Rudolf A. Treumann^{1,2} and Wolfgang Baumjohann^{3}
^{1}Department of Geophysics and Environmental Sciences, Ludwig-Maximilians-Universität München, Munich, Germany

^{2}International Space Science Institute Bern, Bern, Switzerland

^{3}Space Research Institute, Austrian Academy of Sciences, Graz, Austria

Received: 20 Apr 2017 – Accepted: 02 May 2017 – Published: 24 May 2017

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 |Δ*q*| ∼ *c**t* is comparable to the product of observation time and light velocity, a situation which is typical in cosmic physics and astrophysics.

**Citation:**
Treumann, R. A. and Baumjohann, W.: Causal kinetic equation of non-equilibrium plasmas, Ann. Geophys., 35, 683-690, doi:10.5194/angeo-35-683-2017, 2017.