x
>
=
∞−∞
dt
′
χ
(
t
−
t
′
)
f
(
t
′
)+
O
(
||
f
||
2
). The function
χ
(
t
−
t
′
) is stationary and satisﬁes the Kramers-Kr¨onig relations. If
f
is time varying with a monochromic frequency,
f
=
Re
[
f
ω
exp(
iωt
) +
f
∗
ω
exp(
−
iωt
)], the system absorbs ”energy”, with the absorp-tion spectrum
abs
(
ω
)
∝
ω
2
∞
0
dt
f
T ω
<
(
x
t
−
<
x
>
(
x
0
−
<
x
>
)
T
>
f
∗
ω
cos(
ωt
)[24].
2 Special cases and numerical tests
Here we will show that several versions of the previously derived generalizedF-D relations are special cases of the above results. We will also consider sev-eral pedagogical examples, especially biochemical networks, to demonstrate thevalidity of our results.
Case 1
: The ﬁrst one is an analytically solvable irreversible linear chemicalnetwork (Fig. 1a). The system is initially at a state with
k
0
+
δk
. Then at time 0it changes to
k
0
,
i.e.
, the inﬂow ﬂux varies. The system relaxes to a new steady-state. The perturbation is
δ
G
= (
δk,
0
,
0), and the corresponding Hamiltonianperturbation term is
δH
=
−
(
x
1
,x
2
,x
3
)
M
δ
G
. The dynamic equations are,
ddt
x
1
x
2
x
3
=
−
2 0 01
−
1 11 0
−
1
x
1
x
2
x
3
+
k
0
00
+
zζ
1
(
t
)
zζ
2
(
t
)
zζ
3
(
t
)
(10)For simplicity of the following discussions, we write the above equation in theform
d
x
/dt
=
Kx
+
b
+
zζ
(
t
). The predicted relaxation function through theﬂuctuation-dissipation relation is given by,(¯
x
i
(
t
)
−
< x
i
>
)
FD
∝
C
·
M
·
δ
G
(11)where (
C
ij
(
t
)) =
<
(
x
i
(
t
)
−
< x
i
>
)(
x
j
(0)
−
< x
j
>
)
>
. We also tested anincomplete relaxation function by setting
M
as an identity matrix,(¯
x
i
(
t
)
−
< x
i
>
)
incomplete
∝
C
·
δ
G
(12)Analytical solutions are given in the appendix. Fig. 1 shows that ∆
x
i
(
t
) calcu-lated from the F-D relation (Eqn11) reproduces the exact result given by Eqn21(solid line). However, without the term
M
, the incomplete F-D relation Eqn12predicts an initial increase of
x
2
(and similar for
x
3
). Physically, sponta-neous ﬂuctuation of
x
1
is anti-correlated with those of
x
2
and
x
3
concentrationsthrough reactions 1 and 3. The incomplete F-D relation erroneously attributesthis mechanism to the relaxation dynamics.In the second example, we still use the network shown in Fig. 1a, but assumethat all the reactionsexcept reaction0 followthe Michaelis-Mentenkinetics,
e.g.
,
dx
1
dt
=
k
0
−
v
1
x
1
K
1
+
x
1
−
v
3
x
1
K
3
+
x
1
+
zζ
1
(
t
) (13)4

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