m-Particle correlations in theN-particle McKean model

Abstract

This work extends the comparison between exact and approximate solutions of the McKean model to finite particle numbers. We derive the coupled linear equations of motion for them-body densities (BBGKY hierarchy) and the corresponding nonlinear equations for them-body correlation functions. We calculate the stable fixed points and the subspace admitting a probabilistic interpretation for both descriptions of the model. Neglecting higher correlations withm>n, we obtain approximate solutions, which are compared to the exact one. In this way various truncation effects can be studied, such as the appearance of saddle points and unphysical trajectories. Finally, we linearize the truncated equations for the correlations about the stable fixed point, and calculate the relaxation times up toO(N ⁻¹).