Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

Abstract

Abstract. We consider the system of N (≥ 2) hard balls with masses m 1 , . . . , m N and radius r in the flat torusWe prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection (m 1 , . . . , m N ; L) of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case ν = 2. The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.

Lyapunov Modes in Hard-Disk Systems

“…In our simulations we keep the spanning vectors orthonormal. Since (in the experiment) the two fields ϕ L and ϕ P are also (nearly) orthogonal 20 , the matrix Q(t) is close to a rotation matrix (that is c −b, d a, and a 2 + b 2 1). Therefore, the dynamics in the two-dimensional subspace is well described by a phase φ(t) = arctan(b(t)/a(t)).…”

“…The definitions given above allow us to apply the same concepts also to the LP-dynamics of systems with reflecting boundaries, although they do not show traveling but standing waves. 20 The scalar product ϕ L · ϕ P = N j =1 cos(k x q j,x ) sin(k x q j,x )p j,x a priori does not vanish. However, as the simulations show, it is of the same order as N j =1 cos(k x q j,x ) sin(k x q j,x ), which is also small and non-vanishing due to the uneven spacing of the particles.…”

Eigenstate thermalization hypothesis

“…To be more precise, ergodicity has only been proven rigorously in some special cases that limit the number of spheres, or for systems where all of the masses are arbitrary, and then with the caveat that the proof will not hold for a zero measure set of mass ratios[23][24][25].…”

Book Review: Chaotic billiards

“…However, in attempting to do that, one encounters "enormous technical difficulties", as the authors correctly write, and this book does a very good job in presenting a clear and detailed exposition of fundamentals of these techniques [28], [29], [25]. On the other hand, for billiards in polygons the fundamental mirror formula (2) gives nothing because the curvature of wave fronts does not change at the reflections.…”

“…This enormously complicates the analysis of this system. However, the developments in recent years in the sharpening of these techniques gives hope that the full proof of BH is within reach [24], [25]. For nonuniformly hyperbolic dynamical systems, a theory has been developed which allows one to deduce from ergodicity stronger ergodic properties, such as mixing, K-property and B-property [29], [28], [20], [15].…”