If for Coulomb collisions the so-called Rosenbluth potentials,

are exploited (Rosenbluth et al., 1957), the related frictional acceleration and diffusion terms can concisely be written as: Only here we used two indices to indicate the two species involved in the binary collision. The gamma factor is defined as , with the Coulomb logarithm being given via the plasma parameter (i.e., number of particles in the Debye sphere) through , with the total particle density . The Debye shielding length is given by with the individual species Debye length defined as . As one can see in Equation (15), the collision frequency is defined as the negative velocity gradient of the Rosenbluth potential. Hernandez and Marsch (1985) evaluated the collisional times scale for temperature and velocity exchange between drifting bi-Maxwellians, and Livi and Marsch (1986) and Marsch and Livi (1985a,b) calculated the collisional relaxation process and the associated rates for non-thermal solar wind VDFs and self-similar and kappa VFDs. This issue is also addressed in a previous review by Marsch (1991a). In the case of ion-electron collisions with electron thermal speed obeying , we have the collision frequency , showing the well-known strong speed dependence of the collisional friction on the relative speed of the colliding partners.In multi-fluid theory, the dynamic equations for each species are separately obtained (we suppress an index labelling the species) by taking the zeroth, first, second, etc., moment of Equation (10). The zeroth order moment gives the continuity equation for the density, , which reads:

Note that collisions do not change the particle number, i.e. , which is obvious from the form of Equation (13). Here the angular brackets indicate velocity space integration. Similarly, the momentum equation can be written: The couplings between the particles appear through the collisional (or wave-particle) momentum transfer rate, which is given by the first moment of the collision operator, . By taking the other moments of the velocity distribution function, one obtains the zero mean random velocity, the pressure (or stress) tensor, and the heat flux vector: The thermal pressure (and kinetic temperature ), and the thermal stress tensor are derived as follows: Here denotes the trace, is the unit tensor, and is Boltzmann’s constant. By definition the trace of vanishes. In terms of these quantities the internal energy or temperature equation is written as: The left side contains the adiabatic changes (of the entropy) owing to advection, and the right side the dissipation terms related with velocity shear (viscosity), heat conduction and collisional friction (Ohmic heating) at a volumetric rate , with .In principle, one can continue this scheme, by taking ever higher moments of (10), which will lead to an infinite chain. For practical purposes, this chain must be terminated, and a way of closure be found. Transport theory is expected to provide this closure, yielding transport coefficients that link the unknown moments , , and with the fluid variables , , and and their spatio-temporal variations. Transport relations providing closure are explicitly given in the review of Dum (1990). The general closure problem of the Vlasov–Boltzmann equation was recently revisited by Chust and Belmont (2006), with emphasis on the behaviour of a collisionless plasma. Yet, the potential relevance of this work for the solar wind remains to be demonstrated.

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