Abstract
The classical Lorentz model for charged noninteracting point particles in a perpendicular magnetic field is reconsidered in 2D. We show that the standard Boltzmann equation is not valid for this model, even in the Grad limit. We construct a generalized Boltzmann equation which is, and solve the corresponding initial value problem exactly. By an independent calculation, we find the same solution by directly constructing the Green function from the dynamics of the model in the Grad limit. From this solution an expression for the diffusion tensor, valid for arbitrary short-range forces, is derived. For hard disks we calculate the diffusion tensor explicitly. Away from the Grad limit a percolation problem arises. We determine numerically the percolation threshold and the corresponding geometric critical exponents. The numerical evidence strongly suggests that this continuum percolation model is in the universality class of 2D lattice percolation. Although we have explicitly determined a number of limiting properties of the model, several intriguing open problems remain.
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It is with great pleasure we include this paper in the issue honoring Matthieu Ernst, who not only shares our love for kinetic theory, but who also contributed so much to its modern development.
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Bobylev, A.V., Maaø, F.A., Hansen, A. et al. There is more to be learned from the Lorentz model. J Stat Phys 87, 1205–1228 (1997). https://doi.org/10.1007/BF02181280
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DOI: https://doi.org/10.1007/BF02181280