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 N˙a≡∂tNa+[HNa,Na]=0, where it is assumed that the particle number Na of species a is conserved (along all dynamical phase space orbits). Otherwise the right-hand side would contain the difference of number sources and losses Sa-La. This equation, under the assumptions made, is completely general, applying to any system consisting of Na≫1 particles in interaction with an external as well as with their mutual fields, of which they function as sources. These fields are contained in the Hamiltonian and act via the Poisson bracket […].

In view of an application to plasmas, the relevant field is the electromagnetic field E,B, with the particles carrying electric charges ea=∓e≡ae (with a=-, +) being the sources of the field. For simplicity, in the following, we restrict ourselves to electrons and ions (protons) of mass ma, and gravity can be neglected on all scales small enough for the electromagnetic fields to dominate. We also assume global quasi-neutrality and the absence of any external fields. Then E,B=(E,B)m is the set of microscopic electromagnetic fields produced solely by the microscopic charge and current densities of the interacting particle components a, which serve as their sources:

ρam(q,t)=ea∑i=1Naδ(q-qai(t))=ea∫d3pNa(p,q,t),jam(q,t)=eama∑i=1Napai(t)δ(q-qai(t))=eama∫d3ppNa(p,q,t), where the exact 6-D phase space density is defined through Na(p,q,t)=∑i=1Naδ(p-pai(t))δ(q-qai(t)) and qai(t),pai(t) are the spatial and momentum phase space trajectories which the particle ai performs in the phase space under the action of the complete microscopic electromagnetic field (E,B)m, which it feels at its location q=qai(t) at time t. Liouville's equation for the exact phase space density can then be written in the form ∂Na∂t+pma⋅∇qNa+ea{[Em(q,t)+pma∧Bm(q,t)]⋅∂Na∂p}=0. This is Klimontovich's equation for the exact microscopic phase space density Na(p,q,t) in 6-D phase space . 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. The microscopic fields it contains are given by Maxwell's equations in differential form: ∇q∧Bm=μ0jm+μ0ϵ0∂tEm,∇q⋅Bm=0,∇q∧Em=-∂tBm,∇q⋅Em=1ϵ0∑aρam. The solution of this set of equations is not possible as it requires knowledge of all microscopic particle orbits. One can, however introduce some coarse graining procedure and define integrated distribution functions which ultimately 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.

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 ρam(q,t), jm(q,t), which are assumed to be known at any instant t in all space points q. Obtaining this knowledge is impossible as it requires instantaneous measurements at time t of all positions q and momenta p of the particles present in real space. Instead, the information must be synchronized among all locations. This is taken care of in the Liénard–Wiechert potentials, which explicitly account for the transport of information from point q′ to point q. In this case in the Lorentz gauge Em=-∇qϕm(q,t)-∂tAm(q,t),Bm=∇q∧Am(q,t), the correct scalar and vector potentials are to be expressed by the retarded charge and current densities ϕm(q,t)=14πϵ0∑a∫d3q′ρam(q′,t′)|q-q′|,Am(q,t)=μ04π∑a∫d3q′jam(q′,t′)|q-q′| taken at the retarded time t′=t-|q-q′|/c,q≠q′ of arrival of all the signals emitted at t′ from all the particles at spatial distance |q-q′| from the location of particle ai at q and at time t. This also implies that in the expressions for the charge and current densities Na→Na(p,q′,t′) is a function of the retarded time t′.

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 |Δq|≪cΔt. Clearly, this condition will readily be violated in large volumes of cosmic and astrophysical size, where one must refer to the above precise potentials and the fields resulting from them in reference to the Lorentz gauge.

For single-particle–particle interactions, this problem has been discussed in depth in seminal papers by . They showed that in a closed system where no information is lost to the outside eliminating any self-interaction of a particle with its proper electromagnetic field implies that the fields are properly described via retarded potentials only as done above. These potentials account for the emission of a signal by one particle and the absorption of the signal after some travel time by the target particle, causing this particle to interact. The emitted signals belong to advanced potentials which, when correctly included, subtract out, thereby restoring the required real-world causality. It is incorrect to assume that the information arrives microscopically instantaneously at the target, causing this to act. The electromagnetic fields following from the Lorentz gauge in the microscopic domain are Em(q,t)=14πϵ0∫d3q′[(ρm(q′,t′)|q-q′|3+∂tρm(q′,t′)c|q-q′|2)(q-q′)-∂tjm(q′,t′)c2|q-q′|],Bm(q,t)=μ04π∫d3q′[jm(q′,t′)|q-q′|3+∂tjm(q′,t′)c|q-q′|2]∧(q-q′), which were first given, independently, by and . One should note that in these expressions the charge ρm and current densities jm are summed over all particle species a.

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 q of the particle under consideration. Since the microscopic charge and current densities are functionals of the phase space density, these expressions contain the latter, though in a more involved manner than when using the differential forms of the electrodynamic equations, which do not show where and whether the retardation of the signal is taken into account. It is clear from these expressions that particles which are far away from the target do not affect it. The main effect will always come from close neighbours.

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: ρm(q,t)=14π∑a∇q⋅∫d3q′[(ρam(q′,t′)|q-q′|3+∂tρam(q′,t′)c|q-q′|2)(q-q′)-∂tjam(q′,t′)c2|q-q′|],jm(q,t)=14π∑a{∇q∧∫d3q′[jam(q′,t′)|q-q′|3+∂tjam(q′,t′)c|q-q′|2]∧(q-q′)-∂t∫d3q′[(ρam(q′,t′)|q-q′|3+∂tρam(q′,t′)c|q-q′|2)(q-q′)-∂tjam(q′,t′)c2|q-q′|]}.

These are the correct forms of the charge and current densities summed over species a which have to be used in Maxwell's equations in order to account for the retarded transfer of information between the particles in the plasma. These expressions are implicit for both the charge and current densities. In order to relate them to the exact microscopic phase space distribution Na as defined in Eq. (), one refers to the representations (Eq. ) of the charge and current densities. This shows that the functional dependence of the phase space density is itself implicit. It depends on itself, taken at all the positions q′ and retarded times t′.

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: Em(q,t)=∑aea4πϵ0∫d3pd3q′[(Na(p,q′,t′)|q-q′|3+∂tNa(p,q′,t′)c|q-q′|2)(q-q′)-p∂tNa(p,q′,t′)c2|q-q′|],Bm(q,t)=∑aeaμ04π∫d3pd3q′[pNa(p,q′,t′)|q-q′|3++p∂tNa(p,q′,t′)c|q-q′|2]∧(q-q′). These are the expressions of the electromagnetic field which have to be used in the microscopic Liouville Eq. () for the microscopic N-particle phase space density. Not only do they couple the different particle species, thus leading to a coupling between their phase space distributions, they also make each microscopic distribution Na a functional of the distributions taken at all different phase space locations which are causally accessible via their retarded times of signal propagation t′. Clearly, this is a substantial complication, which is introduced into kinetic theory by the requirement of causality.

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. () is an equation for Na. Thus, putting a→b in the last expressions, which means that we sum over all particle species b including also b=a (with self-interaction excluded by the definition of the retarded time), we have

∂Na∂t+pma⋅∇qNa+∑beaeb4π∫d3p′d3q′{1ϵ0[(Nb(p′,q′,t′)|q-q′|3+∂tNb(p′,q′,t′)c|q-q′|2)(q-q′)-p′∂tNb(p′,q′,t′)c2|q-q′|]-μ0map∧[(q-q′)∧(p′Nb(p′,q′,t′)|q-q′|3+p′∂tNb(p′,q′,t′)c|q-q′|2)]}⋅∂∂pNa(p,q,t)=0. Here Na(p,q,t), while Nb(p′,q′,t′) depends on the dummy coordinates of all particles b of integration and on the retarded time t′=t-|q-q′|/c,q≠q′. Thus, in the q′ integration all particles of sort a are also included, with the exception of the particle located at q=q′ at time t.

The above Eq. () is the causal Liouville equation acting on the microscopic Na-particle phase space density Na(p,q,t) in the presence of a large number of charged particles interacting via their self-consistently generated electromagnetic fields. It extends Klimontovich's equation to the correct inclusion of the retardation effect of transmission of information between the particles via electromagnetic fields.

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 c. Immersed in the comparably dense environment of all the other charged particles, any light or radiation experiences radiation transport, which is dominated by scattering, reflection, transmission and absorption, processes that occur due to the active response of the environment to the presence of radiation and depend on the capabilities of the medium to let electromagnetic signals pass. In these processes various proper electromagnetic modes excited in the medium become involved. These are solutions of the dispersion properties of the matter. Hence, correctly accounting for the signal transport becomes rather involved. For this reason the theory even in this complex version applies to sufficiently dilute media to allow the assumption of signal propagation in a vacuum.

In the following we will proceed along the same lines as but will in the end refer to the above field equations. This means that in defining the average distributions, we will consider Liouville's equation without explicit reference to the fields.

Dealing with the causal N-particle kinetic Eq. () is impractical. One wants to reduce it to an equation for a one-particle distribution function in 6-D phase space for indistinguishable particles of sort a. This is done by integrating out in Eq. () all particle coordinates i>1. Defining phase space coordinates x=(p,q),xai(t)=(pai(t),qai(t)), the N-particle density becomes Na(x,t)=∑i=1Naδ(x-xai(t)). Following , let us define the one-particle distribution of sort a of indistinguishable particles by fa(xa1,t)=Va∫fNd6xa2…d6xaNa6××∏b≠ad6xb1…d6xbNb6. The N-particle probability distribution fN depends on all the particle coordinates in phase space which have been integrated out in the last expression, including xa1, and V is the spatial volume of particle a1, i.e. the volume all indistinguishable particles occupy. With its help the averaged phase space density yields directly NaVafa(p,q,t)=〈Na(p,q,t)〉. Here, the right-hand side is the ensemble-averaged one-particle phase space density 〈Na(p,q,t)〉, which is a functional only of the indistinguishable dynamics of the particles indexed by i=1. Accordingly, averaging the product of two phase space densities Na(x,t) and Nb(x′,t) yields 〈Na(x,t)Nb(x′,t)〉=naδabδ(x-x′)fa(x,t)+nanbfab(x,x′,t), where the partial densities are defined as na=Na/Va,nb=Nb/Vb and fab(x,x′,t) is the two-particle distribution function . In the same way, higher-order average products of phase space densities can be reduced to sums of distribution functions.

This procedure must be applied to the causal N-particle kinetic Eq. (). This is a formidable task if using the N-particle kinetic equation in its explicit form. As stated earlier, it is more convenient to remain with the implicit versions of the Lorentz gauge (Eq. ) and the retarded potentials in which we replace the charge end current densities by the general expressions given in Eq. (). From Eq. (), this yields ϕm(q,t)=∑aea4πϵ0∫d3pd3q′Na(p,q′,t′)|q-q′|,Am(q,t)=∑aeaμ04π∫d3pd3q′pNa(p,q′,t′)|q-q′|, with the time taken as the retarded time t′ thus depending on the spatial coordinate q′, which is to be integrated out.

These expressions are to be used in the Lorentz gauge (Eq. ) when expressing the electromagnetic fields in the N-particle kinetic Eq. (). Formally, this is the same as if we used Eq. () directly in deriving the corresponding causal equation for the one-particle distribution function of indistinguishable particles of sort a. It is only the electromagnetic fields in Eq. () which depend on the retarded time. Therefore, one can formally calculate the average to obtain ∂〈Na〉∂t+pma⋅∇q〈Na〉+ea〈[Em(q,t)+pma∧Bm(q,t)]⋅∂Na∂p〉=0. The last term in this equation contains particles of kind a and b as well as the retarded time coordinate. Nevertheless, by carefully ordering the different contributions and variables of integration one can bring it into a more convenient form. For this we indicate all integration variables by primes ′ and rename the retarded time variable by a superscript R. Then t′→tR=t-|q-q′|/c. After expressing the last term in angular brackets for the average phase space density, this yields : ∂〈Na〉∂t+pma⋅∇q〈Na〉-∑beaeb4πϵ0∫d3p′d3q′{(∇q+p′mbc2∂∂t)-1mbc2∇q∧p′}⋅∂∂p(〈Na(p,q,t)Nb(p′,q′,tR)〉|q-q′|)=0. In this version of the phase space (ensemble) averaged equation for the time and one-particle phase space evolution of the (ensemble) averaged one-particle phase space density 〈Na(p,q,t)〉, the retarded time appears only in the averaged product. This equation is the master equation for constructing the kinetic equation for the particle distribution function. Defining the fluctuation of particle number density as δNa(x,t)=Na(x,t)-〈Na(x,t)〉 and referring to the correlation function gab(x,x′,t) defined through the average of the product of the fluctuations 〈δNa(x,t)δNb(x′,t)〉=nanbgab(x,x′,t)+naδabδ(x-x′)fa, we finally arrive at the desired causal kinetic equation: ∂fa∂t+pma⋅∇qfa-∑bnbeaeb4πϵ0∫d3p′d3q′{(∇q+p′mbc2∂∂t)-1mbc2∇q∧p′}⋅∂∂p(fa(p,q,t)fb(p′,q′,tR)|q-q′|)=Ca(p,q,t). The interaction term on the right-hand side arises from the various interparticle collisions which are mediated by the electromagnetic field. From the above definition of the fluctuations and correlations, it is given by Ca(p,q,t)≡1na∑bnbeaeb4πϵ0∫d3p′d3q′{(∇q+p′mbc2∂∂t)-1mbc2∇q∧p′}⋅∂∂p(〈δNa(p,q,t)δNb(p′,q′,tR)〉|q-q′|).

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 , with the main difference being that here they are written in terms of the full electromagnetic field and contain the spatial integration over all the remote particle space. Referring to the full electromagnetic fields is necessary because of their role in the information transport and maintenance of causality in absorber theory. The expressions above are, however, very different from Klimontovich's because they account for the necessary causal relation between the interacting particles, which is contained in their dependence on the retarded time tR, by which the particles respond to the transport of information. As a result of this response, the spatial integral appearing in these expressions contains an integration over tR and thus also the primed space coordinate q′. This complicates the calculation substantially and in an analytical treatment possibly requires the introduction of further approximations. Nevertheless, the above final equation with the implicitly given collision term extends Klimontovich's theory to the explicit reference to causality.

Referring to Eq. () the collision term can also be expressed via the fluctuations of the phase space density δNa=Na-〈Na〉 and the fluctuations of the electromagnetic fields (δE,δB)=(E,B)m-〈(E,B)m〉. This yields Ca(x,t)=-eana∫d3q′d3p′(∂∂p〈δEδNa〉-pma⋅∂∂p∧〈δBδNa〉), where the average refers to the integration over all particle space i>1, and all quantities still depend on the retarded time tR which requires integration with respect to q′. It is, however, more convenient to make use of the representation via the correlation function, in which case, from Eqs. () and () we have

Ca(x,t)≡∑bnbeaeb4πϵ0∫d3p′d3q′{(∇q+p′mbc2∂∂t)-1mbc2∇q∧p′}⋅∂∂pgab(x,x′,tR)|q-q′|. This is the general causal collision integral term including the interactions between particles indexed by i=1 and i=2.

From all these expressions one can again obtain an equation for the fluctuation of phase space density δNa as well as for the fluctuating fields expressed through the space charge density and current fluctuations.

With knowledge of the collision term on the right or some of its approximations, Eq. () provides the basis for a linearized kinetic theory to any order, including particle and time-retarded interaction effects. For this, one defines the fluctuations of the one-particle distribution function in the usual way as δfa=fa-f¯a, where f¯a is the one-particle “equilibrium” distribution around which the fluctuations occur. The distribution is either an equilibrium solution of the stationary kinetic Eq. () or some of its large-scale solutions with scales exceeding those of the fluctuations such that the average of the fluctuation taken over these scales Δf‾a=0 vanishes. Neglecting the collision term by putting Ca=0 and subtracting the fluctuation averaged kinetic equation, the causal collisionless kinetic equation for the fluctuations then becomes ∂δfa∂t+pma⋅∇qδfa-∑bnbeaeb4πϵ0∫d3p′d3q′{×(∇q+p′mbc2∂∂t)-1mbc2∇q∧p′}⋅∂∂p(×δfa(x,t)fb(x′,tR)‾+fa(x,t)‾δfb(x′,tR)-δfaδfb‾|q-q′|)=0. This equation contains the correlations of the fluctuations δfaδfb‾, which are neglected in a linearized theory.

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. (), which is the lowest-order electromagnetic field-charged particle interaction term.