Checking the Validity of Truncating the Cumulant Hierarchy Description of a Small System
We analyze the behavior of the first few cumulant moments in an array with a small number of coupled identical particles. Desai and Zwanzig (J Stat Phys 19(1):1 1978) studied noisy arrays of nonlinear units with global coupling and derived an infinite hie
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tract We analyze the behavior of the first few cumulant moments in an array with a small number of coupled identical particles. Desai and Zwanzig (J Stat Phys 19(1):1 1978) studied noisy arrays of nonlinear units with global coupling and derived an infinite hierarchy of differential equations for the cumulant moments. They focused on the behavior of infinite size systems using a strategy based on truncating the hierarchy. In this work we explore the reliability of such an approach to describe systems with a small number of elements. We carry out an extensive numerical analysis of the truncated hierarchy as well as numerical simulations of the full set of Langevin equations governing the dynamics. We find that the results provided by the truncated hierarchy for finite systems are at variance with those of the Langevin simulations for large regions of parameter space. The truncation of the hierarchy leads to a dependence on initial conditions and to the coexistence of states which are not consistent with the theoretical expectations based on the multidimensional linear Fokker-Planck equation for finite arrays.
1 Introduction The description of nonlinear stochastic systems can hardly be carried out without approximations due to the interplay of noise and nonlinearity. In some problems, the stationary distribution for the relevant variables is available in analytical form, but in general very little information can be obtained without approximations. A convenient way of describing the system dynamics is in term of cumulant moments satisfying an infinite set of coupled ordinary differential equations [1]. For all practical purposes, this infinite hierarchy needs to be truncated in order to M. Morillo () J. Gómez-Ordóñez J.M. Casado Facultad de Física, Área de Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain e-mail: [email protected] R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__19, © Springer International Publishing Switzerland 2014
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obtain a finite set of closed equations. A very much used approximation consists in truncating the infinite hierarchy at the Gaussian level by neglecting cumulants of order three and higher. For a stochastic variable Y .t/, Marcinkiewicz [2] indicated that the characteristic function ./ D hexp.i Y .t//i can be expressed as ./ D exp.P .; t// where P .; t/ is a polynomial of first or second degree in . Consequently, as pointed out by Hänggi and Talkner [3], the truncation of the hierarchy at levels higher than two is problematic, although, as these authors emphasize, it is not an empty concept. In their own words “it only means that the neglect of cumulants beyond a given order cannot be justified a priori”. In this work we consider the statistical mechanical description of a stochastic array containing a small finite number N of coupled identical elements. The dynamics will be given by a set of N coupled n
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