RESUMO

We introduce the violation fraction υ as the cumulative fraction of time that a mesoscopic system spends consuming entropy at a single trajectory in phase space. We show that the fluctuations of this quantity are described in terms of a symmetry relation reminiscent of fluctuation theorems, which involve a function Φ, which can be interpreted as an entropy associated with the fluctuations of the violation fraction. The function Φ, when evaluated for arbitrary stochastic realizations of the violation fraction, is odd upon the symmetry transformations that are relevant for the associated stochastic entropy production. This fact leads to a detailed fluctuation theorem for the probability density function of Φ. We study the steady-state limit of this symmetry in the paradigmatic case of a colloidal particle dragged by optical tweezers through an aqueous solution. Finally, we briefly discuss possible applications of our results for the estimation of free-energy differences from single-molecule experiments.

Assuntos

RESUMO

We define the violation fraction ν as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends on both the number of degrees of freedom N and the duration of the time interval τ. In the large-τ and large-N limit we show that, for ergodic and microreversible systems, the mean value of ν scales as 〈ν(N,τ)〉 ∼ (τN(1/1+α))(-1). The exponent α is positive and generally depends on the protocol for the external driving forces, being α = 1 for a constant drive. As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. In this case we show that the scaling of 〈ν〉 with N and τ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.

RESUMO

We extend the definition of nonadiabatic entropy production given for Markovian systems by Esposito and Van den Broeck [Phys. Rev. Lett. 104, 090601 (2010)], to arbitrary non-Markov ergodic dynamics. We also introduce a notion of stability characterizing non-Markovianity. For stable non-Markovian systems, the nonadiabatic entropy production satisfies an integral fluctuation theorem, leading to the second law of thermodynamics for transitions between nonequilibrium steady states. This quantity can also be written as a sum of products of generalized fluxes and forces, thus being suitable for thermodynamics. On the other hand, the generalized fluctuation-dissipation relation also holds, clarifying that the conditions for it to be satisfied are ergodicity and stability instead of Markovianity. We show that in spite of being counterintuitive, the stability criterion introduced in this work may be violated in non-Markovian systems even if they are ergodic, leading to a violation of the fluctuation theorem and the generalized fluctuation-dissipation relation. Stability represents then a necessary condition for the above properties to hold and explains why the generalized fluctuation-dissipation relation has remained elusive in the study of non-Markov systems exhibiting nonequilibrium steady states.

RESUMO

We study the probability distribution function (PDF) of the position of a Lévy flight of index 0 < α < 2 in the presence of an absorbing wall at the origin. The solution of the associated fractional Fokker-Planck equation can be constructed using a perturbation scheme around the Brownian solution (corresponding to α = 2) as an expansion in ε = 2-α. We obtain an explicit analytical solution, exact at the first order in ε, which allows us to conjecture the precise asymptotic behavior of this PDF, including the first subleading corrections, for any α. Careful numerical simulations, as well as an exact computation for α = 1, confirm our conjecture.

Assuntos

RESUMO

We study, using exact numerical simulations, the statistics of the longest excursion l(max)(t) up to time t for the fractional Brownian motion with Hurst exponent 0 proportional to variantQ(infinity)t, where Q(infinity) identical with Q(infinity)(H) depends continuously on H. These results are compared with exact analytical results for a renewal process with an associated persistence exponent theta=1-H. This comparison shows that Q(infinity)(H) carries the clear signature of non-Markovian effects for H not equal 1/2. The preasymptotic behavior of is also discussed.

RESUMO

Any decomposition of the total trajectory entropy production for Markovian systems has a joint probability distribution satisfying a generalized detailed fluctuation theorem, when all the contributing terms are odd with respect to time reversal. The expression of the result does not bring into play dual probability distributions, hence easing potential applications. We show that several fluctuation theorems for perturbed nonequilibrium steady states are unified and arise as particular cases of this general result. In particular, we show that the joint probability distribution of the system and reservoir trajectory entropies satisfy a detailed fluctuation theorem valid for all times.