Typicality analysis for the NewtonianN-body problem on in the limit

Abstract

A two-dimensional version of the classical NewtonianN-body problem isstudied in which N identical particles move on the two-sphere and interact mutually with an attractive logarithmic pair potential. It is assumed that for largeN the only relevant integrals of motion are the HamiltonianH and the angularmomentum vector L of the system, where the vector notation refers to the embedding . The typical behavior of the N-body system is computed from Boltzmann’s ergodic (sub)ensemble in the limit with values of the integrals of motion scaling as follows: and in leading order when N is large. It is shown that the ergodic sub-ensemble entropy has the asymptotic expansion ; here, the NlnN term is combinatorial in origin, while is the system-specific Boltzmann entropy per particle. It is also shown that is the minimum of Boltzmann’sH function(al) for the kinetic density functionf of a perfect self-gravitating gas of energyε and angularmomentum vector λ. Furthermore it is shown that the minimizers of are typical states for the underlyingN-body systems when . A symmetry-breaking second-order phase transition between different kinds of typical states is foundfor λ = 0 at a critical energy which can be computed by the famous Jeans criterion, and for by abstract arguments. Lastly it is stressed that orbits of dynamical typical states may exist when|λ| is sufficiently large. No regularization of the ergodic sub-ensemble measure is invoked, andthe construction is valid irrespectively of whether sub-ensemble equivalence holds ornot. Boltzmann’s holodic and ergodic ensembles are treated also, as warm-upproblems.