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Checking the validity of truncating the cumulant hierarchy description of a small system

We analyze the behavior of the first few cumulant in an array with a small number of coupled identical particles. Desai and Zwanzig (J. Stat. Phys., {\bf 19}, 1 (1978), p. 1) 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.


I. 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 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 Γ(λ) = exp(iλY (t)) can be expressed as Γ(λ) = exp(P (λ, t)) where P (λ, t) is a polynomial of first or second degree in λ.
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 nonlinear stochastic differential equations for the degrees of freedom characterizing the elements of the array. Equivalently, we can describe the system in terms of an N -dimensional joint probability distribution. We will assume that this joint probability distribution satisfies an N -dimensional linear Fokker-Planck equation (FPE).
Obtaining dynamical information from the FPE is plagued with difficulties due to the nonlinear character of the dynamics. As mentioned above, a way of dealing with them is to consider the cumulant moments satisfying an infinite hierarchy of ordinary differential equations. In the case of the arrays studied in this work, Desai and Zwanzig 4 derived from the FPE such an infinite hierarchy. Two types of cumulants appear: diagonal cumulant moments associated to a single degree of freedom, M i (t), and cross cumulant moments, µ ijk... (t) involving two or more variables. For an infinite system, the off-diagonal cumulant moments are negligible if they were zero at the initial preparation (see 4 ). In other words, off-diagonal cumulant moments are not generated by dynamical evolution in the case of infinitely large systems. But even then, for all practical purposes, the hierarchy has to be approximated by truncating it at a certain level. The results of truncating at different levels the hierarchy of diagonal cumulant moments in the N → ∞ limit was already discussed by Desai and Zwanzig. As also indicated in 3 , the cumulant truncation is a succesful strategy in that limit.
By contrast with the previous work of Desai and Zwanzig, the present work focuses on systems with a small number N of elements. In this case, both the diagonal and off-diagonal cumulant moments have to be taken into account. It is then an open question whether truncating the infinite hierarchy is an adequate approximation strategy. It is not possible to solve analytically the truncated set of equations at any level. So we will rely on a numerical treatment of the set. Our goal is to elucidate whether truncation of the infinite hierarchy of cumulant moments at different levels provides a reliable approximation for small finite arrays. To this end, we will compare the results for the first few cumulant moments obtained from the truncated hierarchy with those obtained by numerically solving the set of the N coupled Langevin equations. Besides the numerical simulations of the Langevin dynamics, the results of the truncated hierarchy will also be contrasted with the exact results expected from the stationary solution of the N -dimensional Fokker-Planck equation, as well as the global stability H-theorem. We will see that the simulation results agree with the exact known information, while the truncation strategy might lead to results incompatible with the exact results.

II. THE MODEL
We consider a global coupling model that can be viewed as a set of nonlinear "oscillators", each of them described by a "coordinate" x i . The dynamics of the system is given by the set of coupled Langevin equations (in dimensionless form) where θ is the strength of the global coupling term. This parameter will be taken to be either positive or negative. The terms ξ i (t), i = 1, 2, . . . N represent N uncorrelated Gaussian white noises with zero averages and ξ i (t 1 )ξ j (t 2 ) = 2Dδ ij δ(t 1 − t 2 ). This model was introduced by where U is the potential energy relief, with the single particle potential The term V (x i ) describes a symmetrical potential with two wells of equal depths separated by a hump at x i = 0. The interaction part of the full potential modifies it in such a way that for θ > 1 the two wells blend into a single minimum at x i = 0. For θ < 1, the two wells exist, but their locations and the barrier height depend on θ. Note that for θ > 0, the interaction energy contribution to the full potential favors that any pair x i and x j should have the same sign (both either positive or negative), while for θ < 0, the opposite happens and the interaction tends to favor configurations with positive and negative values of the variable.
The only explicit solution of Eq. (2) is its long time stationary one, given by where Z is a normalization function. This solution is independent of the initial condition, and the system will necessarily relax to it, although the time it takes to do it might be, depending on the system parameters, extremely large. It should be noted that we keep the finite number of particles N fixed, while we take the long time limit. Had we have taken the limit N → ∞ first, as done in the infinite size limit studies, and afterwards the long time limit, the result would differ. while the zero value becomes unstable. The regions are separated by a transition line whose shape depends on whether θ > 1 or 0 < θ < 1. As we will see in the next section, the truncated hierarchy of cumulants for small systems might yield several stable stationary moments depending upon the initial preparation, as obtained within the N → ∞ limit. Nonetheless, those results are incompatible with the unicity of the solution of a multidimensional linear FPE and with the independence from initial condition of the long-time results, required by the H-theorem.

III. NUMERICAL SIMULATIONS
We have carried out numerical simulations of the whole set of Langevin equations, Eq. (1). Using the procedure detailed in Ref. 6 , we have integrated the Langevin equations for a large number of noise realizations (typically 5000 realizations). Averaging over them, we estimate the first two cumulant moments of a single variable by and where N indicates the total number of trajectories and x (α) (t) indicates the numerically obtained single particle trajectory in the α noise realization.
Let us first investigate what happens for parameter values such that, in the infinite size limit, Desai and Zwanzig obtained a single stable stationary first moment. In Fig. 1  indicating the limitations of the truncated hierarchy.
We now turn our attention to the model with a negative global coupling parameter θ < 0.

IV. CONCLUSION
In conclusion, the work presented here indicates that care must be taken when using an approximation to the dynamical behavior in a chain of interacting identical objects, based on truncating the infinite hierarchy of cumulant moments. Even for very large systems, if the parameter values considered are such that the truncated hierarchy leads to two stable coexisting solutions, the approximation is not correct. The results of the numerical simulations of the Langevin equations and the exact properties of the Fokker-Planck equation for finite systems of any size indicate that the coexistence of two stationary solutions is an artifact of the truncation. On the other hand, when the truncated hierarchy has a single stationary stable solution, it provides a reliable approximation of the system dynamics even for systems of very modest size. Although our conclusions are based on the study of a particular model, we think that they are qualitatively relevant for other cases. Langevin dynamics with additive noise are good general representations of dynamical systems in contact with a thermal environment. Actually, Langevin dynamics are more realistic than their deterministic limits often used to study dissipative nonlinear systems, where dissipation is included but all fluctuations are neglected. Global interactions are also quite general. The quartic nonlinearity we study here is a particular case. But having a different nonlinearity should not change our qualitative conclusions. The nonlinearity considered includes already the nontrivial effects associated to the dynamical coupling between noise and nonlinearity.