In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Abstract

Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F ¼ ÀgV 2 ; (iii) F ¼ ÀgV l with l 6 ¼ 1 and l 6 ¼ 2 and; (iv) F ¼ ÀgV, where g is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined. We revisit the problem of non-interacting particles in a time dependent Lorentz gas. We describe the model by using a four dimensional nonlinear map. As it is known, the phase space of the Lorentz gas with static scatterers is fully chaotic and the velocity of the particle is constant. However, when a time dependent perturbation is introduced to the boundary, the energy is no longer conserved and the unlimited energy growth is observed for the conservative case. On the other hand, our results show that when dissipation is introduced into the system, either by collisional dissipation or dissipation during the flight, the unlimited energy growth is no longer observed. Depending on the strength of the dissipation and considering short time, the average velocity can either grows and reaches a constant plateau or decays until the particle reaches the state of rest. For the cases, when the dynamics does not stop between the scatterers, the behaviour of the average velocity is described by using scaling arguments and once the scaling exponents are known, classes of universalities are defined.

Analytical description of critical dynamics for two-dimensional dissipative nonlinear maps

“…It is relevant to recall that, by the use of scaling arguments, the universality of the ratio I 2 n /I 2 0 was deduced in [10,12,[37][38][39]; there, n CO was also reported to be proportional to I 2 0 K −2 , in agreement with Eq. (25).…”

A dynamical phase transition for a family of Hamiltonian mappings: A phenomenological investigation to obtain the critical exponents

“…7(b) after the transformations I → I/ α and n → n/ z . The procedure discussed in this section can be applied to other systems where such a scaling was observed also in different billiards [35][36][37][38] and under different situations. The break of symmetry of the probability distribution function can be described using a more robust formalism, particularly related with the diffusion equation [34].…”

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

“…The conjecture itself claims that if chaos is present in the dynamics of a particle in a static version of the billiard, then this is a sufficient -but not necessary -condition to observe Fermi acceleration when a time perturbation to the boundary is introduced. Many different billiards exhibit Fermi acceleration under time perturbation to the boundary including the Lorentz gas [23,24], oval billiard [25], stadium [26] and other shapes [27]. The elliptical billiard is an exception and the LRA conjecture does not apply to it.…”