Fluctuation-dissipation relations under Lévy noises This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2012 EPL 98 50006 (http://iopscience.iop.org/0295-5075/98/5/50006) Download details: IP Address: 139.184.30.133 The article was downloaded on 23/08/2012 at 14:30 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Fluctuation-dissipation relations under Lévy noises
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2012 EPL 98 50006
(http://iopscience.iop.org/0295-5075/98/5/50006)
Download details:
IP Address: 139.184.30.133
The article was downloaded on 23/08/2012 at 14:30
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
Fluctuation-dissipation relations under Levy noises
Bart�lomiej Dybiec1,2(a)
, Juan M. R. Parrondo3(b) and Ewa Gudowska-Nowak
2(c)
1 Center for Models of Life, Niels Bohr Institute, University of Copenhagen - Blegdamsvej 17,2100 Copenhagen Ø, Denmark, EU2Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research,Jagiellonian University - ul. Reymonta 4, 30-059 Krakow, Poland, EU3Departamento de Fısica Atomica, Molecular y Nuclear and GISC, Universidad Complutense de Madrid28040-Madrid, Spain, EU
received 9 January 2012; accepted in final form 15 May 2012published online 13 June 2012
PACS 05.10.Gg – Stochastic analysis methods (Fokker-Planck, Langevin, etc.)PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamicsPACS 05.40.Fb – Random walks and Levy flights
Abstract – For systems close to equilibrium, the relaxation properties of measurable physicalquantities are described by the linear response theory and the fluctuation-dissipation theorem(FDT). Accordingly, the response or the generalized susceptibility, which is a function ofthe unperturbed equilibrium system, can be related to the correlation between spontaneousfluctuations of a given conjugate variable. There have been several attempts to extend the FDTfar from equilibrium, introducing new terms or using effective temperatures. Here, we discussapplicability of the generalized FDT to out-of-equilibrium systems perturbed by time-dependentdeterministic forces and acting under the influence of white Levy noise. For the linear and Gaussiancase, the equilibrium correlation function provides a full description of the dynamic properties ofthe system. This is, however, no longer true for non-Gaussian Levy noises, for which the secondand sometimes also the first moments are divergent, indicating absence of underlying physicalscales. This self-similar behavior of Levy noises results in violation of the classical dissipationtheorem for the stability index α< 2. We show that by properly identifying appropriate variablesconjugated to external perturbations and analyzing time-dependent distributions, the generalizedFDT can be restored also for systems subject to Levy noises. As a working example, we test theuse of the generalized FDT for a linear system subject to Cauchy white noise.
Introduction. – The fluctuation-dissipation theorem(FDT) connects correlation functions to linear responsefunctions and constitutes a useful tool in investigations ofphysical properties of systems at thermodynamic equilib-rium [1]. By virtue of the FDT, measurable macroscopicphysical quantities like specific heats, susceptibilities orcompressibilities can be related to correlation functionsof spontaneous fluctuations. For systems weakly displacedfrom equilibrium, the FDT allows one to express the linearresponse of physical observables to time-dependent exter-nal fields in terms of time-dependent correlation functions.Accordingly, departures from the FDT can be expectedfor far-from-equilibrium situations and have been demon-strated in various aging, glassy and biological media [2–5].
On the other hand, the wealth of theoretical, experi-mental and numerical research indicate that the FDT isa special case of more general fluctuation relations thatremain valid also in a specific class of non-equilibriumsystems [5–13]. Following a former generalization ofFDT [14] based on the identity derived by Hatano andSasa [15], we discuss here an extension of the fluctua-tion theorem to stochastic models obeying Markoviandynamics and driven by white α-stable noises. We applythe generalized fluctuation-response theorem to thiscase and analyze the regime in which linear responsetheory becomes invalidated. We illustrate our resultswith the simple example of an oscillator coupled to anon-equilibrium bath whose action is represented by aCauchy white noise.
Let us first review the generalized FDT introducedin [14] and extended to arbitrary observables in [7].The theorem applies to any Markov process x(t) whose
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Bart�lomiej Dybiec et al.
dynamics depends on a set of parameters �λ and for whicha well-defined (non-equilibrium) stationary state exists.We study the linear response of the system to perturba-tions �λ(t) = �λ0+ δ�λ(t) around a reference stationary state
corresponding to constant parameters �λ0 and resultingprobability density function (PDF) ρss(x;�λ0). Given anarbitrary observable A(x), the response (evaluated to firstorder in the perturbation) can be written as
〈A(t)〉− 〈A〉0 �∫ t0
χA,γ(t− t′)δλγ(t′)dt′, (1)
where A(t)≡A(x(t)) and the brackets 〈. . .〉0 indicate an
average over the reference state ρss(x;�λ0); summation overthe repeated index γ is assumed and χA,γ is the time-dependent susceptibility of variable A with respect tovariations of λγ (i.e., perturbations of the γ component
of �λ). The FDT relates this susceptibility to correlationsmeasured in the reference unperturbed state [7,14]
χA,γ(t− t′) =d
dt〈A(t)Xγ(t
′)〉0, (2)
where Xγ(x) is the variable conjugate to the perturbationλγ and is defined as
Xγ(x) = −∂ ln ρss(x;�λ)
∂λγ
∣∣∣∣∣�λ=�λ0
=∂φ
∂λγ. (3)
In this definition φ≡−ln ρss stands for a non-equilibriumpotential [7,14]. If the reference state is the Gibbs equi-librium state corresponding to a temperature kT = β−1
and a Hamiltonian H(x;�λ), the stationary PDF ρss(x;�λ)
assumes the form ρss(x;�λ) = exp[−βH(x;�λ)]/Z(β;�λ) andthe conjugate variable reads
Xγ(x) =1
kT
∂[H(x;�λ)−F (β;�λ)
]∂λγ
∣∣∣∣∣∣�λ=�λ0
, (4)
where F =−kT lnZ stands for the free energy. Accord-ingly,Xγ can be interpreted as the fluctuation of the quan-
tity ∂H(x;�λ0)∂λγ
≡ ∂H(x;�λ)∂λγ
∣∣∣�λ=�λ0
:
Xγ(x) =1
kT
[∂H(x;�λ0)
∂λγ−⟨∂H(x;�λ0)
∂λγ
⟩0
]. (5)
For instance, if λγ is a force coupled to a coordi-nate xγ , i.e., if the control parameter appears in theHamiltonian as −λγxγ , then the conjugate variableXγ =−(xγ −〈xγ〉)/(kT ) represents fluctuations of xγ .
In general, if the reference state ρss(x;�λ0) is not an equi-librium state, the conjugate variables defined by eq. (3) donot have any straightforward physical interpretation [7,14].In this letter, we examine the generalized FDT, eqs. (1)and (2), for a system obeying non-equilibrium Markoviandynamics and driven by Levy white noise. The system
of this type may be conceived as a generalization ofBrownian motion: the particle undergoing Levy superdif-fusion is performing motion with random jumps and steplengths following a power-law distribution. As a result, thewidth of the distribution of particles grows superlinearlywith time [16,17] signaling anomalous dynamics. Notably,unlike in case of a standard Brownian motion, in Levysuperdiffusion large fluctuations of the position occurwith probability higher than for linear systems subjectedto Gaussian uncorrelated noise. Consequently, it is rathercounterintuitive to expect linear response of the system,even for weak perturbations. The divergence of first andsecond moments of some Levy PDFs indicates absence ofunderlying physical scales and is usually interpreted as thescale invariance, characteristic of self-similar (or fractal)behavior. For the standard Fokker-Planck equation (FPE)equivalent to the Langevin equation driven by Gaussianwhite noise, as well as for the subdiffusive fractional FPEone finds the generalized Einstein relation, connecting thefirst moment in the presence of the perturbing force to thesecond moment in the absence of the force [18,19]. This isno longer true for the Levy flight [19], when only in theBrownian limit α= 2, this relation is satisfied (providedthat the proper amplitude of the noise interpreted asthermal fluctuations is considered). In general though, dueto a diverging mean square displacement, the generalizedEinstein relation does not hold leading to a violation ofthe classical fluctuation-dissipation theorem.
Signatures of Levy noise and anomalous transport havebeen found ubiquitous in nature [16,20] and serve assuitable models describing atmospheric turbulence [21],transport in turbulent plasmas [22], activation kineticsby non-thermal baths [23], transport in fracturedmaterials [24], epidemic spreading [25], dispersal ofbanknotes [26] or light scattering in heterogeneous dielec-tric media [27]. In what follows we address foundations oflinear response and FDT in systems perturbed by Levynoises.
Linear system driven by Levy white noises. – Weproceed to discuss response properties of an overdampedLevy-Brownian particle moving in a parabolic potentialthat is subject to a deterministic time-dependent force f(t)and a white Levy noise ζ(t) resulting from the fluctuatingenvironment. The corresponding Langevin equation reads{
x(t) =−ax+ f(t) + ζ(t),
x(0) = x0.(6)
The white Levy noise ζ(t) is defined as the time derivativeof a stationary Levy process [16,28], i.e., the integral overtime
Lα,β(t)≡∫ t0
ζ(s)ds= z(t) (7)
represents a stochastic process with independent incre-ments whose probability density pα,β(z, t) is a stableLevy distribution. Consequently, the Fourier transform of
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Fluctuation-dissipation relations under Levy noises
the probability density (characteristic function) ϕ(k, t) =∫∞−∞ e
ikz(t)pα,β(z, t;σ0, µ)dz reads
ϕ(k, t) = exp[ikµ0t−σα0 |k|αt
(1− iβ sign(k) tan
πα
2
)](8)
for α �= 1 and
ϕ(k, t) = exp
[ikµ0t−σ0|k|t
(1 + iβ
2
πsign(k) ln |k|
)](9)
for α= 1 [28]. Here α∈ (0, 2] is the Levy (stability) index,β ∈ [−1, 1] is the skewness parameter (for β = 0 the distri-butions are symmetric), σ0 > 0 represents the noise inten-sity, and µ0 ∈R is a location (shift) parameter. For α=2 and β = 0 a standard Gaussian distribution is recov-ered with µ0t indicating the mean of the random vari-able z(t) and σ20t staying for its variance. For α< 2,stable probability densities exhibit heavy tails and diver-gent moments: the asymptotic (large z) behavior of thecorresponding PDF is then characterized by a power lawpα,β(z, t;σ0, µ0)∝ |z|−(1+α). Under those circumstances,eq. (6) is associated with the space-fractional Fokker-Planck-Smoluchowski equation (FFPE) [18,29]:
∂p(x, t)
∂t=− ∂∂x
[µ0− ax+ f(t)] p(x, t)
+σα0∂α
∂|x|α p(x, t) +σα0 β tanπα
2
∂
∂x
∂α−1
∂|x|α−1 p(x, t). (10)
Here, the fractional (Riesz-Weyl) derivative is defined by
its Fourier transform F[∂α
∂|x|α f(x)]=−|k|αF [f(x)] [18,29].
Accordingly, eq. (10) has the following Fourier represen-tation:
∂p(k, t)
∂t= −ak ∂
∂kp(k, t) + ik [µ0+ f(t)] p(k, t)
−σα0 |k|α[1− iβ sign(k) tan
πα
2
]p(k, t), (11)
where p(k, t) =F [p(x, t)]. In what follows, we adhere tothe analysis of strictly α-stable random variables [28], i.e.,those for which µ0 = 0, and additionally β = 0 if α= 1.
Since our original Langevin equation (6) is linear, itssolution depends linearly on the stable process Lα,β(t).Accordingly, the probability density of the solution,p(x, t|x0, 0), has the form of an (α, β)-stable Levy distri-bution with time-dependent location µ(t) and scale σ(t)parameters [19]. By analogy, its characteristic function isgiven by (cf. eqs. (8) and (9))
p(k, t) = exp[ikµ(t)−σα(t)|k|α
(1− iβ sign(k) tan
πα
2
)].
We insert this ansatz into FFPE (11). Since the derivativewith respect to k appears multiplied by k in (11), the non-analyticity of |k|α at k= 0 does not create any singularityin the equation. The real part of eq. (11) yields thefollowing evolution equation for the scale parameter σ(t):
−ασα−1σ= aασα−σα0 , (12)
whereas the imaginary part gives
[µ+ aµ− f(t)] k =[−ασα−1σ− aασα+σα0
]×β tan
πα
2|k|α sign(k). (13)
The RHS of eq. (13) vanishes due to eq. (12). From LHSone gets the evolution equation for the location parameter:
µ=−aµ+ f(t). (14)
The evolution equations (12) and (14) are completedwith the initial conditions µ(0) = x0 and σ(0) = 0 (we arecalculating probability densities conditioned to x(0) = x0).The solution of these differential equations are
µ(t) = e−atx0+ e−at∫ t0
easf(s)ds (15)
and
σ(t) = σ0
[1
aα
(1− e−aαt)]1/α , (16)
where σ0 is the scale parameter of the corresponding α-stable density. For a constant force f(t)≡ f , the long timeasymptotics of the above equations are limt→∞µ(t) = f/aand limt→∞σ(t) = σ0/(aα)1/α.
The conjugate variable. – To determine the conju-gate variable to the external force, we need the station-ary distribution in position space for a constant forcef . Despite the characteristic functions of stable distribu-tions assume closed expressions, the corresponding PDFshave a known simple analytical form [28,30] only in a fewcases: For α= 2 and β = 0 the resulting distribution isGaussian; for α= 1, β = 0 one gets the Cauchy distrib-ution; finally, for α= 1/2, β = 1 the Levy-Smirnoff distri-bution is obtained. Here, we derive explicit expressions forthe conjugate variable for these three cases.
For α= 2 and β = 0, the time-dependent solution of thecorresponding Langevin equation (6) is
p2,0(x, t|x0, 0) =1√
2πσ2(t)exp
[− (x−µ(t))
2
2σ2(t)
](17)
with µ(t) and σ(t) given by eqs. (15) and (16). Thestationary solution pss(x) for a constant force f is obtainedby replacing µ(t) and σ2(t) by their stationary values, f/aand σ20/(2a), respectively. We then get the non-equilibriumpotential φ≡−ln pss(x) and the conjugate variable can beeasily derived as
XG =− ∂ ln pss(x)
∂f
∣∣∣∣f=0
=∂φ
∂f
∣∣∣∣f=0
=−2x
σ20, (18)
which is proportional to x, as expected, since the Gaussiancase corresponds to a Brownian particle in equilibrium.
For α= 1 and β = 0, the time-dependent solution ofthe corresponding Langevin equation (6) is the Cauchydistribution
p1,0(x, t|x0, 0) =σ(t)
π
1
[x−µ(t)]2
+σ2(t)(19)
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Bart�lomiej Dybiec et al.
and the stationary solution for a constant force f isobtained replacing µ(t) and σ(t) by their stationary values,f/a and σ0/a, respectively. The corresponding conjugatevariable takes now the form
XC =− 2x
a [x2+ (σ0/a)2](20)
which is proportional to x only for small values of x andbecomes proportional to 1/x for large x. This large xbehavior ensures the convergence of all the moments ofXC, whereas for the Cauchy case |x|ν exists only if ν < 1,see [16,28].
Finally, for α= 1/2 and β = 1 the solution to eq. (6) isthe Levy-Smirnoff PDF
p1/2,1(x, t|x0, t0) =
√σ(t)
2π [x−µ(t)]3 exp
[− σ(t)
2(x−µ(t))
](21)
for x> µ(t) and p1/2,1(x, t|x0, 0)≡ 0 for x� µ(t). Thestationary values of µ(t) and σ(t) are in this case f/a and4σ0/a
2, respectively. Inserting these values to eq. (21), onecan easily obtain pss(x) and the conjugate variable
XL-S =4σ0− 3a2x
2a3x2; x> µ(t). (22)
Susceptibility and response. – The main objectiveof the current work is to compare the response of thesystem to external perturbation as calculated directlyfrom the definition
〈X(t)〉=∫ ∞−∞X(x)p(x, t)dx, (23)
or, otherwise determined by the generalized susceptibilityχ(t) = d
dt 〈X(t)X(0)〉0 within linear response theory:
〈X(t)〉LR =
∫ t0
χ(t− s)f(s)ds. (24)
For that purpose, we restrict our analysis to the fullyanalytically solvable Cauchy case, α= 1, β = 0, and iden-tify the conjugate variable as X ≡XC, with XC given byeq. (20). In this case, the time-dependent average (23) canbe calculated exactly with the probability density:
p(x, t) =
∫ ∞−∞p(x, t|x0, 0)p(x0)dx0, (25)
where
p(x0) =σ0
aπ
1
x20+ (σ0/a)2(26)
and p(x, t|x0, 0) is given by eq. (19).On the other hand, the FDT relates the susceptibility
with the autocorrelation of the conjugate variables inthe reference state, i.e., for f = 0. The autocorrelation is
defined as
〈X(t)X(0)〉0 =
∫∫2x
a[x2+ (σ0/a)2]
2y
a[y2+ (σ0/a)2]
× σ(t)
π [(x−µ(t))2+σ2(t)]
× σ0
aπ [y2+ (σ0/a)2]dxdy
where µ(t) = e−aty and σ(t) = σ0[(1− e−at)/a]. The finalresult is surprisingly simple:
〈X(t)X(0)〉0 =1
2σ20e−at. (27)
From the above, the generalized susceptibility can bederived by differentiation with respect to time (see eq. (2)):
χ(t) =d
dt〈X(t)X(0)〉0 =− a
2σ20e−at. (28)
In further calculations, for the sake of simplicity, it isassumed that a= 1 and σ0 = 1, so that 〈X(t)X(0)〉= 1
2e−t
and χ(t) =− 12e−t.In order to test the linear response theory for our
dynamic Markov system subjected to Levy white noise,we calculate the response of the conjugate variable X totwo different time dependent perturbations: the sum ofa small periodic and a linearly increasing force, f1(t) =sin(t)/10 + t/100; and a periodic force with increasingamplitude, f2(t) = t sin(t)/100. Figure 1 displays the exactevolution of 〈X(t)〉 and the result obtained from thelinear response theory. For small perturbations, (i.e., shorttimes, t� 50) in both cases, linear response theory yieldsan accurate estimation of the response. In the case of f1(t),for large times the (exact) response 〈X(t)〉 is insensitiveto the sinusoidal component of the force, which is smallcompared with the linear part. This is due to the peculiarform of the conjugate variable X given by eq. (20). For aconstant force f , the mean value of X is
〈X〉 = − σ0aπ
∫ ∞−∞
dx
[x− f/a]2+ (σ0/a)22x
a [x2+ (σ0/a)2]
= − 2f
f2+ 4σ20,
which yields 〈X〉=−0.5 for σ0 = 1 and f = 2 (at t= 200,f1(t)� 2). A similar saturation effect is not observed forthe sinusoidal force f2(t).
We can apply the generalized FDT to any func-tion A(x) with finite average. Due to the heavytails of stable distributions, only moments 〈|x|ν〉with ν < α converge (ν < 1 in the case of Cauchydistributions). Moreover, those moments are evenfunctions of x and, for symmetry reasons, the correla-tion with X vanishes: 〈|x(t)|νX(0)〉= 0. Consequently,the deviation of 〈|x(t)|ν〉 with respect to its refer-ence value is non-linear in the perturbation f(t). Onthe other hand, we can obtain non-trivial results forfractional moments A(x) = sign[x(t)]|x(t)|ν , whose
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Fluctuation-dissipation relations under Levy noises
0.25
-1
-0.5
0
0 50 100 150 200
<X>
t
lrtexact
-1
-0.5
0
0.5
1
0 50 100 150 200
<X>
t
lrtexact
Fig. 1: Response of 〈X(t)〉 to external drivings f1(t) =sin(t)/10+ t/100 (upper plot) and f2(t) = t sin(t)/100 (lowerplot). The solid and dotted lines present an exact result(eq. (23)) and a result constructed by use of the linear responsetheory (eq. (24)), respectively.
average in the reference state vanishes 〈A〉0 = 0. Thecorresponding correlation function reads
〈A(x(t))X(0)〉=− ν
sin [πν/2]e−t (29)
and the generalized susceptibility is given by
χA(t) =ν
sin [πν/2]e−t. (30)
In the spirit of the former definition, see eq. (23), the exactvalue of 〈A(x(t))〉 can be calculated as
〈A(x(t))〉 =
∫ ∞−∞A(x(t))p(x, t)dx
=
∫ ∞−∞A(x(t))p(x, t|x0, 0)p(x0)dx, (31)
where p(x0) and p(x, t|x0, 0) are given by eqs. (19)and (26), respectively. Figure 2 displays the compar-ison of the exact evolution 〈sign[x(t)]|x(t)|1/2〉 with
the linear response approximation∫ t0χA(t− s)f(s)ds.
As previously, linear response theory is valid for weakperturbation up to f � 0.5. However, for the fractionalmoment and the linearly increasing force (upper plot), wedo not observe saturation.
-0.25
0
0.5
1
1.5
0 50 100 150 200
<sg
n(x)
|x|1/
2 >
t
lrtexact
-1
-0.5
0
0.5
1
0 50 100 150 200
<sg
n(x)
|x|1/
2 >
t
lrtexact
Fig. 2: Response of the 〈sign[x(t)]|x(t)|1/2〉 to f(t) =sin(t)/10+ t/100 (upper plot) and f(t) = t sin(t)/100 (lowerplot). The solid line presents exact results, see eq. (31), whilethe dotted line results constructed by use of the linear responsetheory, see eqs. (24) and (29).
Summary and conclusions. – We have shown thatthe generalized FDT can be applied to linear systemsdriven by Levy noise. The FDT allows one to calcu-late the susceptibility of any observable and then theresponse to any small time-dependent perturbation. Fora noise distributed according to the Cauchy distribution,we have calculated the susceptibility of the conjugate vari-able XC and the susceptibility of odd fractional moments〈sign[x(t)]|x(t)|ν〉, which have a simple exponential behav-ior. From these susceptibilities it is easy to get simpleanalytical expressions for the response of the system usingeq. (24). We have to notice that, although the exactresponse can be calculated analytically using eq. (23), thecorresponding integrals are cumbersome and can be onlysolved numerically in the simplest cases. Therefore, thegeneralized FDT is shown to be a useful analytical tool todeal with these types of systems.
It is still not obvious whether the conjugate variablesthat we have calculated for the Cauchy and Levy-Smirnoffnoises have any physical meaning, besides the one providedby the generalized FDT itself. The generalized FDTshows that these conjugate variables represent the changein the probability distribution of the system under the
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Bart�lomiej Dybiec et al.
perturbation. In equilibrium, this change is also relatedwith the energy that the system absorbs from the pertur-bation. On the other hand, for non-equilibrium systems,the lack of conserved quantities prevents such an interpre-tation. For instance, in the case of the harmonic oscillatordriven by a Cauchy-Levy noise, 〈x〉 and higher momentsdiverge1. Consequently, both the potential energy of thesystem in the harmonic potential and the work done by theexternal force f(t) also diverge. The system is plagued bydivergent quantities and exhibits anomalous work fluctu-ations [8,9] which imply that large positive fluctuations ofwork are asymptotically as likely to be observed as nega-tive fluctuations of equal magnitude. However, the conju-gate variable XC given by eq. (20) has finite moments andstill captures the dynamical response of the Levy particle.Summarizing, although systems driven by α-stable noisesmight significantly differ from their Brownian (equilib-rium) counterparts [31–33] due to their heavy tail asymp-totics, we have shown that in such far-from-equilibriumsituations some concepts from weakly perturbed equilib-rium systems can still be used. One of the drawbacks of thegeneralized FDT derived in [14] is the difficulty to find theconjugate variable, since it requires the knowledge of thestationary state. We have been able to find this stationarystate for a linear system. An interesting open question iswhether this state, or some slight modification, can still beused to calculate susceptibilities in the presence of weaknon-linearities.
∗ ∗ ∗
We are grateful to Jordan Horowitz for discussionsand useful comments. The authors acknowledge thesupport by the European Science Foundation throughEPSD/PESC program. JMRP acknowledges financialsupport from grants MOSAICO (Spanish Government)and MODELICO (Comunidad de Madrid). BD acknowl-edges the Danish National Research Foundation forfinancial support through the Center for Models of Life.
REFERENCES
[1] Kubo R., Rep. Prog. Phys., 29 (1966) 255.[2] Hayashi K. and Takano M., Biophys. J., 93 (2007) 895.[3] Crisanti A. and Ritort F., J. Phys. A: Math. Gen., 36(2003) R181.
[4] Calabrese P. and Gambassi A., J. Phys. A: Math.Gen., 38 (2005) R133.
[5] Marconi U. M. B., Puglisi A., Rondoni L. andVulpiani A., Phys. Rep., 461 (2008) 111.
[6] Chetrite R., Falkovich G. and Gawedzki K., J. Stat.Mech. (2008) P08005.
1With a more general definition of integrals, i.e., by using a“principal value” integral, the first moment of a Cauchy distributioncan be evaluated as 〈x〉= lima→∞ 1
π
∫ a−a
x1+x2
dx and is equal tozero.
[7] Seifert U. and Speck T., EPL, 89 (2010) 100.[8] Touchette H. and Cohen E. G. D., Phys. Rev. E, 80(2009) 011114.
[9] Touchette H. and Cohen E. G. D., Phys. Rev. E, 76(2007) 020101.
[10] Chechkin A. V. and Klages R., J. Stat. Mech. (2009)L03002.
[11] Dubkov A., Chem. Phys., 375 (2010) 364.[12] Allegrini P., Bologna M., Fronzoni L., Grigolini
P. and Silvestri L., Phys. Rev. Lett., 103 (2009)030602.
[13] Ebeling W. and Sokolov I. M., Statistical Thermody-namics and Stochastic Theory of Nonequilibrium Systems(World Scientific Publishing, Singapore) 2005.
[14] Prost J., Joanny J.-F. and Parrondo J. M. R., Phys.Rev. Lett., 103 (2009) 090601.
[15] Hatano T. and Sasa S., Phys. Rev. Lett., 86 (2001) 3463.[16] Dubkov A. A., Spagnolo B. and Uchaikin V. V., Int.
J. Bifurcat. Chaos Appl. Sci. Eng., 18 (2008) 2649.[17] Dybiec B. and Gudowska-Nowak E., Phys. Rev. E, 80
(2009) 061122.[18] Metzler R. and Klafter J., Phys. Rep., 339 (2000) 1.[19] Jespersen S., Metzler R. and Fogedby H. C., Phys.
Rev. E, 59 (1999) 2736.[20] Barndorff-Nielsen O. E., Mikosch T. and Resnick
S. I. (Editors), Levy Processes: Theory and Applications(Birkhauser, Boston) 2001.
[21] Shlesinger M. F., Klafter J. andWest B. J., PhysicaA, 140 (1986) 212.
[22] Sanchez R., Newman D. E., Leboeuf J.-N., Decyk
V. K. and Carreras B. A., Phys. Rev. Lett., 101 (2008)205002.
[23] Dybiec B. and Gudowska-Nowak E., Phys. Rev. E, 69(2004) 016105.
[24] Lee C.-C., Lee H.-C., Yeh H.-F. and Lin H.-I.,Environ. Earth Sci., 63 (2010) 1199.
[25] Dybiec B., Kleczkowski A. and Gilligan C. A., J. R.Soc. Interface, 6 (2009) 941.
[26] Brockmann D., Hufnagel L. and Geisel T., Nature(London), 439 (2006) 462.
[27] Barthelemy P., Bertolotti J. and Wiersma D.,Nature (London), 453 (2008) 495.
[28] Janicki A. and Weron A., Simulation and ChaoticBehavior of α-stable Stochastic Processes (Marcel Dekker,New York) 1994.
[29] Yanovsky V. V., Chechkin A. V., Schertzer D. andTur A. V., Physica A, 282 (2000) 13.
[30] Penson K. A. and Gorska K., Phys. Rev. Lett., 105(2010) 210604.
[31] Chechkin A. V., Klafter J., Gonchar V. Y.,
Metzler R. and Tanatarov L. V., Chem. Phys., 284(2002) 233.
[32] Dybiec B. and Gudowska-Nowak E., J. Stat. Mech.(2009) P05004.
[33] Dybiec B., Chechkin A.V. and Sokolov I. M., J. Stat.Mech. (2010) P07008.
