Explaining a changeover from normal to super diffusion in time-dependent billiards

Abstract

The changeover from normal to super diffusion in time dependent billiards is explained analytically. The unlimited energy growth for an ensemble of bouncing particles in time dependent billiards is obtained by means of a two dimensional mapping of the first and second moments of the velocity distribution function. We prove that for low initial velocities the mean velocity of the ensemble grows with exponent ∼ 1/2 of the number of collisions with the border, therefore exhibiting normal diffusion. Eventually, this regime changes to a faster growth characterized by an exponent ∼ 1 corresponding to super diffusion. For larger initial velocities, the temporary symmetry in the diffusion of velocities explains an initial plateau of the average velocity. PACS numbers: 05.45.-a, 05.45.Pq, 05.40.FbAs coined by Enrico Fermi [1] Fermi acceleration (FA) is a phenomenon where an ensemble of classical and non interacting particles acquires energy from repeated elastic collisions with a rigid and time varying boundary. It is typically observed in billiards [2-4] whose boundaries are moving in time [5][6][7][8][9]. If the motion of the boundary is random and the initial velocity is small enough [10], the growth of the average velocity is proportional to n 1/2 , with n denoting the number of collisions. If the initial velocity is larger, a plateau of constant velocity is observed in a plot V vs. n which is explained from the symmetry of the velocity diffusion [11]. The symmetry warrants that part of the ensemble grows and part of it decreases in such a way the growing parcel cancels the portion decreasing. As soon as such symmetry is broken the constant regime is changed to a regime of growth. For deterministic oscillations of the border, the scenario is different. Breathing oscillations preserve the shape but not the area of the billiard. It is known that the average velocity evolves in a sub-diffusive manner with a slope of the order of 1/6 [12,13]. For oscillations preserving the area but not the shape of the billiard there are two regimes of growth. For short time the diffusion of velocities is normal passing to super diffusion regime for large enough number of collisions [14]. This changeover is, so far, not yet explained and our contribution in this letter is to fill up this gap in the theory. This is achieved by studying the momenta of the velocity distribution function, noticing that the dynamical angular/time variables have an inhomogeneous distribution in phase space.The results presented in this letter are illustrated by a time-dependent oval-billiard [15] whose phase space is mixed when the boundary is static. The boundary of the billiard is written as R b (θ, t) = 1 + ǫ [1 + a cos(t)] cos(pθ) where R b is the radius of the boundary in polar coordinates, θ is the polar angle, ǫ controls the circle deformation, p > 0 is an integer number [16] given the shape of the boundary, t is the time and a is the amplitude of oscillation of the boundary. Figure 1 shows a typical scenario of the boundary and three collisions illustr...

Dynamical thermalization in time-dependent billiards

“…Notice for elastic collisions γ → 1, the stagnation speed diverges, which is consistent with the phenomenon of Fermi acceleration, where there is an unlimited growth of energy as the time evolves [9,12,16].…”

“…As detailed in [12], if we make a second-order expansion of the W (V ) around the mean speed V n of the speed distribution function ρ n (V ), we obtain…”

“…2(a,b). As discussed in [12], to determine the behavior of the mean quadratic speed it is not necessary to describe the evolution of the global distribution function, but only to know the evolution of its first momenta. Therefore, replacing the Eq.…”

“…The physics behind the unlimited energy growth is understood and is mainly related to the diffusion of velocities as a function of time [9][10][11][12]. Different regimes of growth are related to different shapes of the speed distribution function.…”

Propriedades Estatísticas e Termodinâmicas de Bilhares Clássicos

“…(3.61). Esse perfil não homogêneo de F (α, θ)é uma característica comum de sistemas mistos [68,69], o que pode sugerir que além da correlação, uma fator importante para a observação da super difusão em bilhares do tipo não breathingé a estrutura mista do espaço de fases.…”