First-passage time for upper bounds on fluctuations of trajectory observables

What is it about?

This article examines the properties of fluctuations in statistical systems. It has recently been established that thermodynamic uncertainty relations provide lower bounds on the magnitude of fluctuations (random deviations from average values). Were also proven the existence of general upper bounds on the size of fluctuations of any linear combination of fluxes (including all time-integrated currents or dynamical activities) for continuous-time Markov chains. These “inverse thermodynamic uncertainty relations” (iTURs) are valid for all times and not only in the long-time limit. The paper investigates the first-passage time (FPT) of the observable A, which are obtained by calculating the moments and FPT of the process of upper bound. Particular attention is paid to the problem of first-passage time for processes with independent increments in risk theory. The article obtained the average values and variances of general upper bounds on the size of fluctuations and of any linear combination of fluxes (including all time-integrated currents or dynamical activities) for continuous-time Markov chains. But the random variables A themselves can reach arbitrary values x (with certain probabilities), exceeding the average values. Therefore, in the expressions for the average values and dispersion of FPT on the size of fluctuations A, arbitrary values of the levels of achievement x appear, both positive and negative. It is possible to express the average values and variances of general upper bounds on the size of fluctuations A and general upper bounds of FPT through changes in entropy.

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Why is it important?

This paper examines the possibilities of achieving both positive and negative levels for the process A(t), which characterizes the size of the fluctuations limited from above. It is shown that FPT for the upper fluctuation boundaries are minimal. Physically, this is explained by the fact that with increasing fluctuations, the random process reaches the boundary values faster. The moments of the fluctuation size A and their FPT provide complete and direct information about the size and behavior of the fluctuations, as does the rate limiting function associated with the dynamical entropy or the relative error. Finding FPT for the process of upper bound is a nontrivial task. Perhaps more interesting than the moments of fluctuation size are the FPT achievements of positive and negative levels. Just as FPTs have a large number of applications in a wide variety of fields, upper bounds for FPTs should prove important in a wide variety of applications. From Figs. of the article, it is clear that the average values of FPT reach positive and negative levels quickly decrease with increasing thermodynamic quantity γ, conjugate to FPT. Previously a statistical distribution is introduced and considered that contains the internal energy of the system and the FPT as random thermodynamic variables. The parameter γ is present in this distribution as a thermodynamic parameter conjugate to the FPT. At γ=0 the system is in equilibrium with the Gibbs distribution. As the parameter γ increases the system moves away from equilibrium. The parameter γ can be considered as a measure of distance from equilibrium. As one moves away from equilibrium, fluctuations increase and their FPT decreases.

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