Brownian Dynamics of charged particles in a constant magnetic field

where, as usual, m, v and r are, respectively, the mass, velocity and position of a Brownian particle, whereas F is a systematic (deterministic) force coming from external fields and/or from inter-particle interactions within the system. What is different from Newton’s equations is the appearances of dynamical friction, -γv, and random, or Brownian acceleration, A(t). These two force components represent two complementing effects of a single, sub-scale phenomenon: numerous, frequent collisions of the Brownian particle with molecules in the surrounding medium.

While the friction represents an average effect of these collisions, the random acceleration represents fluctuations due to discreteness of collisions with molecules, and is generally assumed to be a delta-correlated Gaussian white noise. The friction and random acceleration are related through a fluctuation-dissipation theorem which includes the ambient temperature, therefore guaranteeing that a Brownian particle can ultimately reach thermal equilibrium within the medium [8].

The Langevin equations, Eq. ( 1), can be numerically integrated in a manner similar to Newton’s equations in the MD simulation, which gives rise to several algorithms for performing BD simulation, such as the Euler-like [1], Beeman-like [2,6], Verlet-like [3], and Gear-like Predictor-Corrector (PC) methods [15], as well as a wide class of Runge-Kutta-like algorithms (see, e.g., [4,5]). All those methods were used successfully to study problems in various dispersed systems, such as polymer solutions [9], colloidal suspensions [10] and, in particular, complex (dusty)

plasmas [11,12,13,14,15,16].

Recently, there has been a growing interest in studying the dynamics of dust particles and dust clouds in both unmagnetized [17,18] and magnetized dusty [19,20,21,22] plasmas. The topics studied so far include, besides Brownian motion of a dust particle in an unmagnetized plasma [18], also the gyromotion of a single dust particle and rotations [19] of dust clouds [20,21,22] and clusters [23,24,25,26] in a magnetized plasma. There have also been several theoretical proposals for studying waves and collective dynamics [27] in such magnetized plasma systems, in which dust particles are fully magnetized [28,29,30,31,32]. However, exploring those proposals in the laboratory does not seem to be quite feasible as yet, due to many constrains [28,29,30,32]. Therefore, it is desirable to have algorithms for numerical experiments that can validate the existing theories, on one hand, and that can serve as a guide for future laboratory experiments, on the other hand. Since there are no such algorithms, to the best of our knowledge, we propose here a few new BD algorithms, which treat an external magnetic field in the simulation in a manner consistent with the Langevin dynamics, Eq. ( 1).

The manuscript is organized in the following fashion. In Sec. II, we present the general formula for a BD simulation with a constant magnetic field. Detailed implementations to the Euler-, Beeman-and Gear-like methods are given, respectively, in Sections III, IV, and V. Concluding remarks are contained in Sec. VII.

The dynamics of a charged Brownian particle in a constant external magnetic field B is described by introducing the Lorentz force in the Langevin equation [7,8]

where Q is the charge on the particle and c is the speed of light in vacuum.

As usual, certain assumptions must be made about the deterministic force F in order to construct a meaningful algorithm for a many-particle simulation from the above equation. A common approach [1,2,3,6] is to assume that F is only an explicit function of time t. Thus, a Taylor series of F or, equivalently the deterministic acceleration, a ≡ F/m, can be written as

where a (n) represents the nth-order time derivative of a. There are various ways to derive formulas for conducting a BD simulation from the Langevin equation, Eq. ( 2), based on the above Taylor series. We shall adopt the strategy outlined in Refs. [1,2,3,6], and more recently implemented in Refs. [7,8], as it is simple and straightforward, especially for readers with some simulation background but without much background, or interest in stochastic calculus.

The Langevin equations, Eq. ( 2), may be integrated analytically in a short time, based on an adopted truncation rule for the series in Eq. ( 3), thereby giving an updating formula for a BD simulation. (We note that detailed technique for integrating the Langevin equation is particularly well described in Refs. [7,8].) The resultant for

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