Universal limits of nonlinear redistribution processes

and their applications

R.Teodorescu

University of South Florida

Tampa, FL 33620-5700, USA

Deriving the time evolution of a distribution of probability (or a probability density matrix) is a problem encountered frequently in a variety of situations: for physical time, it could be a kinetic reaction study, while identifying time with the number of computational steps gives a typical picture of algorithms routinely used in quantum impurity solvers, density functional theory, etc. Using a truncation scheme for the expansion of the exact quantity is necessary due to constraints of the numerical implementation. However, this leads in turn to serious complications such as the Fermion Sign Problem (essentially, density or weights will become negative). By integrating angular degrees of freedom, and reducing the dynamics to the radial component, the time evolution is reformulated as a nonlinear integral transform of the distribution function. A canonical decomposition into orthogonal polynomials leads back to the original sign problem, but using a characteristic-function representation allows us to extract the asymptotic behavior, and gives an exact large-time limit, for many initial conditions, with guaranteed positivity.

 

The notion of coarse graining in statistical physical models (or field theory), introduced by Migdal and Kadanoff [1, 2], is essential to many fundamental concepts and results, like the continuum limit of lattice models, or the universality of scaling behavior near a phase transition, to name two of the most celebrated consequences.

From the perspective of mathematical statistics, the analysis of such coarse graining processes is straightforward due to the fact that the elementary operation of the process is averaging: at step , we create a new random variable from two variables defined at step -1 by . Then by repeated application of this operation, the result after sufficiently many steps is simply given by the central limit theorem (CLT).

In this work, we consider another class of coarse-graining processes, where at each step we take the absolute difference between variables, rather than their average. This is justified by a number of relevant physical problems, but also by classical issues from decision theory or economics.

As in the case of standard coarse graining, there is a limiting distribution (in fact, a whole class) which will be reached after arbitrarily many steps. Unlike in the standard case, the particular limit is chosen from this class based on the asymptotic properties of the initial distribution (more precisely, the radius of convergence of its moment-generating function). This is an example of the extreme selection criterion, which characterizes other important stochastic processes, such as the Fisher-Kolmogorov evolution.

The structure of this paper is the following: in the first section we present the difference coarse-graining procedure, as well as some of its realizations, and derive its universal long-time asymptotic behavior. In the second section, we consider some particular types of distributions which are useful examples for the general results. The last section is a discussion on possible applications of this new universal limiting behavior.

In research it is shown, that coarse graining of difference processes represents a fundamental generalization of standard coarse graining with averaging. This procedure is a natural description for relevant phenomena ranging from multi-species stochastic processes to socio-economics. In this paper, we have identified the class of stead-states for this process, and shown how a particular, universal limiting distribution is chosen, as well as the convergence rate towards the steady-state.

 

1.       A.A.Migdal. Phase transitions in gauge and spin-lattice systems. Soviet Journal of Experimental and Theoretical Physics, 42:743, October 1975.

2.       L.P.Kadanoff. Notes on Migdal's recursion formulas. Annals of Physics, 100:359--394, September 1976.

3.       P.L.Krapivsky and E.Ben-Naim. Aggregation with multiple conservation laws. Phys. Rev. E, 53:291--298, January 1996.

4.       C.M.Bender and E. Ben-Naim. FAST TRACK COMMUNICATION: Nonlinear integral-equation formulation of orthogonal polynomials. Journal of Physics A Mathematical General, 40:F9-F15, January 2007.