PHYSICAL REVIEW E 84, 061118 (2011)

Fluctuation relations with intermittent non-Gaussian variables

Adrian A. BudiniConsejo Nacional de Investigaciones Cientıficas y Tecnicas (CONICET), Centro Atomico Bariloche,

Avenida E. Bustillo Km 9.5, 8400 Bariloche, Argentina(Received 17 August 2011; revised manuscript received 1 November 2011; published 12 December 2011)

Nonequilibrium stationary fluctuations may exhibit a special symmetry called fluctuation relations (FRs). Here,we show that this property is always satisfied by the subtraction of two random and independent variables relatedby a thermodynamiclike change of measure. Taking one of them as a modulated Poisson process, it is demonstratedthat intermittence and FRs are compatible properties that may coexist naturally. Strong non-Gaussian featurescharacterize the probability distribution and its generating function. Their associated large deviation functionsdevelop a “kink” at the origin and a plateau regime respectively. Application of this model in different stationarynonequilibrium situations is discussed.

DOI: 10.1103/PhysRevE.84.061118 PACS number(s): 05.70.Ln, 05.40.−a, 45.70.−n, 82.70.Dd

I. INTRODUCTION

Intermittency is a phenomenon that appears in a wideclass of situations such as chaotic dynamics [1], atomic[2] and nanoscopic [3] fluorescent systems, single-moleculereaction dynamics [4], biological self-organized models [5],and fluid turbulence [6], just to name a few. It consistsof a random switching among qualitatively different systemdynamical regimes. Its stochastic behavior may be ergodic ornot. Complexity, non-Gaussian statistics, and nonequilibriumdynamics are closely related with its development. In thislast context, Gallavoti-Cohen fluctuation relations (FRs) [7]address the distribution of entropy production in far-from-equilibrium steady states. They relate the probability ofobserving a certain entropy production rate to the probability ofobserving the corresponding entropy consumption rate [7–12].Theoretical and experimental results confirm the validity ofFRs in diverse nonequilibrium systems [13–22] . Interestingly,the symmetry imposed by FRs has also been found valid fornonthermodynamic variables [23].

As FRs and intermittence are intrinsically related withnonequilibrium dynamics, it is natural to ask about thepossible coexistence of both properties. The main goal of thiscontribution is to give a positive answer to this issue. Wedemonstrate that there may exist nonequilibrium steady stateswhose (non-Gaussian) fluctuations are intermittent and satisfythe FRs. An explicit construction of a stochastic variablewith the required symmetry defines the basis of the presentanalysis. It is shown that FRs can always be satisfied by thesubtraction of two independent stochastic variables related byan exponential change of measure. Hence, intermittence isintroduced by choosing one of them as a random modulatedPoisson process. Special attention is paid to the asymptoticregime, where FRs can be analyzed through the large deviationfunctions (LDFs) [7–9] of the probability distribution and itsassociated generating function. In the parameter regime whereintermittence arises, they develop a kink around the originand a plateau regime, respectively. These strong non-Gaussianfeatures are similar to those found in different observablessuch as the entropy production rate of a colloidal particle [22]and the (time-average) velocity of a polar granular rod [23].While the specific analysis of these systems is beyond thescope of this contribution, the present results suggest that

intermittence under the constraint of FR symmetries maybe a central ingredient when studying nonequilibirum statescharacterized by non-Gaussian fluctuations.

II. FLUCTUATION RELATIONS WITH INDEPENDENTSTOCHASTIC VARIABLES

Our results rely on the following analysis. Let there be thearbitrary stochastic variable xst, whose probability distributionP (x) satisfies a FR defined as

P (−x) = P (x) exp[−ζx], (1)

where ζ is a given real positive constant. This symmetryimplies that negative values are exponentially less probablethan positive values. In terms of the generating functionZ(λ) = 〈e−λxst〉 = ∫ +∞

−∞ dxP (x)e−λx, Z(0) = 1, the FR reads

Z(λ) = Z(−λ + ζ ). (2)

In order to satisfy the FR, we write xst as the subtraction oftwo statistically independent variables, x±

st ,

xst = x+st − x−

st . (3)

Then, Z(λ) = Z+(λ)Z−(−λ), where Z±(λ) are the generatingfunctions of x±

st and whose distributions are P±(x). Thecondition (2) is satisfied by demanding the relation

Z−(λ) = Z+(λ + ζ )

Z+(ζ ), (4)

which in turn implies

P−(x) = P+(x)e−ζx

〈e−ζx〉+ , (5)

where 〈e−ζx〉+ = Z+(ζ ). This is one of the central resultsof this paper. It tells us that the FR symmetry (1) is alwayssatisfied by Eq. (3) whenever the change of measure (5) isimposed. By reading x as an energy and ζ as an inversetemperature, its thermodynamiclike structure is self-evident.Similar transformations were introduced in Refs. [24–26]. Thestructure of Eq. (5) can also be read as a variation of the changeof measure used in umbrella sampling [27].

Notice that in Eq. (5) no conditions are required for P+(x),while P−(x) is determined from it. Alternatively, one can

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ADRIAN A. BUDINI PHYSICAL REVIEW E 84, 061118 (2011)

choose P−(x) as the independent distribution. In fact, Eq. (2)can also be satisfied by taking

Z+(λ) = Z−(λ − ζ )

Z−(−ζ ), (6)

delivering the relation

P+(x) = P−(x)e+ζx

〈e+ζx〉− , (7)

where 〈e+ζx〉− = Z−(−ζ ). These last two expressions followstraightforwardly by inverting Eqs. (4) and (5). Nevertheless,what is important to realize is that, in general, both kinds ofsolutions lead to different families of solutions if one assumesthe same property for each “independent” distribution.

The relations (5) and (7) do not involve time. Therefore, theproposal Eq. (3) guarantees the fulfillment of the symmetry (1)at any time.

A. Time-average variables

When studying a given nonequilibrium steady state, thestochastic variable of interest may be defined through a timeaverage,

μτ (t) ≡ 1

τ 〈v〉∫ t+τ

t

dt ′vst(t′) = xst(t + τ ) − xst(t)

τ 〈v〉 , (8)

where (d/dt)xst(t) = vst(t). This definition makes sense whenthe average of the integral contribution grows linearly withtime. For example, xst(t) may represent entropy production andvst(t) its rate [22], or, respectively, the position and velocity ofa self-propelled particle [23]. 〈v〉 is the stationary mean valueof vst(t).

Usually the regime of interest is the stationary one, t =∞. Hence, we write μτ = limt→∞ μτ (t). For this stationaryvariable, instead of Eq. (1), we write the FR

P (−μτ ) = P (μτ ) exp[−αμτ τ ]. (9)

This condition can also be satisfied in the present approachbecause the statistical properties of μτ follow from those of thestationary increments of xst(t), that is, x∞

τ ≡ limt→∞[xst(t +τ ) − xst(t)]. By writing x∞

τ as the subtraction of two indepen-dent stochastic variables related by the change of measures(5) [or Eq. (7)], it follows that P (−x∞

τ ) = exp[−ζx∞τ ]P (x∞

τ ).The connection between x∞

τ and μτ implies the probabilitiesrelation P (μτ )dμτ = P (x∞

τ )dx∞τ [28], which leads to FR

Eq. (9) with α = ζ 〈v〉. Consequently, the proposal based onindependent stochastic contributions, as in the previous case[Eq. (1)], warrants the fulfilment of symmetry (9) for any valueof τ.

B. Large deviation functions

In general, a (rate) variable [Eq. (8)] that characterizes agiven nonequilibrium state only satisfies the relation (9) in along time regime, τ → ∞. In fact, the Gallavoti-Cohen FRis an asymptotic relation in time [7–9]. In this regime, whenthe probability distribution adopts the asymptotic structure(limτ→∞) P (μτ ) ≈ exp[−τϕ(μτ )], the statistics can be ana-lyzed through a large deviation theory [24]. Consistently, thegenerating function Z(λ) = 〈e−λτμτ 〉 scales in the same way,

Z(λ) ≈ exp[−τ�(λ)]. Both ϕ(μτ ) and �(λ) define the LDFsof the problem. They completely characterize the asymptoticregime. In terms of ϕ(μτ ), the FR reads

1

τlog

[P (+μτ )

P (−μτ )

]≈ −ϕ(+μτ ) + ϕ(−μτ ) = αμτ . (10)

Through a saddle-point approximation, both LDFs can berelated by a Legendre-Fenchel transformation [24]:

ϕ(μτ ) = maxλ

[�(λ) − λμτ ], �(λ) = minμτ

[ϕ(μτ ) + λμτ ].

(11)

These relations and Eq. (10) lead to the equivalent formulationof the FR symmetry [9]:

�(λ) = �(−λ + α). (12)

Conditions (10) and (12) are well-known expressions ofGallavoti-Cohen FR symmetry [7–9]. As the splitting basedon independent variables [Eq. (3)] allows one to fulfill the FR(9) at any time, trivially it can also be utilized in the long timeregime. After explicitly writing μτ = μ+

τ − μ−τ , Eq. (4) and

the asymptotic structure of Z(λ) allow us to write

�(λ) = �+(λ) + �+(−λ + α) − �+(α), (13)

where �+(λ) is the LDF corresponding to the generatingfunction of μ+

τ . By knowing (an arbitrary) �+(λ) the previousexpression defines �(λ), which in turn through Eq. (11)provides the LDF ϕ(μτ ). By construction, the fulfillment ofconditions (10) and (12) is guaranteed. Alternatively, fromEq. (6) we can also write

�(λ) = �−(−λ) + �−(λ − α) − �−(−α), (14)

where now �−(λ) is the LDF associated with the generatingfunction of μ−

τ . The relations (13) and (14) define the secondmain result of this paper. They allow one to characterize thelong time regime in terms of asymptotic properties of theindependent stochastic contributions.

III. INTERMITTENT VARIABLES

The previous results allow one to build up a variable thatsatisfies the FR Eq. (9) after knowing the statistical propertiesof an arbitrary one. The regime of interest is the stationary one,where the FR symmetry is characterized by Eq. (13) or (14).For example, one can assume Gaussian or Poissonian statistics.In this last case, if one takes �+(λ) = γ (1 − e−λ), whichcorresponds to the LDF of an unidirectional Poisson processwith rate γ, from Eq. (13) we get �(λ) = γ (1 − e−λ) +γ e−α(1 − eλ). We note that this expression corresponds to theLDF for the entropy production of an asymmetric randomwalk [22]. Hence, while the previous analysis seem to berather abstract, there exist nontrivial dynamics where theyapply.

The random walk model is able to fit some non-Gaussianproperties found in Ref. [22]. Here, motivated by the experi-mental results of Ref. [23], we introduce a similar generalizedmodel able to develop intermittence. On the basis of previous

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analysis, the velocity [see Eq. (8)] is defined as vst(t) =v+

st (t) − v−st (t), where

v±st (t) = x0

∑i

δ(t − t±i ). (15)

The constant x0 introduces the right units of vst(t), δ(t) isthe Dirac δ function, and t±i are successive random times.Consistently, xst(t) = x+

st (t) − x−st (t), and

x±st (t) = x0n

±st (t), (16)

where n±st (t) is the number of (±δ) events in the interval (0,t).

Hence, they are positive (discrete) random variables. Theirgenerating functions are denoted as Z±

n (s,t) = 〈e−sn±st (t)〉 =∑∞

m=0 q±m (t)e−sm, where {q±

m (t)}∞m=0 are the respective count-ing probabilities. Independently of their statistics, by choosingn+

st (t) as the “free” variable, this model satisfies the FRsymmetry (9) after demanding [see Eq. (5)] the change ofmeasures:

q−m (t) = q+

m (t)e−s0m

〈e−s0n+st (t)〉 , (17)

where 〈e−s0n+st (t)〉 = Z+

n (s0,t) and s0 is an arbitrary positivedimensionless constant. Hence, n−

st (t) can be read as the “s0

ensemble” associated with n+st (t) [26]. After a change of

variables based on Eqs. (8) and (16), the constant α reads

α = s0δI ≡ s0(I+ − I−), (18)

where I± ≡ − limt→∞ t−1(∂/∂s)Z±n (s,t)|s=0, or equivalently

limt→∞〈n±st (t)〉 I±t, which in turn implies 〈v〉 = x0(I+ −

I−).In order to close the model, it is necessary to specify the

statistical properties of n+st (t). Its generating function is written

as

Z+n (s,t) = Z+

A (s,t) + Z+I (s,t), (19)

where the evolution of each contribution reads

dZ+A (s,t)

dt= −θsZ

+A (s,t) − �AZ+

A (s,t) + �IZ+I (s,t),

(20)dZ+

I (s,t)

dt= +�AZ+

A (s,t) − �IZ+I (s,t),

with θs ≡ γ (1 − e−s). These dynamics allow us to read n+st (t)

as a modulated Poissonian (counting) process [26], whose rateat random times adopts the values γA = γ (active regime)and γI = 0 (inactive regime). The switching between bothstates is governed by a classical master equation with transitionrates �A and �I . The asymmetric random walk model [22] isrecovered from Eq. (20) by taking �A/I = 0.

The generating function Z−n (s,t) can be obtained from

Eq. (4) after knowing Z+n (s,t). Hence, the (lattice) distribution

[28] of the process xst(t) can be obtained by finding Z+n (s,t)

from Eq. (20) and a posterior numerical Fourier inversionof Zn(s,t) = Z+

n (s,t)Z−n (−s,t) in the s = −ik variable. By

using the Markovian property of Eq. (20), the probabilityof x∞

τ ≡ limt→∞ xst(t) follows from that of xst(t) by takingstationary initial conditions for the rate fluctuations. The

FIG. 1. (Color online) (a) Probability distribution P (μτ ) thatfollows from Eq. (20) (see text) for different values of τ. (b)Linear dependence of (1/τ ) ln[P (+μτ )/P (−μτ )] corresponding tothe distributions shown in panel (a). The value of the slope α wasobtained from Eq. (18). The parameters [Eq. (20)] are s0 = 0.2,γ = 100 s−1, �A = 0.54 s−1, and �I = 4 s−1.

distribution P (μτ ) of μτ follows from the change of variablesdefined by Eq. (8).

In Fig. 1(a) we plot the distribution P (μτ ) for differentvalues of τ. For all times the distributions satisfy the symmetryEq. (9) [Fig. 1(b)]. As expected, we confirmed that thisproperty is valid for any value of the rate parameters thatdefine the evolution (20). On the other hand, the short timebehavior of P (μτ ) strongly depends on the chosen parametervalues. Nevertheless, for increasing τ the distributions becomesimilar and, consistently, develop an increasing peak aroundμτ = 1. In this regime [τ � 0.3 in Fig. 1(a)] a large deviationtheory applies. Hence the problem can be analyzed throughthe corresponding LDFs.

A. Large deviations functions

The long time behavior of Z+n (s,t) completely defines

the asymptotic statistical properties of the counting processn+

st (t). Before characterizing the LDF functions from it,we notice that in the long time limit some results canbe established for Z−

n (s,t). As demonstrated in Ref. [26],n+

st (t) can be mapped with a renewal process characterizedby a shift closure property, which implies that asymptoti-cally n−

st (t) also becomes a renewal process with a renor-malized waiting time distribution. Hence, asymptoticallyZ−

n (s,t) has the same structure and dynamics as Z+n (s,t)

but with renormalized rates. After some hard calculationsteps, which are not relevant for the following analysis, weobtained γ → γs0 = γ e−s0 and more complex expressionsfor the hopping rates �A/I → �A/I (s0). Furthermore I+ =γ�I/(�A + �I ), and similarly I− = γs0�I (s0)/[�A(s0) +�I (s0)]. In the case of an asymmetric random walk model[22], �A/I = 0, both sets of probabilities {q±

m (t)}∞m=0 be-come Poissonian counting processes with I+ = γ andI− = γ e−s0 .

By working Eqs. (19) and (20) in a Laplace domain, f (ξ ) =∫ ∞0 f (t)e−ξ t , it is possible to write Z+

n (s,t) as a superpositionof two exponential functions scaled by the roots [Q(ξ ) = 0]of the characteristic polynomial

Q(ξ ) = ξ 2 + ξ (θs + �A + �I ) + θs�I . (21)

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FIG. 2. (Color online) (a) LDF �(λ) [Eq. (13)] for two differentvalues of α. (b) and (c) LDF ϕ(μτ ) [Eq. (11) )] for α = 43.7 s−1 andα = 65.5 s−1, respectively. (d) and (e) Realizations of μτ (t) [α =43.7 s−1] for τ = 0.025 s and τ = 0.1s, respectively. In all cases,the dotted lines follow from approximation (23). The insets showthe skewness (S) and kurtosis (K) as functions of τ , indicating thestrongly non-Gaussian nature of μτ . The parameters [Eq. (20)] for thecurves with α = 43.7 s−1 are s0 = 1.56, γ = 40 s−1, �A = 0.54 s−1,

and �I = 1.25 s−1. For α = 56.3 s−1 they are the same except for�I = 5 s−1.

From its definition, the smaller solution [22], after the changeof variable s → λ/δI, provides �+(λ). We get

�+(λ) = θ ′λ + �A + �I

2−

[�Aθ ′

λ + 1

4(�A + �I − θ ′

λ)2

]1/2

,

(22)

where θ ′λ ≡ θλ/δI = γ (1 − e−λ/δI ). This result, jointly with

Eq. (13) and the transformation (11), completely characterizesthe asymptotic statistics and the LDFs of μτ .

In Fig. 2(a) we plot the LDF �(λ) for two different sets ofparameter values {s0,γ,�A,�I }. In both cases the symmetry(12) is satisfied. The values of α were determined fromEq. (18). Notice that �(λ) cannot be well approximated bya quadratic polynomial, which in turn implies the presence ofstrong non-Gaussian features. Figures 2(b) and 2(c) confirmthis fact. Both LDFs ϕ(μτ ) were obtained numerically throughthe Legendre transformation (11). The same results followfrom the asymptotic behavior of the exact distribution P (μτ )obtained by Fourier inversion of its generating function.Consistently, the LDFs and the asymptotic exact distributionsatisfy the FR (10). In the insets, we plot the time dependenceof the skewness S = 〈δμ3

τ 〉/σ 3 and kurtosis K = 〈δμ4τ 〉/σ 4 − 3,

obtained from the generating function of P (μτ ), where δμτ =μτ − 〈μτ 〉 and σ 2 = 〈δμ2

τ 〉. These objects also confirm thestrong non-Gaussian nature of P (μτ ).

The functions ϕ(μτ ) not only depart from a quadraticpolynomial, but also develop a “kink,” that is, and abruptchange around the origin. This characteristic arises when �(λ)presents a plateau regime centered around α/2. For the chosenparameter values, the curves and these special features are

FIG. 3. (Color online) (a) LDF �(λ) [Eq. (13)] with α =12.5 s−1. (b) LDF ϕ(μτ ) [Eq. (11)]. (c) Probability distribution of μτ

for τ = 0.22 s and τ = 0.70 s. (d) and (e) Realizations of μτ (t) forτ = 0.1 s and τ = 0.3 s, respectively. The inset shows the skewness(S) and kurtosis (K) as functions of τ , indicating the tendency tobecome more Gaussian at higher time τ. In all curves the parameters[Eq. (20)] are s0 = 0.83, γ = 25 s−1, �A = 0.3 s−1, and �I = 20 s−1.The dotted lines follow from approximation (23).

similar [29] to those found in the numerical and experimentalresults of Refs. [22,23]. In the present model, a physical effectcan be associated with these properties: the kink and theplateau regime are closely related with the development ofintermittence in the stochastic realizations of μτ . This is thethird main result of this contribution.

The stochastic realizations of n+st (t) can be obtained

from Eq. (20) by using standard Monte Carlo methods.Alternatively, a more simple algorithm follows by mappingEq. (20) with a renewal process [26]. On the other hand,the realizations of n−

st (t) can be obtained from a conditionalscheme defined in Ref. [26], where realizations of n+

st (t)with m events are selected as one of n−

st (t) with probabilitye−s0m and discarded with probability (1 − e−s0m). Statisticallyindependent realizations of n±

st (t) allow one to generate thestochastic trajectories of μτ [see Eqs. (8) and (16)].

In Figs. 2(d) and 2(e) we plot two realizations of μτ ,

corresponding to α = 43.7 s−1, for two different values ofτ. Consistently with the definition (8) they fluctuate aroundμτ = 1. For the chosen parameter values, the trajectoriesswitch among periods of time where μτ is positive, negative,or null. As expected, for increasing τ the fluctuations arediminished. Furthermore, for higher averaging times (notshown) the realizations only assume positive values. FromEq. (20), it can immediately be deduced that when γ (�A,�I ), the realizations of n+

st (t) develop intermittence. Thisis not a surprising result. What is novel and not trivial is thatone can build up the complementary process n−

st (t) such thattheir (normalized) subtraction [Eq. (8)] satisfies the FR. Thecloseness between the kinks of Figs. 2(b) and 2(c) and theintermittence shown in Figs. 2(d) and 2(e) is also supportedby the results shown in Fig. 3. Here, �I �A. Therefore,

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FLUCTUATION RELATIONS WITH INTERMITTENT NON- . . . PHYSICAL REVIEW E 84, 061118 (2011)

the inactive regime is statistically (dynamically) inhibited,which in turn implies that neither n+

st (t) nor n−st (t) develops

any intermittence. As shown by the plots, the LDFs convergeto quadratic polynomial functions, while the probabilitydistribution becomes Gaussian. Consistently, the trajectories[Figs. 3(d) and 3(e)] for any value of τ do not develop thephenomenon of intermittence.

Equations (13) and (22) provide an analytical expressionfor the LDF �(λ) that is very complicated. Furthermore,the transformation, Eq. (11), which delivers the LDF ϕ(μτ ),is only manageable numerically. Nevertheless, when �A �(γ,�I ), from Eq. (21) it is simple to deduce the expression

lim�A→0

�+(λ) = min[θ ′λ,�I ]. (23)

This result, jointly with Eqs. (11) and (13), allows us to obtainsimple analytical expressions for the LDFs �(λ) and ϕ(μτ )(see the Appendix). They correspond to the dotted lines inFigs. 2 and 3. Evidently, they provide a very good fit tothe plotted curves. Nevertheless, what is more relevant is thepossibility of getting an analytical description of the relationbetween intermittence and the non-Gaussian features of bothLDFs.

In the limit �A → 0, both �(λ) and ϕ(μτ ) present pointswhere they are not derivable functions with respect to theirarguments. In particular, when intermittence develops, �I �γ, the LDF ϕ(μτ ) has a linear behavior around the originwith different slopes for μτ ≶ 0 [see dotted lines (μτ ≷ 0) inFigs. 2(b) and 2(c)]. The addition of the slopes of ϕ(μτ ) aroundthe origin is equal to −α [see Eq. (A12)]. The experimentaldata of Ref. [23] seem to be consistent with this relation. Onthe other hand, in the intermittence regime �(λ) develops aplateau regime [see dotted lines in Figs. 2(a)] with value �I

[Eq. (A11)], which is similar to that found in the numericalresults of Ref. [22].

Finally, let us remark that the discontinuities of thederivative of �(λ) can be read as phase transitions betweendifferent dynamical regimes of μτ [25,26]. These features,as well as our main proposal, Eq. (5), suggest an interestingbridge between FRs and equilibrium thermodynamics.

IV. CONCLUSIONS

In conclusion, we have demonstrated that FRs are alwayssatisfied by the subtraction of two independent stochasticvariables whose probability distributions are related by athermodynamiclike change of measure. This relation leavescompletely arbitrary one of the distributions, this degree offreedom being the basis of the present approach. By choosingone of them as a random modulated Poisson process, thedistribution of interest develops strong non-Gaussian features,which in turn arise in the parameter regime where intermittencedevelops. Hence, variables such as entropy production ratesmay satisfy FRs and develop intermittence.

Depending on the model’s parameter values, the LDFsdevelop a rich variety of functional dependencies, which in turnare similar to those found in recent numerical and experimentalresults [29]. In particular, in the intermittence regime the LDFsof the probability distribution and the generating functionare characterized by a kink at the origin and a plateau

regime, respectively. While our analysis does not rely on anyspecific (nonlinear many body) dynamical model, it stronglysuggests that intermittence may be a central ingredient innonequilibrium steady states characterized by non-Gaussianfluctuations.

Due to the large class of systems where intermittence devel-ops, added to the central role of FRs in nonequilibrium states,the characterization of dynamical conditions that guarantee thecoexistence of these two properties becomes a very interestingopen issue, to which this paper intends to contribute. On otherhand, the inclusion of statistical correlations that preserve theFRs in the two-variable model and the analysis of these ideasin the context of deterministic thermostated systems (phasespace contraction conjecture) [13] are additional open issuesthat may deserve extra analysis.

ACKNOWLEDGMENTS

This work was supported by CONICET, Argentina, underGrant No. PIP 11420090100211.

APPENDIX: LDF IN THE LIMIT �A → 0

Here we obtain the LDFs associated with Eq. (23). �(λ)follows from Eq. (13). Hence, ϕ(μτ ) is obtained throughthe Legendre transformation (11). In order to simplify theexpressions, we write

�(λ) = �(s = λ/δI ), (A1)

and similarly we write ϕ(μτ ) as

ϕ(μτ ) = ϕ(k = μτδI ), (A2)

where δI = s0(I+ − I−) [Eq. (18)]. The functions �(s) andϕ(k) are the LDFs of the stochastic variable kst(t) = n+

st (t) −n−

st (t)]/t. Furthermore, we define the parameters

s+ ≡ − ln

(1 − �I

γ

), (A3)

s− ≡ s0 + ln

(1 − �I

γ

), (A4)

and we define, respectively,

k+ ≡ γ e−s+ = γ

(1 − �I

γ

), (A5)

k− ≡ γ e−s− = γe−s0

1 − �I

γ

. (A6)

The LDF of a Poisson process with rate γ is denoted as

θ (s) ≡ γ (1 − e−s), (A7)

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and its (symmetrized) Legendre transform as

ϕp(k) ≡ γ

{1 − |k|

γ

[1 − ln

( |k|γ

)]}. (A8)

Finally, we introduce the function

ϕrw(k) ≡ γ

{1 + e−s0 −

(4e−s0 + k2

γ 2

)1/2

+ k

γln

[1

2

(4e−s0 + k2

γ 2

)1/2

+ k

2γ

]}. (A9)

Both �(s) and ϕ(k) must be defined in different parameterregimes.

i) In the parameter regime

0 <�I

γ〈(1 − e−s0/2), (A10)

it follows s+ < s−, with

�(s) =

⎧⎪⎨⎪⎩

θ (s), s < s+,

�I , s+ < s < s−,

θ (−s + s0), s− < s,

(A11)

and, respectively,

ϕ(k) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

ϕp(k) − ks0, k < −k+,

�I − ks−, −k+ < k < 0,

�I − ks+, 0 < k < k+,

ϕp(k), k+ < k.

(A12)

This is one of the more interesting parameter regimes, which infact corresponds to the intermittence one (Fig. 2). Notice thatin the plateau regime of �(λ) (s+δI < λ < δIs−) it assumesthe value �I . On the other hand, around the origin, ϕ(μτ ) has alinear behavior with different slopes for μτ ≷ 0. This propertygives rise to the characteristic kink shown in Figs. 2(b) and 2(c).From Eqs. (A2) and (A12), we deduce that the addition of theslopes is equal to −α [Eq. (18)]. Furthermore, from Eq. (A12),it is possible to demonstrate that the origin is the unique pointat which the derivative of ϕ(μτ ) is a discontinuous function.Remarkably, in the following parameter regimes ϕ(μτ ) has acontinuous derivative. Hence, a kink related to a discontinuousderivative only arises in the present case.

ii) In the parameter regime

(1 − e−s0/2) <�I

γ〈(1 − e−s0 ), (A13)

it follows s− < s+, with

�(s) =

⎧⎪⎨⎪⎩

θ (s), s < s−,

θ (s) + θ (−s + s0) − �I , s− < s < s+,

θ (−s + s0), s+ < s,

(A14)

and by defining the parameters

k∗ ≡ k− − k+, � ≡ �I − θ (s0), (A15)

(k∗ > 0), we write

ϕ(k) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ϕp(k) − ks0, k < −k−,

(γ − k−) − ks+, −k− < k < −k∗,ϕrw(k) − �, −k∗ < k < k∗,(γ − k−) − k(s0 − s+), k∗ < k < k−,

ϕp(k), k− < k.

(A16)

In this intermediate regime, the non-Gaussian propertiesare gradually lost. The plateau regime of �(λ) becomesbend. On the other hand, we remark that even when ϕ(μτ )is defined by parts, its first derivative is a continuousfunction.

iii) In the parameter regime

(1 − e−s0 ) <�I

γ< 1, (A17)

with s− < s+, we get

�(s) =

⎧⎪⎨⎪⎩

θ (s) + �, s < s−,

θ (s) + θ (−s + s0) − θ (s0), s− < s < s+,

θ (−s + s0) + �, s+ < s,

(A18)

and

ϕ(k) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ϕp(k) − ks0 + �, k < −k−,

(γ − k−) − ks+ + �, −k− < k < −k∗,ϕrw(k), −k∗ < k < k∗,(γ − k−) − k(s0 − s+) + �, k∗ < k < k−,

ϕp(k) + �, k− < k.

(A19)

Figure 3 falls in this regime, where both LDFs approachquadratic functions. In fact, while the derivative of �(λ)presents smooth discontinuities, they are far beyond the origin[lim(�I /γ )→1 s± = ±∞]. As in the previous case, ϕ(μτ ) has acontinuous derivative. Its dependence is mainly defined by thefunction ϕrw(k) [lim(�I /γ )→1 k∗ = ∞].

iv) In the parameter regime

1 <�I

γ< ∞, (A20)

for any value of s, we get

�(s) = θ (s) + θ (−s + s0) − θ (s0). (A21)

Therefore, in this approximation �(λ) corresponds to the LDFof an asymmetric random walk. Furthermore, it follows thatϕ(k) = ϕrw(k) [Eq. (A9)]. As in the previous regime, therate fluctuations do not induce any appreciable non-Gaussianfeatures because their characteristic time is comparable to thatof the individual events.

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FLUCTUATION RELATIONS WITH INTERMITTENT NON- . . . PHYSICAL REVIEW E 84, 061118 (2011)

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[29] Notice that Figs. 2(b), 2(c), and 3(c) are, respectively, verysimilar to Figs. 2(c), 2(d), and 3(b) of Ref. [23]. In addition,Fig. 1 of Ref. [22] can be recovered with other parametervalues. Besides these similitudes, it is important to notice thatour analysis concentrates only on the asymptotic statisticalproperties of μτ . Hence, there may exist different processes, v(t)and x(t), that lead to the same long time statistical behaviors.On the other hand, the description of specific nonstationaryproperties, where in general the FRs are not satisfied, plusthe relation between the parameters of the model and theexperimental variables are interesting open problems whosesolutions depend on each specific situation. These issues arebeyond the scope of this contribution.

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