Classical diamagnetism, magnetic interaction energies, and repulsive forces in magnetized plasmas (original) (raw)
Related papers
2010
The Bohr-van Leeuwen theorem is often summarized as saying that there is no classical magnetic susceptibility, in particular no diamagnetism. This is seriously misleading. The theorem assumes position dependent interactions but this is not required by classical physics. Since the work of Darwin in 1920 it has been known that the magnetism due to classical charged point particles can only be described by allowing velocity dependent interactions in the Lagrangian. Legendre transformation to an approximate Hamiltonian can give an estimate of the Darwin diamagnetism for a system of charged point particles. Comparison with experiment, however, requires knowledge of the number of classically behaving electrons in the sample. A new repulsive effective many-body force, which should be relevant in plasmas, is predicted by the Hamiltonian.
Classical Diamagnetism Revisited
Modern Physics Letters B, 2010
The well-known Bohr–van Leeuwen Theorem states that the orbital diamagnetism of classical charged particles is identically zero in equilibrium. However, results based on real space–time approach using the classical Langevin equation predicts non-zero diamagnetism for classical unbounded (finite or infinite) systems. Here we show that the recently discovered Fluctuation Theorems, namely, the Jarzynski Equality or the Crooks Fluctuation Theorem surprisingly predicts a free energy that depends on magnetic field as well as on the friction coefficient, in outright contradiction to the canonical equilibrium results. However, in the cases where the Langevin approach is consistent with the equilibrium results, the Fluctuation Theorems lead to results in conformity with equilibrium statistical mechanics. The latter is demonstrated analytically through a simple example that has been discussed recently.
Yet another surprise in the problem of classical diamagnetism
2009
The well known Bohr-van Leeuwen Theorem states that the orbital diamagnetism of classical charged particles is identically zero in equilibrium. However, results based on real space-time approach using the classical Langevin equation predicts non-zero diamagnetism for classical unbounded (finite or infinite) systems. Here we show that the recently discovered Fluctuation Theorems, namely, the Jarzynski Equality or the Crooks Fluctuation Theorem surprisingly predict a free energy that depends on magnetic field as well as on the friction coefficient, in outright contradiction to the canonical equilibrium results. However, in the cases where the Langevin approach is consistent with the equilibrium results, the Fluctuation Theorems lead to results in conformity with equilibrium statistical mechanics. The latter is demonstrated analytically through a simple example that has been discussed recently.
2010
Recently [EPL, 86, (2009) 17001], we had simulated the classical Langevin dynamics of a charged particle on the surface of a sphere in the presence of an externally applied magnetic field, and found a finite value for the orbital diamagnetic moment in the long-time limit. This result is surprising in that it seems to violate the classic Bohr-van Leeuwen Theorem on the absence of classical diamanetism. It was indeed questioned by some workers [EPL, 89, (2010) 37001] who verified that the Fokker-Planck (FP) equation derived from our Langevin equation, was satisfied by the classical canonical density in the steady state, obtained by setting ∂/∂t = 0 in the FP equation. Inasmuch as the canonical density does not contain the magnetic field, they concluded that the diamagnetic moment must be zero. The purpose of this note is to show that this argument and the conclusion are invalid-instead of setting ∂/∂t = 0, one must first obtain the fundamental time-dependent solution for the FP equation, and then calculate the expectation value of the diamagnetic moment, and finally consider its long-time limit (i.e., t → ∞). This would indeed correspond to our numerical simulation of the dynamics. That this is indeed so is shown by considering the simpler analytically solvable problem, namely that for an unbounded plane for which the above procedure can be carried out exactly. We then find that the limiting value for the expectation of the diamagnetic moment is indeed non-zero, and yet the steady-state FP equation obtained by setting ∂/∂t = 0 is satisfied by the canonical density. Admittedly, the exact analytical solution for the sphere is not available. But, the exact solution obtained for the case of the unbounded 2D-plane illustrates our point all right. We also present some further new results for other finite but unbounded surfaces such the ellipsoids of revolution.
On the nature of the plasma equilibrium
We calculate the energy of a homogeneous one component plasma and find that the energy is lower for correlated motions of the particles as compared to uncorrelated motion. Our starting point is the conserved approximately relativistic (Darwin) energy for a system of electromagnetically interacting particles that arises from the neglect of radiation. For the idealized model of a homogeneous one component plasma the energy only depends on the particle canonical momenta and the vector potential. The vector potential is then calculated in terms of the canonical momenta using recent theoretical advances and the plasma Hamiltonian is obtained. The result can be understood either as due to the energy lowering caused by the attraction of parallel currents or, alternatively, as due to the inductive inertia associated with the flow of net current.
A Classical Description of the Electrons in Plasma
حوليات العلوم و التكنولوجيا, 2013
In this work, we calculated the effective potential of an electron in a plasma. This potential, which is obtained by solving a non-linear integral equation, is a sum of three contributions: the first is the interaction energy between the electron plasma and a test charge regarded as an impurity. We have taken this interaction equal to screened "Kelbg". The second is the Coulomb interaction energy between the electron in question and the other electrons plasma, that we calculate using a Maxwell-Boltzmann distribution. The third is Coulomb interaction energy between the electron and ions plasma uniformly distributed. The effective potential, is obtained, in the first stage, we have calculated the distribution of electric microfield created by the electrons on the impurity. In the second stage we calculated the time autocorrelation function of the electric microfield. The results are compared with those given by molecular dynamics simulation.
Evaluating the Adiabatic Invariants in Magnetized Plasmas Using a Classical Ehrenfest Theorem
Entropy
In this article, we address the reliance on probability density functions to obtain macroscopic properties in systems with multiple degrees of freedom as plasmas, and the limitations of expensive techniques for solving Equations such as Vlasov’s. We introduce the Ehrenfest procedure as an alternative tool that promises to address these challenges more efficiently. Based on the conjugate variable theorem and the well-known fluctuation-dissipation theorem, this procedure offers a less expensive way of deriving time evolution Equations for macroscopic properties in systems far from equilibrium. We investigate the application of the Ehrenfest procedure for the study of adiabatic invariants in magnetized plasmas. We consider charged particles trapped in a dipole magnetic field and apply the procedure to the study of adiabatic invariants in magnetized plasmas and derive Equations for the magnetic moment, longitudinal invariant, and magnetic flux. We validate our theoretical predictions us...
Barkas effect in strongly magnetized plasmas
Physics of Plasmas
Strongly magnetized plasmas, which are characterized by the particle gyrofrequency exceeding the plasma frequency, exhibit novel transport properties. For example, recent work showed that the friction force on a test charge moving through a strongly magnetized plasma not only consists of the typical stopping power component but also includes components perpendicular to the test charge's velocity. However, these studies only considered test charges that have the same sign as the charge of the plasma particles. Here, we extend these calculations to the case of charges with opposite signs (such as an ion interacting with strongly magnetized electrons). This is done with both a novel generalized Boltzmann kinetic theory and molecular dynamics simulations. It is found that the friction force changes dramatically depending on the sign of the interacting charges. Likewise, the stopping power component for oppositely charged particles decreases in magnitude compared with like-charged pa...