Communications in Nonlinear Science and Numerical Simulation
Short communicationFractional dynamics in the Rayleigh’s piston
Introduction
During the last decades several papers addressed a conceptual example of statistical mechanics known as the “Rayleigh piston” [1], [2]. This classical prototype system consists of a one-dimensional array of particles separated by means of an adiabatic piston. The particles in the two cylinders have non-zero random velocities and collide sporadically with the piston provoking its motion. While a very simple system, a kind of conceptual paradox occurs and considerable debate took place about the steady state operating conditions. Nevertheless, most of the technical literature addresses the relationship of the system final equilibrium conditions and the study of the complex dynamics has not attracted relevant attention.
This paper focus the dynamics of the Rayleigh piston in the perspective of Fractional Brownian motion (fBm) and Fractional Calculus (FC). The fBm was introduced by Kolmogorov [3]. Later Mandelbrot adopted the concept of fBm to model phenomena with self-similarity and long range effects [4]. The fBm is also called 1/f noise [5], where f denotes frequency, because its spectrum is given by 1/f α, α > 0. The fBm is interpreted as a signature of complexity [6] and has been observed in many distinct areas [7], namely in economics and finance [8], [9], geophysics [10], [11], [12], [13], [14], [15], music and speech [16], [17], [18], biology [19], [20], [21], [22], [23] and others. During the last years the relation between fBm and FC was studied by some researchers [24], [25], [26], [27]. FC emerged with the ideas of Leibniz and several important mathematicians contributed to its development [28], [29], [30], [31], [32]. However, only in the last decades [33], [34] FC was recognized to be an important tool to study systems with long range memory phenomena [35], [36], [37], [38], [39], [40], [41], [42], [43], [44]. FC generalizes the operations of integration and differentiation to non-integer orders and constitutes an efficient mathematical tool for describing natural phenomena with long-range memory effects and power law description. This paper addresses the Rayleigh piston and its characterization by means of fBm and FC concepts.
Having these ideas in mind, this paper focus on the fBm in the perspective of FC and is organized as follows. Section 2 introduces the “Rayleigh piston”, develops the analysis in the Fourier domain, extracting several power-law parameters, and discusses the results in the perspective of dynamical systems. Finally, Section 3 outlines the main conclusions.
Section snippets
Preliminary concepts
The Rayleigh’s piston is a system consisting of two cylinders, to be denoted as 1 and 2, containing some type of fluid, and separated by an adiabatic movable piston (Fig. 1). A brake maintains the piston at rest until time . The two fluids are in equilibrium with pressure, volume and temperature {pi(0), Vi(0), Ti(0)}, . The piston with mass M undergoes random one-dimensional collisions with particles of mass m. Furthermore, there are ni, particles per unit volume, with Maxwell
Conclusions
This paper studied the dynamical properties of Rayleigh piston. The novel contribution was in the viewpoint of fBm and FC. Several numerical experiments with distinct values for the system parameters, such as number of particles, their masses and their velocities, were conducted. The transient and steady-state behavior was characterized in the Fourier domain by means of power law approximations. The results demonstrated that fBm and FC are useful tools for investigating the complexity present
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