The ratio of the average particle velocity to the driving force (field strength) is the Lorentz-gas conductivity. To simulate a nonequilibrium steady-state current (particle velocity) in a periodic, two-dimensional, classical Lorentz gas. Together with a Gaussian thermostatting constraint forceF Additional evidence supporting further the above findings is given by examining the GALI(k) indices, which generalize SALI (=GALI(2)) to the case of k>2 deviation vectors and depend on the complete spectrum of Lyapunov exponents of the tangent flow about the reference orbit. This alerts us to the importance of the Deltalambda=lambda(1)-lambda(2) variations in that regime and helps us identify the energy range over which "melting" occurs as a multistage diffusion process through weakly chaotic layers in the phase space of the microplasma. During the melting phase, SALI exhibits a peculiar stairlike decay to zero, reminiscent of "sticky" orbits of Hamiltonian systems near the boundaries of resonance islands.
In its lower energy regime, we first detect macroscopically the transition from a "crystallinelike" to a "liquidlike" behavior, which we call the "melting transition." We then proceed to study this transition using a microscopic chaos indicator called the smaller alignment index (SALI), which utilizes two deviation vectors in the tangent dynamics of the flow and is nearly constant for ordered (quasiperiodic) orbits, while it decays exponentially to zero for chaotic orbits as exp, where lambda(1)>lambda(2)>0 are the two largest Lyapunov exponents. We present results demonstrating the occurrence of changes in the collective dynamics of a Hamiltonian system which describes a confined microplasma characterized by long-range Coulomb interactions. On the contrary, we prove that the PM has a conformally symplectic structure, which is the basis for establishing the hyperbolicity of such a hybrid dynamical system. The analysis developed in this paper shows how the Weyl-flow technique has failed for NEEG, revealing the necessity to develop new strategies in order to obtain hyperbolicity. Despite numerical investigations supporting the idea of their dissipative dynamics, the hyperbolicity of these billiards has not been yet established.
Recently, different mathematical methods have been developed for establishing hyperbolicity in thermostatted billiards, among these, the Weyl-flow and the conformally symplectic structure techniques.This paper deals with analytical investigations on the possible hyperbolic nature of two thermostatted billiards: The nonequilibrium Ehrenfest gas (NEEG) and the pump model (PM). The onset of hyperbolic dynamics (deterministic chaos) in such a billiard has evidenced an interesting stabilization of the transport properties, especially in microporous media. The highly complex nature of the transport in thermostatted billiards has been of interest in the last few decades because of industrial and medical applications.
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