Poincaré models, rigorous justification of the second element of thermodynamics on the basis of mechanics, and the Fermi acceleration mechanism

Abstract

Illinois 601 15. USC M S . received 14th M a y 1970Ahtract. Starting from bare-ion pseudopotentials, net crystal potentials are evalutiied in direct space by (a) the usual linear screening technique and (b) the Thomas-Fermi method. For the metals considered (Na and AI), the potentials are quite similar, crossing outside the cores and never differing by more than 0.02 a.u. in the case of Na nor by more than 0.2 a.u. in the case of Al. The similarity of the results based on (a) and (b) suggest the use of the pseudopotential concept in conjunction with the rather general method (b) under conditions when (a) is questionable, that is whenever the zeroth order uniform electron gas solution is a poor physical approximation to the actual system under study (specific examples being molecules, insulating solids and interfaces).Methods (a) and (bj both yield shallower wells than those most often used as input information in band structure calculations and obtained by superposing atomic data.

Numerical test of Born–Oppenheimer approximation in chaotic systems

“…Besides the considerations of various regimes, several varieties of the Fermi accelerator have also been proposed, including relativistic models, of the type which embodies thermodynamic considerations [15] and mechanical Fermi accelerator with damping [16]. Our model can however be contrasted with the previous ones in terms of energy conservation.…”

“…See [15]) Accordingly, this model may be called an 'Autonomous Fermi Accelerator'(AFA), and the property of energy conservation enables us to calculate the exact eigenstates. With this set of eigenstates, we analyze the dynamics of the AFA and derive the condition for the validity of the BornOppenheimer approximation in the chaotic regime.…”

Chaotic Dynamics in an Impact Problem

“…The proof goes by induction onn using (10). Writing Q = (j + 1) + (Q − j − 1) and applying (16) with n = j + 1 and n = Q − j − 1 we get the desired estimate…”

“…This is the map (up to changes of systems of references) considered by Pustyl'nikov [15,16]. He proved that if the motion of the racket is periodic and there exists an instant in which the velocity is sufficiently large, precisely if…”

Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

“…The study of instability properties of oscillators was pioneered in [Lit66a,Lit66b,Lev92]. Other papers establishing instability in oscillators are [AO98,Ort97,LY97,Ort04] The papers [Pus77a,Pus77b,Dou89,KPT95,Pus95] study instability in systems with collisions. The papers [Dou88,DLC83] construct examples of instability near elliptic points.…”