On the motion of charged particles subject to random forces and fields

Abstract

In plasma physics, it is often necessary to study the motion of a charged particle in a magnetic field, modelling the effect of collisions by additive random forces. Following Langevin, the force is taken to be the sum of a friction term, proportional to the velocity, and a fluctuating component referred to as “noise”. This stochastic term is assumed to be a Gaussian stationary process with vanishing mean and a covariance matrix characterized by correlation times which may vanish (“white noise”) or be finite (e.g. “coloured noise”). This problem, for the case of a uniform field, leads to two systems of equations, one for the ID longitudinal motion along the direction of the field and another for the 2D transverse motion perpendicular to it. These two systems can be studied independently. Because of the linearity of the equations, formal solutions are written and expressions for the expectation values of powers of the position and/or the velocity of the particle are obtained. For white noise, the problem has been studied analytically and numerically. However, the case of a stochastic term with finite correlation time has not been considered in detail. Here, the influence of non-white noises is studied and numerical simulations based on two different approaches are presented. In the first approach, by adding an equation for the evolution of the stochastic term, the problem is reduced essentially to that of white noise. However, this technique is not to be adopted in general since it can only reproduce specific kinds of noise. In the second approach the forcing term is modelled by a Fourier series with random, uniformly distributed, phases and may be considered as more general. Simulations with the latter method are presented for a random force with the correlation function of coloured, Gaussian or Lorentzian noise. To judge the extracted accuracy from the computational approaches, the numerical results have also been compared with available analytical solutions. In addition, an extensive parametric study has been performed with respect to various involved parameters including the correlation time and the friction coefficient. This numerical investigation serves for validating and benchmarking purposes of the implemented computational schemes and will be used in more complex problems. The problem of motion of a charged particle inside a space dependent electric field has also been investigated. The governing system of equations is non-linear and can only be solved numerically using the Fourier series approach because the explicit form of the field is not available. Results show good qualitative agreement with benchmark problems. In the future, random motion in electromagnetic fields and inhomogeneous magnetic fields will also be considered.