Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach

Leonel, Edson D.; Galia, Marcus Vinicius Camillo; Barreiro, Luiz Antônio; Oliveira, Diego F. M.

doi:10.1103/physreve.94.062211

Cited by 12 publications

(12 citation statements)

References 35 publications

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“…The explanation of the initial plateau is related to the behavior of the velocity distribution function [17]. For an initial velocity larger than the maximum moving wall velocity, say V max = aǫ, the following is observed: (1) Part of the ensemble of particles acquires energy leading to such portion of the ensemble to grow energy; (2) However, another part of the ensemble leads to decreasing of velocity.…”

mentioning

confidence: 99%

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

Hansen¹,

Ciro²,

Caldas³

et al. 2018

EPL

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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...

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confidence: 99%

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

Hansen¹,

Ciro²,

Caldas³

et al. 2018

EPL

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“…As discussed in Ref. [24], the behavior of the V rms (n) can be summarized as: (i) for short n, V rms (n) ∝ n β ; (ii) for large enough n, it is observed that V sat ∝ (1−γ) α1 (ηǫ) α2 , (iii) finally the crossover iteration number is written as n x ∝ (1 − γ) z1 (ηǫ) z2 . Doing the same procedure we made along on the paper we end up with the following set of critical exponents α 1 = −0.5, z 1 = −1, α 2 = 1, z 2 = 0 and β = 0.5 as earlier obtained in Ref.…”

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confidence: 94%

“…Doing the same procedure we made along on the paper we end up with the following set of critical exponents α 1 = −0.5, z 1 = −1, α 2 = 1, z 2 = 0 and β = 0.5 as earlier obtained in Ref. [24] by using the thermodynamical approach.…”

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Application of the diffusion equation to prove scaling invariance on the transition from limited to unlimited diffusion

Leonel,

Kuwana,

Yoshida

et al. 2020

Preprint

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The scaling invariance for chaotic orbits near a transition from unlimited to limited diffusion in a dissipative standard mapping is explained via the analytical solution of the diffusion equation. It gives the probability of observing a particle with a specific action at a given time. We show the diffusion coefficient varies slowly with the time and is responsible to suppress the unlimited diffusion. The momenta of the probability are determined and the behavior of the average squared action is obtained. The limits of small and large time recover the results known in the literature from the phenomenological approach and, as a bonus, a scaling for intermediate time is obtained as dependent on the initial action. The formalism presented is robust enough and can be applied in a variety of other systems including time dependent billiards near a transition from limited to unlimited Fermi acceleration as we show at the end of the letter and in many other systems under the presence of dissipation as well as near a transition from integrability to non integrability.

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“…In a regular situation, a gas of non-interacting particles with an initial low temperature T 0 will increase its energy if introduced in a, pre-viously empty, recipient with walls at ambient temperature T a > T 0 . The opposite will happen if the gas is at an initially larger temperature T 0 > T a [13]. This thermalization process, in general, manifests as a monotonic change in temperature as time advances, leading to an asymptotic state of thermal equilibrium.…”

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confidence: 99%

Dynamical thermalization in time-dependent billiards

Hansen

Ciro

Caldas

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Numerical experiments of the statistical evolution of an ensemble of non-interacting particles in a time-dependent billiard with inelastic collisions, reveals the existence of three statistical regimes for the evolution of the speeds ensemble, namely, diffusion plateau, normal growth/exponential decay and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Further, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for thermalization, where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space towards a stationary distribution function with a kind of reservoir temperature determined by the boundary oscillation amplitude and the restitution coefficient.

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Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach

Cited by 12 publications

References 35 publications

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

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

Application of the diffusion equation to prove scaling invariance on the transition from limited to unlimited diffusion

Dynamical thermalization in time-dependent billiards

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