Nondiffusive fluxes in a Brownian system with Lorentz force

Abstract

The Fokker-Planck equation provides complete statistical description of a particle undergoing random motion in a solvent. In the presence of Lorentz force due to an external magnetic field, the Fokker-Planck equation picks up a tensorial coefficient, which reflects the anisotropy of the particle's motion. This tensor, however, can not be interpreted as a diffusion tensor; there are antisymmetric terms which give rise to fluxes perpendicular to the density gradients. Here, we show that for an inhomogeneous magnetic field these nondiffusive fluxes have finite divergence and therefore affect the density evolution of the system. Only in the special cases of a uniform magnetic field or carefully chosen initial condition with the same symmetry as the magnetic field can these fluxes be ignored in the density evolution.

Statistical physics of flux-carrying Brownian particles

“…(3) and ( 4) show the generation of a disipationless vortex flow in the single-particle scenario. This stands in contrast to the conventional Brownian motion in presence of an external magnetic field, in which there is no fluxes at thermal equilibrium [43,44,47]. We notice that the small strength of both the steady flow density and the fluid vorticity is in agreement with the subsidiary condition (45), which establishes that the flux-carrying effects must remain perturbative in comparison with the dissipative effects (otherwise the quantum kinetics (58) would deviate from the low-lying description provided by (1) [25,31]).…”

“…For sake of simplicity, we consider a radially symmetric initial Gaussian state with a simple covariance matrix V (0) = I 4 (this corresponds to the extensively studied coherent state in quantum optics). The reason to focus the attention on this kind of states is because it retrieves a purely diffusive flow (which is parallel to the particle density gradient) in the case of the conventional Brownian motion subject to an uniform magnetic field [43,44]. Hence, the vortex flow shown here is exclusive to the flux-carrying Brownian motion [25] (it is a direct consequence of the broken time reversal and parity symmetries).…”

“…with sgn(t) denoting the usual sign function. Notably, the flux-carrying term in (44) closely resemblances the ordinary Hall response of two-dimensional particles found in the dissipative Hofstadter model [54,55]. Actually, its Fourier transform in the frequency domain identically coincides with the fluctuations of an antisymmetric 1/f noise of power spectrum S(ω) = (iω) −1 (to see this one must realize in (22) that coth ( ωβ/2) renders an inverse scaling for the power spectrum of the transversal correlations, while the spectral density contributes with a constant value in the Breit-Wigner approximation).…”

“…which manifests that the flux-carrying effects challenge with the dissipative effects to establish a non-vanishing vorticity. Certainly, the first term in the above equation pinpoints that the flux-carrying Brownian particles will eventually spin with an angular speed of strength determined by Ω flux in absence of particle density or temperature gradients, which is in contrast to diffusive systems subject to a Lorentz force [43,44]. As the vorticity flux is the fluid counterpart of the magnetic flux definition, the flux-carrying coupling Ω flux can be thought of as the magnitude of a coarse-grained pseudomagnetic flux traversing the plane.…”

Drift-diffusion processes resulting from elimination of fast variables under inhomogeneous conditions

“…Thirty years ago, van Kampen [1] investigated the effects of environmental inhomogeneity in models of random motion of passive particles. In general, the problem continues to attract interest [2][3][4][5][6][7][8], and has been especially revived for active matter [9][10][11][12][13]. Cates and Tailleur [14] and Vuijk et alia [15], in particular, studied variants of a model to be taken up here.…”