Diffusion phenomena in a mixed phase space

Abstract

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in phase space, the method of images and use of the classical diffusion equation. The method is demonstrated on the simplified Fermi-Ulam accelerator model, which has a mixed phase space with chaotic seas, invariant tori and Kolmogorov-Arnold-Moser (KAM) islands. The calculated average velocities agree well with numerical simulations and with an earlier empirical theory. The procedure can readily be extended to other systems including time-dependent billiards.

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

“…At such windows of I 0 and n, an additional crossover is observed when n ′ x ∼ = 2 I 2 0 ǫ 2 . This crossover had already been observed in [1] when a phenomenological approach was proposed and confirmed analytically in [19].…”

A Survey on Macroeconomic Data in the Eurozone and a Control Dashboard Model Based on the Kam and Nekhoroshev Theorems and the Hénon Attractor

“…The conjugated action of economic and monetary policy, at the Eurozone level, expresses a macroeconomic dynamic that can be represented by quasi-periodic motions, at least for limited levels of perturbations. The aim of this contribution is to model this phenomenon, first by borrowing the theoretical KAM emanations (Kolmogorov-Arnold-Moser) (Wang et al, 2018), (Palmero and Diaz, 2020).…”

Estudo do comportamento da entropia em bilhares

“…The growth over time (t) of the standard deviation (σ = Ct ∆ ) is characterized by two variables, a diffusion coefficient (C) and a diffusion exponent (∆), the most important being the exponent since it allows us to distinguish between anomalous diffusion, when the exponent is different from 1/2, and normal diffusion, when the exponent is 1/2. Even though this method gives good results in some cases, like in an almost completely chaotic sea [22,23], it fails in some other cases, since it is difficult to find the appropriate generalized momentum that it is better suited to describe a specific behaviour in phase space [24,25]. A particular physical scenario where the momentum method fails, is when exists coexistence of dynamics, regions of regular trajectories arranged in complex structures called Kolmogorov-Arnold-Moser (KAM) islands [26,27,28], surrounded by a sea of chaotic trajectories.…”

“…From [50,22,23] we know that for an ensemble of points near to V = and small number of iterations, the diffusion process is mainly uniform in the V direction with diffusion exponent δ near to 0.5. Of all curves shown in figure 3.3 the ones that give a δ closer to 0.5 in the diffusive regime, are shown in figure 3 In figure 3.4 we show the entropy curves that give a fitted δ near to 0.5, as expected from theory [50,22,23]. These curves are the ones with I, J = 1024 (that gives δ = 0.527) and I, J = 512 (that gives δ = 0.489).…”

“…From [23], we know that the standard deviation of the fluctuations in the V direction is √ 2 , from [22] we know that the diffusion range goes from V min = 0 to V max = 2 √ . The division 2 √ √ 2 , i. e. diffusion range over fluctuation size, gives the value 44.72, then, when I, J are one order bigger than this value (I, J = 512 in this case), the grid box size is appropriate to calculate the diffusion exponent.…”