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. The diffusion tensor follows. Away from the Grad limit a percolation problem arises. We study numerically its critical properties.PACS numbers: 05.60.+w, 05.20.Dd, 64.60.Ak, 73.50.Jt The Boltzmann equation (BE) has been used to study classical transport problems for more than a century.In its linear version, as introduced by Lorentz [1], it provides precise kinetic models, underpinning classical Drude theory of diffusive electron transport in solids. We now know, of course, that in spite of the partial success of classical models, quantum mechanics is essential for a proper description of many aspects of electron transport.In particular, during the past 15 years magnetotransport in the two-dimensional (2D) electron gas has been the subject of intense investigation, both experimental and theoretical [2]. With all this activity focused on quantum aspects, it is easily taken for granted that classical magnetotransport is a closed chapter, with no surprises in store. In this Letter we shall demonstrate that this is not the case. The Lorentz-Boltzmann model [1,3] is much richer than expected. Proofs exist that, under seemingly mild restrictions, the BE is exact in the socalled Grad limit [4] (to be specified below). In contrast, we shall demonstrate that in 2D, and with a nonzero magnetic field, the BE is no longer valid, even in the Grad limit. Furthermore, we shall formulate a non-Markovian generalized Boltzmann equation (GBE), which is exact in this limit, and we shall solve the corresponding initial value problem for the spatially homogeneous case. The corresponding diffusion tensor follows immediately from this solution. Moving away from the Grad limit, we shall then show that at low but finite dimensionless densities a percolation problem arises. This problem has earlier been studied in a different context in [5]. We improve on the numerical estimate for one of the critical exponents.Our value for this exponent is slightly higher than the one expected from 2D lattice percolation, thus raising the question of whether they are in the same universality class. However, we draw no firm conclusions at present. In this Letter we treat the classical 2D Lorentz model in a perpendicular magnetic field as a model of intrinsic interest in kinetic theory. (Only the basic arguments and principal results are presented; further details will be published elsewhere. ) Whether the model provides a reasonable description of a real system is far from clear. This question raises a number of nontrivial problems +~l f(y, t) = v rit (3/1 dA g(A)Here co = eSjm is the cyclotron frequency (with m the effective mass), v = nvX is the collision frequency (with r = v ' the mean free time between collisions...

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In the low-frequency limit, the displacement currents in the Maxwell equations can be neglected. However, for numerical simulations, a small displacement current should be present to achieve numerical stability. This requirement leads to a large range of propagation velocities with high velocities for the high frequencies and low velocities for the low frequencies. As a consequence, the number of time steps may become large. I show that it is possible to transform mathematically the original physical problem to one that has propagation velocities with less frequency dependence. Hence, the number of time steps necessary for a signal to travel a certain distance with the lowest velocity is significantly reduced. A typical example shows a reduction in computational time by a factor of 40. A comparison of the solutions from plane-layered modeling in the frequency and wavenumber domain and the proposed method shows good agreement between the two. The proposed method can also be used for other systems of diffusive equations.

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