Chance and Stability

Учайкин, В. В.; Золотарев, В. М.

doi:10.1515/9783110935974

Cited by 465 publications

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“…These fluctuations lead therefore to a novel type of fluctuation relation, which differs from the conventional and extended fluctuation relations discussed up until now. Other noises behaving asymptotically like Lévy white noise should give rise to similar results, since Lévy distributions are stable [11]. In this sense, the results obtained here should be representative of a large class of fluctuation relations associated with "power-law" noises having infinite variances.…”

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confidence: 72%

“…This characteristic function is the main mathematical result of this paper, from which P (W τ ) is obtained by an inverse Fourier transform. In the present case, there is no need to explicitly compute the inverse Fourier transform because the form of G Wτ (q) shown in (11) already tells us that P (W τ ) is a symmetric Lévy distribution having the same index α as the noise ξ(t) [11]. The parameter M is the value at which P (W τ ) is centered, while V is a measure of the width of P (W τ ); see Fig.…”

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confidence: 99%

“…In this case, P (W τ ) is not Gaussian anymore. Rather, it is a so-called strict Lévy distribution [11] having, as is well known, power-law tails of the form…”

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confidence: 99%

“…This difference in behavior clearly shows that the fluctuations of the work performed on a particle driven by strict Lévy white noise satisfy an entirely different kind of fluctuation relation when compared with its Gaussian noise counterpart. Following a terminology used in studies of Lévy noise [11], we call this new fluctuation relation, satisfying both (18) and (21), an anomalous fluctuation relation.…”

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confidence: 99%

“…Note that Lévy noise, like Gaussian noise, is a proper form of noise, in the sense that its realizations are always finite [11]. Therefore, forcing an object with a Lévy noise having an infinite variance, or even an infinite mean, does not imply that we supply that object with an infinite amount of energy.…”

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Fluctuation relation for a Lévy particle

Touchette

Cohen

2007

Phys. Rev. E

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We study the work fluctuations of a particle subjected to a deterministic drag force plus a random forcing whose statistics is of the Lévy type. In the stationary regime, the probability density of the work is found to have "fat" power-law tails which assign a relatively high probability to large fluctuations compared with the case where the random forcing is Gaussian. These tails lead to a strong violation of existing fluctuation theorems, as the ratio of the probabilities of positive and negative work fluctuations of equal magnitude behaves in a non-monotonic way. Possible experiments that could probe these features are proposed. The study of fluctuations in systems driven in nonequilibrium steady states has concentrated lately on a special symmetry property referred to as the fluctuation theorem or fluctuation relation. The first studies of this symmetry focused on the entropy production of nonequilibrium systems [1,2], but it was soon realized that the same symmetry holds for other quantities of interest, such as the work W τ performed on a nonequilibrium system during a time τ [3]. In the context of this quantity, we define a fluctuation relation [4] as follows. Assuming that W τ is extensive in τ , we say that the probability density P (W τ ) of W τ satisfies a fluctuation relation or, to be more precise, a conventional fluctuation relation ifwhere c is a constant which depends neither on τ nor w.We also say that P (W τ ) satisfies a conventional fluctuation relation if the above equality holds, when properly scaled (see below), in the limit τ → ∞. In that case, we speak of a stationary conventional fluctuation relation.One important problem about fluctuation relations, which is as yet unresolved, is to determine the class of nonequilibrium systems or, more precisely, the class of nonequilibrium observables [5] whose fluctuations satisfy a conventional fluctuation relation. Our understanding of this issue at this point is that this relation holds for two general classes of systems: (i) finite, deterministic systems having a sufficiently chaotic dynamics [2,6], and (ii) finite, stochastic systems whose evolution is a Markov process [7]. There are subtle points to be taken into account, however. In some cases, boundary conditions or special forms of large deviations may restrict the range of validity of conventional fluctuation relations, and in these cases, corrections or extensions of the conventional fluctuation relation have been proposed [8,9].Our goal in this paper is to expand this picture of fluctuation relations by studying a model of a nonequilibrium * Electronic address: ht@maths.qmul.ac.uk system whose work fluctuations neither obey the conventional fluctuation relation nor the extended fluctuation relation of van Zon and Cohen [8]. Based on this model, we propose a novel type of fluctuation relation, and compare it, from the general point of view of large deviation theory, with the conventional fluctuation relation defined in (1). In the end, we also discuss two experiments for which "anomalous"...

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confidence: 99%

“…In this case, P (W τ ) is not Gaussian anymore. Rather, it is a so-called strict Lévy distribution [11] having, as is well known, power-law tails of the form…”

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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