Abstract
The cumulant representation is common in classical statistical physics for variables on the real line and the issue of closures of cumulant expansions is well elaborated. The case of phase variables significantly differs from the case of linear ones; the relevant order parameters are the Kuramoto-Daido ones but not the conventional moments. One can formally introduce `circular' cumulants for Kuramoto-Daido order parameters, similar to the conventional cumulants for moments. The circular cumulant expansions allow to advance beyond the Ott-Antonsen theory and consider populations of real oscillators. First, we show that truncation of circular cumulant expansions, except for the Ott-Antonsen case, is forbidden. Second, we compare this situation to the case of the Gaussian distribution of a linear variable, where the second cumulant is nonzero and all the higher cumulants are zero, and elucidate why keeping up to the second cumulant is admissible for a linear variable, but forbidden for circular cumulants. Third, we discuss the implication of this truncation issue to populations of quadratic integrate-and-fire neurons [E. Montbri\'o, D. Paz\'o, A. Roxin, Phys. Rev. X, vol. 5, 021028 (2015)], where within the framework of macroscopic description, the firing rate diverges for any finite truncation of the cumulant series, and discuss how one should handle these situations. Fourth, we consider the cumulant-based low-dimensional reductions for macroscopic population dynamics in the context of this truncation issue. These reductions are applicable, where the cumulant series exponentially decay with the cumulant order, i.e., they form a geometric progression hierarchy. Fifth, we demonstrate the formation of this hierarchy for generic distributions on the circle and experimental data for coupled biological and electrochemical oscillators.
Highlights
The problem of cumulant representation and closure of cumulant expansions is likely one of the most generic problems in nonequilibrium statistical physics
For a variable on the circle we have derived that the circular cumulant series has either one nonzero element or an infinite number of them
With two or a larger but finite number of nonzero elements, the high-order Kuramoto-Daido order parameters eimφ tend to infinity, while their absolute value is not allowed to exceed 1. This should be taken into account when one deals with the systems governed by nontrivial macroscopic variables like the firing rate in a network of quadratic integrate-and-fire neurons [32]
Summary
The problem of cumulant representation and closure of cumulant expansions is likely one of the most generic problems in nonequilibrium statistical physics. Closure of the equation chain is a trivial task: For incompressible flows one makes this closure by assuming a constant density and discarding the energy equation (the second cumulant); for compressible flows the third cumulant is neglected and an algebraic relation between pressure and internal energy is adopted These closures are correct for weakly nonequilibrium systems (mathematically, the limit of small Knudsen number), which is relevant for most macroscopic processes on the earth’s surface with excellent accuracy. In statistical physics, one can address the problem of a macroscopic description for exotic systems, where particles are not actual molecules or atoms but macroscopic objects: grains, stones, asteroids, etc For these systems, may the limit of small Knudsen number not be relevant, and the reversibility of interparticle collisions is lost, leading to essentially non-Gaussian distributions of the microscopic velocity [2]. We report the presence of this progression for important generic distributions and demonstrate it with experimental data for coupled biological and electrochemical oscillators
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