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

The starting point of (classical) kinetic theory is Liouville's equation.
Written in terms of the

In view of an application to plasmas, the relevant field is the electromagnetic
field

The problem of the above equations is that they do not account for the fact
that the electromagnetic signal of the presence and motion of the particles is
transferred from the signal-emitting particles to the signal-receiving
particles under consideration, i.e. the absorbers and reactors. Their sources are
the charge and current densities

Since all particles serve both as field sources and actors, excluding their
self-interaction, the use of the instantaneous fields ignores the
time-consuming signal transport and thus cannot be correct. It is an
approximation only that holds for comparably small volumes such that, in the
expression for the retarded time, the spatial difference can be neglected.
Thus, the restriction on the distance between particles is that

For single-particle–particle interactions, this problem has been discussed in
depth in seminal papers by

This explicit representation of the microscopic fields accounts properly for
the time delay between the signal emitted from the total compound of primed
particles to arrive at the location

Taking the divergence of the microscopic electric field and the curl of the
microscopic magnetic field, one readily reads the correct microscopic charge
and current densities when comparing the expressions with the microscopic
Maxwell equations:

These are the correct forms of the charge and current
densities summed over species

The proper way of dealing with this problem is to focus on the microscopic picture for as long as possible. There, all the charged particles can be imagined as moving in a vacuum as long as the medium is sufficiently dilute. By progressing to a coarse-grained picture one may afterwards advance to considering a more continuous medium in which ultimately the propagation properties of the signals will become modified by the collective properties of the matter.

With these results it is convenient to express the microscopic
electromagnetic fields through the microscopic phase space densities of the
particle species:

It is quite inconvenient to deal with all microscopic phase space densities.
We would rather have separate equations for them. This can be
achieved when observing that Eq. (

The above Eq. (

The inclusion of information transport between the interacting particles substantially complicates the basic kinetic equation. It causes a delay in response, and thus refers to a natural measuring process in which the particles are not only generators of the electromagnetic field but also measure its effect over a causal distances accessible to them. The delay must thus necessarily cause decorrelation of the response.

There is another complication with this picture which comes into play when
considering large compounds of particles rather than single particles. Single
charged particles are assumed to move in the vacuum; the signal propagation
between them takes place at light speed

In the following we will proceed along the same lines as

Dealing with the causal

This procedure must be applied to the causal

These expressions are to be used in the Lorentz gauge (Eq.

Formally, these expressions, as claimed in the previous sections, are rather
similar to those which, for the non-retarded interactions, had already been
obtained by

Referring to Eq. (

From all these expressions one can again obtain an equation for the
fluctuation of phase space density

With knowledge of the collision term on the right or some of its
approximations, Eq. (

Clearly, the above equations resemble the well-known approach to plasma
kinetic theory. It should, however, be pointed out that even when dropping
the collision term on the right in a Klimontovich–Vlasov approach in linear
theory, the retardation effect remains in the third term on the left-hand
side in Eq. (

The one-particle kinetic equation (Eq.

Under such conditions Klimontovich–Vlasov theory applies, and the complications introduced by reference to the retarded time can be neglected. On the other hand, in very large volumes like in cosmical and astrophysical applications transport of information is provided by radiation transport and becomes rather slow. Hence, remote volumes will not respond immediately and not even within light-propagation time, which can then be treated again in the simplified theory.

However, the current investigation is necessary as a clarification of two points: Firstly, that the interaction among different volumes in plasma in principle cannot be considered to occur instantaneously. Secondly, the inclusion of retarded times gives a clue to the direction of time – as briefly discussed below – which in many-particle systems has only one direction, forward. Events are delayed by information transport and thus decorrelate even though they become relativistically synchronized by accounting for the information transport. This should necessarily contribute to dissipation because information becomes diffused by passing across the plasma from one particle to another.

Reference to the retarded potentials and the effect of emission and
absorption implies a distinction between advanced and retarded effects. This
in itself unexpectedly brings up the problem of direction of time, this time
not in electrodynamics like in absorber theory, but also and directly in the
microscopic theory of phase space evolution. The delayed and integrated
response of the charge and current densities at location

This question cannot be answered a priori. Absorber theory is restored in the
second case in the causal many-particle theory. When considering the vacuum
as a medium in which the dispersion of electromagnetic waves is described by
a dispersion relation

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.

A number of points may be worth mentioning. First, the result looks simple as
it seems that simple replacement of time with retarded time would have been
sufficient to obtain it. This is true, but it is not proof for the result's
correctness. For this reason we have chosen to follow the derivation step by
step, which is the usual way of confirming a hypothesis. This required using
the basic equations derived by

The same procedure may also be applied to other classical fields since in all interactions the transport of information from the agent to the absorber takes time. This is the case in gases where sound waves or gravity waves can be excited and these transport the information from one fluid element to another place to affect the dynamics of other elements. In these cases it is not the photons but phonons that transport energy and information. Application to these systems lies outside the intention of the present work.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This work was part of a Visiting Scientist Programme at the International Space Science Institute Bern in 2007. The interest of the ISSI Directorate is acknowledged, as, and in particular, is the friendly hospitality of the ISSI staff. Thanks are directed to the ISSI system administrator S. Saliba for technical support and to the librarians Andrea Fischer and Irmela Schweizer for access to the library and literature. The topical editor, E. Roussos, thanks P. Yoon for help in evaluating this paper.