Momentum relaxation of a relativistic Brownian particle

Abstract

The momentum relaxation of a relativistic Brownian particle immersed in a fluid is studied on the basis of the Fokker-Planck equation for the relativistic Ornstein-Uhlenbeck process. An analytical expression is derived for the short-time relaxation rate. The relaxation spectrum has both discrete and continuum components. It is shown that the Fokker-Planck equation under consideration is closely related to the Schrödinger equation for the hydrogen atom. Hence it follows that there is an infinite number of discrete states. The momentum autocorrelation function is calculated numerically for a strongly relativistic particle.

Quasirelativistic Langevin equation

“…Despite these difficulties (and perhaps because of them), many authors prefer to pursue an approach based on a straightforward extension of the nonrelativistic Langevin phenomenology [6][7][8][9][10]. In a simple version, one assumes that the dissipative force on the particle is linear in the particle's velocity V and composes the Langevin equation for the particle's momentum P in the rest frame of the bath in the form [6] dP dt = −ζ V (P ) + ξ(t),…”

Heat Distribution of Relativistic Brownian Motion

“…From a more mathe-matical point of view, these relativistic diffusive systems can be modeled by the Relativistic Brownian motion [29][30][31] which are a particular case of Brownian motion with nonlinear friction term [32]. One well studied case is the Relativistic Orstein-Uhlenbeck [33], where its relaxation properties were studied in [34,35].…”

Nonspectral Relaxation in One Dimensional Ornstein-Uhlenbeck Processes

“…Although the effect of non-spectral relaxation can be observed under quite general conditions, in this Letter we concentrate on simplest, exactly solvable examples of Ornstein-Uhlenbeck processes (OUPs) describing the coordinate of an overdamped particle in a harmonic potential driven by a white noise. Because the OUP generally approximates random processes in the vicinity of a stable stationary point it is a very important analytical tool in many fields of research, from statistical physics [3][4][5], to theoretical neuroscience [6], ecology [7] and economics [8]. Since the assumption of Gaussian fluctuations is violated in many non-equilibrium systems, in recent years the attention has been shifted to non-Gaussian statistics arising from Lévy noise [8,9].…”

“…is given by a Hermitian OperatorĤ [1]. Equation (2) defines a similarity transformation betweenL and −Ĥ and, hence, a transformation ψ =Ŝp from the space of the solutions of the FP equation to the space of the solutions of the Schroedinger-like equation (3). The eigenvalues −λ of the HamiltonianĤ are real valued and the corresponding eigenfunctions ψ λ (x) form a basis in the Hilbert space of square integrable functions.…”

“…Methods of spectral analysis are central in physics, in particular in quantum mechanics and in the theory of oscillations, and are universally employed to the solution of linear problems. Thus the discussion of the spectrum of the FP operator is often the first step in the solution of the FP equation and the investigation of its relaxation properties [1][2][3]. As we proceed to show, this first step might not deliver a complete picture.…”