Brownian motion of a classical harmonic oscillator in a magnetic field

Abstract

In this paper, the stochastic diffusion process of a charged classical harmonic oscillator in a constant magnetic field is exactly described through the analytical solution of the associated Langevin equation. Due to the presence of the magnetic field, stochastic diffusion takes place across and along the magnetic field. Along the magnetic field, the Brownian motion is exactly the same as that of the ordinary one-dimensional classical harmonic oscillator, which was very well described in Chandrasekhar's celebrated paper [Rev. Mod. Phys. 15, 1 (1943)]. Across the magnetic field, the stochastic process takes place on a plane, perpendicular to the magnetic field. For internally Gaussian white noise, this planar-diffusion process is exactly described through the first two moments of the positions and velocities and their corresponding cross correlations. In the absence of the magnetic field, our analytical results are the same as those calculated by Chandrasekhar for the ordinary harmonic oscillator. The stochastic planar diffusion is also well characterized in the overdamped approximation, through the solutions of the Langevin equation.

Optimal performance of periodically driven, stochastic heat engines under limited control

“…[48]. However, in order to describe the Lorentz-force in a thermodynamically consistent way, the momenta of the system must be fully taken into account [5,34,[49][50][51]. These kinetic degrees of freedom can typically not be controlled directly but rather provide only an additional source of dissipation.…”

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

“…In his celebrated 1943 Brownian motion paper [11], Chandrasekhar outlined the method for solving a Brownian particle in a general field of force. It took approximately sixty years to report exact solutions for the Brownian motion of a charged particle in uniform and static electric and/or magnetic fields [71][72][73][74][75][76][77][78][79][80][81][82][83][84][85] (see also some previous related works [86,87]). …”

Hot Brownian carriers in the Langevin picture: Application to a simple model for the Gunn effect in GaAs

“…As a final remark, we comment on the several techniques employed to solve the Brownian Motion Problem in Fields of Forces, following from Kramers original mathematical acrobacies [9]. We mention Chandrasekhar's proposal of six independent first integrals within a Gaussian ansatz [10]; tensorial frictional forces [25]; gauge transformations [27]; a combination of several of the above mentioned techniques [28,45] and the time-dependent rotation matrix method [29][30][31][36][37][38][39]44]. Here, in this paper (as in [47]) we directly apply the Cayley- In this context, we may regard as equivalent all the techniques mentioned above, when pursuing the exact solution of this linear problem.…”