Abstract
Abstract. Statistical plasma theory far from thermal equilibrium is subject to Liouville's equation, which is at the base of the BBGKY hierarchical approach to plasma kinetic theory, from which, in the absence of collisions, Vlasov's equation follows. It is also at the base of Klimontovich's approach which includes single-particle effects like spontaneous emission. All these theories have been applied to plasmas with admirable success even though they suffer from a fundamental omission in their use of the electrodynamic equations in the description of the highly dynamic interactions in many-particle conglomerations. In the following we extend this theory to taking into account that the interaction between particles separated from each other at a distance requires the transport of information. Action needs to be transported and thus, in the spirit of the direct-interaction theory as developed by Wheeler and Feynman (1945), requires time. This is done by reference to the retarded potentials. We derive the fundamental causal Liouville equation for the phase space density of a system composed of a very large number of charged particles. Applying the approach of Klimontovich (1967), we obtain the retarded time evolution equation of the one-particle distribution function in plasmas, which replaces Klimontovich's equation in cases when the direct-interaction effects have to be taken into account. This becomes important in all systems where the distance between two points |Δq| ∼ ct is comparable to the product of observation time and light velocity, a situation which is typical in cosmic physics and astrophysics.
Highlights
The starting point of kinetic theory is Liouville’s equation
The present investigation extends Klimontovich’s approach to kinetic plasma theory to the inclusion of signal retardation effects. It applies to systems of indistinguishable charged particles interacting via their self-consistent electromagnetic fields
One can trivially extend it to the presence of external fields like stationary or variable magnetic fields caused by external sources
Summary
The starting point of (classical) kinetic theory is Liouville’s equation. Written in terms of the Na-particle Hamiltonian HNa (q,p, t) and defining the 6-D phase space density Na(q,p,t) of species a, both functions of space q and momentum p, it becomes. This is Klimontovich’s equation for the exact microscopic phase space density Na(p,q, t) in 6-D phase space (Klimontovich, 1967) It is a tautology because it does not say anything other than that particle number is conserved along all the dynamical orbits of the particles in phase space under the action of their mutual electromagnetic fields. Introduce some coarse graining procedure and define integrated distribution functions which reduce the system to a set of equations known as Klimontovich– Vlasov equations for a one-particle phase space distribution in the presence of the average electromagnetic fields This procedure is very efficient, and we will follow it below in a modified version
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