Modified fluctuation-dissipation and Einstein relation at non-equilibrium steady states

0.21 MB·English

by  Debasish Chaudhuri

Modified fluctuation-dissipation and Einstein relation at non-equilibrium steady states
Debasish Chaudhuri1,∗ and Abhishek Chaudhuri2,3,†
1 FOM Institute for Atomic and Molecular Physics,
Science Park 104, 1098XG Amsterdam, The Netherlands ‡
2 Department of Biomedical Science, University of Sheffield,
Western Bank, Sheffield S10 2TN, United Kingdom
3 The Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom
2
(Dated: January 24, 2012)
1
0 Starting from the pioneering work of G. S. Agarwal [Zeitschrift fu¨r Physik 252, 25 (1972)], we
2
present a unifiedderivation of a numberof modified fluctuation-dissipation relations (MFDR) that

n relate response to small perturbations around non-equilibrium steady states to steady-state corre-
a lations. Using this formalism we show the equivalence of velocity forms of MFDR derived using
J continuum Langevin and discrete master equation dynamics. The resulting additive correction to

2 theEinstein relation is exemplified using a flashing ratchet model of molecular motors.
2
PACSnumbers: 05.70.Ln,05.40.-a,05.60.-k

]
h
c I. INTRODUCTION atedatsteadystate. Forasystemevolvingwithastatis-
e
tical dynamics characterized by the Fokker-Planck (FP)
m
Derived within linear response theory, the fluctuation equation∂tp=L0p,Agarwalshowedthataperturbation
-
t dissipation theorem (FDT) predicts how the response intheoperatorL0 →L0+h(t)L1leadstoaresponsethat
a functionofathermodynamicobservableisrelatedtocor- can be expressed in terms of a correlation function eval-
t
s relationofthermalfluctuationsatequilibrium. Letusas- uated at the unperturbed steady state [10, 11],
.
t sume that an equilibrium system described by a Hamil-
a
m tonian H is perturbed at time t = t1 by an external R (t −t )= δhA(t2)i =hA(t )M(t )i (2)
A 2 1 2 1
force h(t). The FDT predicts a response at a later time δh(t )
- 1
d t2 >t1 [1]
n where the Agarwal term M = [L p ]/p with p denot-
1 s s s
o Req(t −t )= δhA(t2)i =β ∂ hA(t )[−∂ H(t )] i ing the steady-state probability distribution. Through-
c A 2 1 δh(t1) ∂t1 2 h 1 h=0 eq outthis paperby h...i we denote a steady-stateaverage.
[
(1)
Overthelastdecadeaformalismofstochasticthermo-
1 where the correlation is calculated at equilibrium cor- dynamics has been developed that allows description of
v responding to temperature T with β = 1/T. The dif-
energy and entropy along fluctutating trajectories [12–
5 ferential operator ∂ in the above relation denotes the
h 14]. Various fluctuation theorems involving the distribu-
2 scalarderivativeevaluatedattime t . Thus−∂ H isthe
1 h tion of entropy [15–21], and work theorems [22–24] were
5
displacementconjugatetohwithrespecttotheHamilto-
4 discovered. Recently, using an integral fluctuation theo-
nian. Throughout this paper we use Boltzman constant
. rem, a number of these relations were derived in a uni-
1 k = 1, unless otherwise stated. Using the Onsager re-
B fied manner [14, 25]. Important experimental tests in-
0
gression hypothesis the FDT can be interpreted as fol-
clude colloidal particles manipulated by laser traps [26–
2
lows–thedecayofafluctuationisindependentofhowit
1 28],biomoleculespulledbyAFMorlasertweezer[29,30]
has been created, under the influence of a small applied
: andautonomousmotionofmotorproteins[31]. Stochas-
v force orspontaneously by thermalnoise. The FDT is vi-
tic thermodynamics has also been used to derive several
i
X olated away from equilibrium regime and this violation versions of MFDR around NESS [6, 7, 9, 32–36]. Some
has been studied in context of glassy systems, granular
r ofthese predictionswereexperimentallyverified[27,37].
a matter,shearedfluid,stochasticprocesses,andbiological
Inthispaper,wepresentaunifiedderivationofanum-
systems [1–9].
ber of MFDRs based entirely on the Agarwal formal-
In a pioneering study back in 1972 [10], G. S. Agar-
ism [10]. Thus the MFDRs we obtain are intrinsically
wal obtained a modified fluctuation-dissipation relation
equivalent to each other. We show that the Agarwal
(MFDR) that related response functions around non-
term M can be expressed as a velocity excess from a
equilibrium steady states (NESS) to correlations evalu-
local mean velocity using both the continuum Langevin
and discrete master equation dynamics. This interpreta-
tionleadsus to amodified Einsteinrelationthathas the
same additive correction term for the two cases. Finally
∗Electronicaddress: [email protected]
we apply this framework to a flashing ratchet model of
†Electronicaddress: [email protected]
‡Currentaddress: IndianInstituteofTechnologyHyderabad,Yed- molecularmotors[38–40]tocalculatetheMFDRandthe
dumailaram502205, AndhraPradesh,India additive correction in Einstein relation, which shows a
non-monotonic variation with the asymmetry parameter probability. The time evolution ∂ p = L p can be
t 0
of the ratchet. solvedtoobtainthetransitionprobabilityatsteadystate
The structure of this paper is as follows. In Sec. II we w(ς,t|ς′,0) = exp(L t)δ(ς −ς′). Thus the two-time cor-
0
reviewthederivationoftheAgarwalformofMFDR,that relation at steady state takes the form hA(t)B(0)i =
we use throughout this paper to calculate other versions dςA(ς)exp(L t)B(ς)p (ς). Therefore we can write
0 s
ofMFDRexpressedinphysicallyobservableform. Using ERq. 6 as
this result, in Sec. III we present a simple and straight-
R (t −t )=hA(t )M(t )i. (7)
forward derivation of the MFDR in terms of stochastic A 2 1 2 1
entropy production, keeping in mind that this relation This is the Agarwal form of MFDR [10]. The derivation
was used earlier to derive velocity-MFDR for a master presented here used a continuum notation of the phase
equationdynamics [9]. Then, directly using the Agarwal space variable ς. However,the result is general, and can
form, we derive the velocity-MFDR for a system evolv- be derived similarly for a system that evolves through
ingwithcontinuumLangevindynamicsinSec.IV,anda transitions between discrete states (see Eq. 21).
discrete master equation in Sec. V. The velocity-MFDR The Agarwal term in its operator form M(ς) ≡
is used in Sec. VI to derive a modified Einstein relation [L p ]/p requires detailed knowledge of the probability
1 s s
at NESS. In Sec. VII, we study the velocity-MFDR, and distribution at steady state. In the rest of this paper
theviolationoftheEinsteinrelationinaflashingratchet wefocusonexpressingthis terminphysicallyobservable
modelof molecularmotors. Finally inSec. VIII we sum- form.
marise our main results and conclude.
III. MFDR IN TERMS OF STOCHASTIC
II. THE AGARWAL FORM OF MFDR ENTROPY
The probabilitydistribution p(ς,t)of finding a system The definition of non-equilibrium Gibb’s entropy S =
at state ς at time t evolves with time as − dςp(ς,t) lnp(ς,t) ≡ hs(t)i has recently been used
toRget a definition of the stochastic entropy s(t) =
∂ p(ς,t)=L(ς,h)p(ς,t) (3) −lnp(ς,t) [25]. For a master equation based discrete
t
dynamics between states denoted by n(t), the stochas-
whereLisageneraltimeevolutionoperatorthatdepends tic entropy can be written as s(t) = −lnp . Using
n(t)
onexternalforceh(t). Forweakh,Taylorexpandingthe this definition we obtain a simple interpretation of the
operator we get Agarwalterm in terms of stochastic entropy
1 ∂ L(h)p
L(ς,h)=L0(ς)+h(t)L1(ς) (4) M = p L1ps = h p (cid:12)
s (cid:12)h=0
where L1 =[∂hL]h=0. The solution to Eq. 3 is ∂h∂tp (cid:12)(cid:12)
= =−∂ [∂ s] . (8)
p (cid:12) t h h=0
t (cid:12)h=0
p(ς,t)=ps+Z dτeL0(t−τ)h(τ)L1ps(ς) (5) In deriving the above rel(cid:12)(cid:12)ation we assumed that L(h) is
−∞
linear in h. We also used the fact that the steady state
whereps denotesthesteady-statedistributionthatobeys distribution ps = p|h=0. Thus M is expressed as time-
L p =0. Then the response of any observable hA(t)i= evolution of a variable conjugate to the external force h
0 s
dςA(ς)p(ς,t) to a force h(t) is with respect to the stochastic system-entropy s. In this
R sense,sinNESSplaystherolesimilartotheHamiltonian
δhA(t )i δp(ς,t ) in equilibrium FDT. We can now write the MFDR at
2 2
R (t −t ) = = dςA(ς)
A 2 1 δh(t ) Z δh(t ) NESS as
1 1
∂
= Z dςA(ς)eL0(t2−t1)L1ps(ς) RA(t2−t1)= ∂t1hA(t2)[−∂hs(t1)]h=0i. (9)
= dςA(ς)eL0(t2−t1)M(ς)p (ς) Ref.[7,41]foundthisrelationbyconsideringaperturba-
Z s tion that takes the system to a final steady state. Note
(6) that our simple and straightforward derivation does not
require such an assumption, and thus the result is more
whereinthe laststepweusedthe AgarwaltermM(ς)≡ general.
[L p ]/p . By definition, the two-time correlation func-
1 s s
tion is hA(t)B(0)i = dς dς′A(ς)B(ς′)p (ς,t;ς′,0),
2
where p (ς,t;ς′,0) is theR joinRt probability distribution A. Equilibrium FDT
2
of finding the system at state ς′ at time 0 and at
state ς at time t. One can express p (ς,t;ς′,0) = The FDT at equilibrium can easily be derived from
2
w(ς,t|ς′,0)p(ς′,0) where w(ς,t|ς′,0) is the transition Eq. 9. If, even in the presence of external perturbation
2
the system remains at equilibrium, one can write down This is the velocity form of MFDR, which for velocity-
the probability distributions as p = exp[−β(H − F)] response gives
where F is the free energy. This distribution leads to
the relation [∂hp]h=0 = β[(∂hF − ∂hH)p]h=0. Note Rv(t2−t1)=βhv(t2)[v(t1)−ν(t1)]i. (15)
that the equilibrium displacement evalutaed at h = 0
is [∂ F] = 0. Thus we get the identity [∂ s] = Note that the steady state average of ν is the same as
h h=0 h h=0
−[(∂ p)/p] = β[∂ H] , which leads to the equilib- the mean velocity:
h h=0 h h=0
rium FDT Eq. 1.
L/2
hν i= dxp (x)ν (x)=µhFi−D[p ]L/2 =hv i.
s Z s s s −L/2 s
−L/2
IV. VELOCITY MFDR USING LANGEVIN
(16)
EQUATION
L/2
The boundary term [p ] = 0 either by a periodic
LetusconsideraLangevinsystemwherethedynamics s −L/2
boundary condition [34], or by taking the boundaries to
of a particle evolves by
infinity where the probabilities vanish. If the system is
at equilibrium ν = 0, and we get back the well-known
v =µf +η (10)
equilibrium response,
where v = x˙ is the particle velocity, µ is the mobil-
Req(t −t )=βhv(t )v(t )i . (17)
ity, and f denotes total force imparted on the particle. v 2 1 2 1 eq
The total force f(x,t) consists of a force due to inter-
Therefore the non-equlibrium MFDR Eq. 15 can be
action F(x) and an external time dependent force h(t):
viewed as the equilibrium FDT with an additive correc-
f(x,t) = F(x)+h(t). The last term η denotes a ther-
tion −βhv(t )ν(t )i.
mal noise that obeys hηi = 0 and hη(t)η(0)i = 2Dδ(t) 2 1
It is interesting to note that using Eq. 13, we can
with D = µT, the equilibrium Einstein relation. The
arrive at a non-equilibrium MFDR first obtained in Ref.
corresponding FP equation is
[32] for continuous Langevindynamics and subsequently
showntobetruefordiscretespinvariables(aswellas,for
∂ p(x,t) = −∂ j(x,t) (11)
t x
conservedandnon-conservedorderparameterdynamics)
with, j(x,t) = (µf(x,t)−D∂ )p(x,t).
x in Ref. [33]. Defining the position correlation function
C (t ,t )=hx(t )x(t )i and the corresponding response
ThevelocityformofMFDRforaLangevinsystemwas x 2 1 2 1
function 2TR (t ,t ) = hx(t )η(t )i (using Eq. 13), we
originallyderivedinRef.[34]. Herewebrieflyoutlinethe x 2 1 2 1
get the modified MFDR
derivation starting from the Agarwal form. Eq. 11 can
be expressed as,
(∂ −∂ )C (t ,t )=2TR (t ,t )+A(t ,t ) (18)
t1 t2 x 2 1 x 2 1 2 1
∂ p(x,t)=(L +h(t)L )p(x,t),
t 0 1 where A(t ,t ) = hµf(t )x(t ) − µf(t )x(t )i is the
2 1 1 2 2 1
so-called asymmetry which vanishes in the presence of
where, L = −∂ (µF) + D∂2, and L = −µ∂ . Thus
0 x x 1 x time reversal symmetry. Note that causality demands
the Agarwal term M = −µ(∂ p )/p , and TM =
x s s that the response of the system at time t to a per-
−D(∂ p )/p . The definition of the steady state current 2
x s s turbation at time t , R (t ,t ), is nonzero only when
j leadsto the relationD∂ p =µF(x)p (x)−j . Defin- 1 x 2 1
s x s s s t ≥ t . Incorporating time translation invariace and
ingalocalmeanvelocityatsteadystateν (x)=j /p (x) 2 1
s s s time reversal symmetry restores the equilibrium FDT,
we can then rewrite TM = −D(∂ p )/p = ν (x) −
x s s s TR (t ,t )=∂ C (t ,t ). Also note that, the choice of
µF(x). In this relation, using the Langevin equation at x 2 1 t1 x 2 1
the observable V in Ref. [36] as an 1D coordinate x, re-
initial steady state (h = 0), we get TM = ν −v +η.
s duces the second term on the r.h.s of Eq. 13 in Ref. [36]
Thus, the response function
to h(L−L∗)V(s)Q(t)i = h2(j/ρ)∇xQ(t)i = hνQi. Now
setting Q≡v (velocity), leads Eq. 13 in Ref. [36] to Eq.
TR (t −t )=hA(t )[ν(t )−v(t )+η(t )]i. (12)
A 2 1 2 1 1 1 14 in our manuscript, the velocity form of MFDR.
Note that in the Langevin equation µh(t) and η(t) have
the same status, and A(x,t) can be regarded as a func-
V. VELOCITY MFDR USING MASTER
tional of noise history. Then it can be shown that [34],
EQUATION
δhA(t )i 1
2
TR (t −t )=D = hA(t )η(t )i. (13)
A 2 1 δη(t ) 2 2 1 We now focus on a master equation system where the
1
time-evolution occurs via transitions between discrete
Thus we can write Eq. 12 as states. Following Ref. [34], we first derive the discrete
form of the Agarwal term M. Our main contribution in
R (t −t )=βhA(t )[v(t )−ν(t )]i (14) this section is to express M as an excess velocity, and
A 2 1 2 1 1
3
thus arrive at a velocity form of MFDR, similar to the Then fromEq.22 we find the velocityform ofAgwarwal
Langevin system. term,
We begin by considering a set of discrete states {n}
and write down the corresponding master equation for (p )
n s
M = w α − w α
the probability pm(t) of finding the system in a state m m X(pm)s nm nm X mn mn
at time t: n n
(p )
n s
= w (α +βd )− w α
∂tpm(t) = [wnmpn(t)−wmnpm(t)] X(pm)s nm mn nm X mn mn
X n n
n
(p ) 1
≡ Lmnpn(t) (19) = β n swnmdnm− (Jmn)sαmn
Xn Xn (pm)s Xn (pm)s
= β(v −ν ), (26)
where, w represents transition rate from state m to n m m
mn
and is generally dependent on the external force h. The
time evolution operator where
Lmn =wnm−δmn wmk. (20) v = (pn)sw d ,
X m (p ) nm nm
k X m s
n
If the external force h(t) acting on the system is weak, (Jmn)s
βν = α . (27)
Taylor expanding about h=0, we get m (p ) mn
X m s
n
L (h)=(L ) +h(t)(L ) .
mn 0 mn 1 mn These relations lead to the velocity form of MFDR
In this relation
R (t −t ) = A [eL0(t2−t1)p(t )] [β(v −ν )]
A 2 1 m 1 mn m m
(L1)mn =wnmαnm−δmn wmkαmk Xm,n
X
k
= βhA(t )[v(t )−ν(t )]i. (28)
2 1 1
where α =[∂ lnw ] gives the relative change of
mn h mn h=0
rates. Note that the system is prepared in a NESS at Note that Eq. 28 agrees with the results obtained in
h=0 characterizedby the stationary distribution (p ) . Ref.s [7, 9]. In particular, Ref. [9] used the MFDR ex-
n s
Then Eq. 6 can be expressed in the discrete notation as pressed in terms of stochastic entropy of the system s
(Eq. 9) to obtain Eq. 28. They used the total stochastic
R (t −t ) = A [eL0(t2−t1)p(t )] M entropys =s+s where s is the stochasticentropy
A 2 1 m 1 mn n tot m m
Xm,n of the medium and showed
= hA(t )M(t )i (21)
2 1
∂ s˙ (t) = δ(t−τ )d ≡v(t)= δ v
h m m m−1,m n(t),m m
where the Agarwal term is X X
m m
1 ∂ s˙ (t) = ν(t)= δ ν (29)
h tot n(t),m m
M = (L ) (p )
m (p ) 1 mn n s Xm
m s X
n
(p )
= n swnmαnm− wmnαmn. (22) where vm and νm are given by Eq. 27.
(p )
Xn m s Xn NotetheequivalenceofEq.28withEq.14. Indeedthe
analogy of ν described here with the local mean veloc-
Now we use the above relation to derive the velocity
ity [j(x,t)/p(x,t)] in the Langevin system becomes even
form of MFDR. We assume a displacement d associ-
mn more clear when we compare the steady state average
atedwitheachtransitionfromstatemton. Thishasthe
hν i= (p ) ν with hv i= (p ) v and find
property d = −d and gives a definition of velocity s m m s m s m m s m
mn nm P P
v(t)= δ(t−τ )d [42]. A generalized detailed
m m m−1,m
balancePin presence of the external force h hνsi=T Jmnαmn = (pn)swnmdnm =hvsi. (30)
X X
mn mn
w (h) w (0)
mn mn
= exp[βhd ] (23)
w (h) w (0) mn This relation is the same as Eq. 16 obtained for the
nm nm
Langevin system.
leads to the following useful relation Foravelocity-responseEq.28readilyleadsustoEq.15
already obtained in the context of Langevin dynamics.
α −α =βd . (24)
mn nm mn This completes one of the main achievements of this pa-
We also utilize the probability current per – the Agarwal formalism leads to the same form of
velocity-MFDR for discrete master equation and contin-
J =p w −p w =−J . (25) uum Langevin dynamics.
mn m mn n nm nm
4
FIG. 1: Flashing ratchet model: The potential height
switches between W = 0 (off-state) and W0 (on-state). The
asymmetryofthepotentialintheon-stateisdescribedbythe
inequalitya6=b. ω1,2 denotetransition ratesbetweenonand
off states.
VI. EINSTEIN RELATION
FIG. 2: (color online) Velocity correlations and response:
UsingthevelocityMFDR(Eq.15)wefindthemobility velocity response kBTRv(τ), correlation function of the ve-
in NESS locity with the local mean velocity hv(τ)ν(0)i, and velocity
auto correlation function hv(τ)v(0)i as a function of time.
∞ ∞ The parameter-values used to obtain these curves are λ =
µs =Z dτRv(τ)=βZ dτhv(τ)[v(0)−ν(0)]i. (31) 8nm, a/λ = 0.1, D = 0.009µm2s−1, kBT = 4.2pN nm,
0 0 W0 = 18.85kBT and simulation time-step δt = 1.8×10−6s.
On the other hand, the diffusion constant in an NESS The transition rates are chosen to be equal with w1 =
having mean velocity hvsi is 0w.200=843µ5m362ss−−11,. kWBTithµsth=ese0.p00a5ra7mµemte2rs−v1a,luaensdwtehefinvdiolDatsio=n
∞ integral I = 0.0027µm2s−1. The mean velocity in steady
Ds =Z dτh[v(τ)−hvsi][v(0)−hvsi]i. (32) state is hvsi=2.04µm/s.
0
Thusthe mobility µ anddiffusionconstantD atNESS
s s
donotsatisfytheequilibriumEinsteinrelation,i.e.,D −
s of the motor with the polymeric track [43]. Thus a sim-
Tµ =I 6=0. The difference givesus the modification in
s ple two-state approximation of the dynamics of motor-
the Einstein relation in terms of the violation integral
proteins was proposed [43, 44] where the motor encoun-
tersalocallyasymmetricbutgloballyperiodicpotential,
∞
I ≡D −T µ = dτ hv(τ)ν(0)i−hv i2 . (33) whoseheightswitchesbetweenalargeandasmallvalue.
s s Z s
0 (cid:2) (cid:3) We consider a flashing-ratchet model where the sys-
tem swithces between two states, (1) on-state: stochas-
Since the form of velocity-MFDR for Langevin equation
tic motion in an asymmetric piece-wise linear potential,
(Eq. 14) and master equation (Eq. 28) are the same,
(2) off-state: simple one dimensional diffusion (Fig. 1).
we get the same modified Einstein relation for both the
The probability distributions in the two states p (x,t)
cases. 1,2
evolve by [44]
∂ p +∂ j = ω p −ω p
t 1 x 1 2 2 1 1
VII. FLASHING RATCHET MODEL OF
∂ p +∂ j = −ω p +ω p
t 2 x 2 2 2 1 1
MOLECULAR MOTORS
where j = −D∂ p and j = −D[p ∂ (W/T)+∂ p ],
1 x 1 2 2 x x 2
In this section we apply the concepts developed so and ω denote the transition rates. In the on-state,
1,2
far in this paper on a specific realization of the flash- the potential W(x) is periodic W(x) = W(x+λ) with
ing ratchet model of molecular motors [38–40, 43, 44]. periodλ=(a+b). Within oneperiod, W(x)=(W /a)x
0
In particular, we calculate the velocity-MFDR for this if 0≤x<a, and W(x)=(W /b)(λ−x) if a≤x<λ.
0
modelandderivetheviolationintegralofthecorrespond- We perform molecular dynamics simulations of a par-
ing non-equilibrium Einstein relation. ticle moving under the influence of the above-mentioned
A molecular motor, e.g., kinesin, moves along a poly- ratchetpotentialinthepresenceofaLangevinheatbath.
meric track, e.g., microtubule in a strongly fluctuating Weusestochasticswitchingbetweentheonandoffstates
thermalenvironmentutilizingintrinsiclocalassymmetry with a constant switching rate ω = ω . From this sim-
1 2
of the track and chemical energy provided by hydroly- ulation, in Fig. 2, we plot the velocity-response func-
sis of ATP to ADP and a phosphate. The binding and tion k TR (τ) and the related steady-state correlations
B v
hydrolyzing of ATP changes the strength of interaction hv(τ)v(0)i, hv(τ)ν(0)i (Eq. 15). The parameter values
5
we use are enlisted in Fig. 2 and are typical of micro- passing through x. At steady state, this definition is
tubule associated molecular motors [45]. At long time, the same as ν (x) = j /p (x) where the mean current
s s s
bothhv(τ)v(0)i andhv(τ)ν(0)i decorrelatesto hv i2. We is constant everywhere: j = ρhv i with hv i the mean
s s s s
utilize the correlation functions to determine the mobil- velocityatsteadystateandρ=1/λthemeandensity. In
ity µ , diffusion constant D , and the violation integral calculating hv(τ)ν(0)i, the local mean velocity at time t
s s
I =D −k Tµ (see Fig. 2). isobtainedbyidentifyingthevalueofν(x)corresponding
s B s
We calculate the dependence of the steady-state mo- to the position x visited by the particle at that instant.
bility k T µ , diffusion constant D and the violation
B s s
integralI onthe asymmetryparameterα=a/λ (Fig. 3)
where α=1/2 denotes the symmetric ratchet. This cal-
VIII. SUMMARY
culation leads us to the curious result that all the three
quantities have minimum at α = 1/2. The steady-state
Wehavepresentedaunifiedderivationofmodifiedfluc-
diffusion constant D in the flashing-ratchet is always
s
tuationdissipationrelations(MFDR)atnon-equilibrium
suppressed (D < D), and moves closer to the free dif-
s
steadystates(NESS)usingtheAgarwalformalism. Thus
fusion D for the most asymmetric ratchet. Note that
all the various versions of MFDR that we derived in
the violation integral quantifies the difference between
this paper are intrinsically equivalent to each other. We
NESSandequilibrium,withequilibriumrequiringI =0.
showedthattheresponsefunctionaroundanyNESScan
The symmetric ratchet does not generate unidirectional
beexpressedasacorrelationbetweentheobservableand
motion, but the switching between the on and off states
a variable conjugate to the external force with respect
keeps the system out of equilibrium. Thus, though the
to the system’s stochastic entropy production. For both
violation integral reaches its minimum at α = 1/2 it re-
a continuum Langevin and a discrete master equation
mains I 6= 0. Setting switching rates ω ,ω = 0 would
1 2
system, we have shown that the non-equilibrium form of
restore equilibrium with I = 0. The dependence of I on
FDT involving velocity response can be expressed as an
various models and parameter values at different NESS
equilibrium one and an additive correction. The correc-
is yet to be fully understood.
tion in both these cases is a correlation function of the
velocity with a local mean velocity. The resulting mod-
ification of the Einstein’s relation gives the violation in
terms of a time integral over this additive correction.
Using molecular dynamics simulations in presence
of Langevin heat-bath, we studied a flashing ratchet
model within this framework and obtained the re-
sponse function and velocity correlations in the steady
state. We showed that the violation integral varies
non-monotonicallywiththeasymmetryparameterofthe
ratchet and reaches a non-zero minimum for the case
of a symmetric ratchet. We plan to extend our study
to other models of molecular motors [40], stochastic
particle-pumps [46, 47], polymer translocation dynam-
ics [48], and dynamics of self-propelled particles [49].
Acknowledgments
FIG. 3: (color online) Flashing ratchet: diffusion constant
Ds, mobility kBTµs and violation integral I as a function
of asymmetry parameter α = a/λ. All the other parameter DC thanks Bela Mulder for useful comments and a
values are same as in Fig 2. critical reading of the manuscript. The work of DC is
partof the researchprogramof the “Stichting voor Fun-
damenteel Onderzoek der Materie (FOM)”, which is fi-
While calculation of all the other quantities from our nanciallysupportedbythe“Nederlandseorganisatievoor
simulations are straight-forward, ν(t) demands a special Wetenschappelijk Onderzoek (NWO)”. AC acknowl-
mention. The local mean velocity ν(x) is the stochastic edges support by grant EP/G062137/1 from the Engi-
particlevelocityvaveragedoverthesubsetoftrajectories neering and Physical Sciences Research Council, UK.
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