A general method of constructing dissipative equations is developed, following Ehrenfest's idea of coarse graining. The approach resolves the major issue of discrete time coarse graining versus continuous time macroscopic equations. Proof of the H theorem for macroscopic equations is given, several examples supporting the construction are presented, and generalizations are suggested.

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A recently introduced systematic approach to derivations of the macroscopic dynamics from the underlying microscopic equations of motions in the short-memory approximation [Gorban et al, Phys. Rev. E 63, 066124 (2001)] is presented in detail. The essence of this method is a consistent implementation of Ehrenfest's idea of coarse-graining, realized via a matched expansion of both the microscopic and the macroscopic motions. Applications of this method to a derivation of the nonlinear Vlasov-Fokker-Planck equation, diffusion equation and hydrodynamic equations of the fluid with a long-range mean field interaction are presented in full detail. The advantage of the method is illustrated by the computation of the post-Navier-Stokes approximation of the hydrodynamics which is shown to be stable unlike the Burnett hydrodynamics.

We demonstrate that laser beam collapse in highly nonlinear media can be described, for a large number of experimental conditions, by the geometrical optics approximation within high accuracy. Taking into account this fact we succeed in constructing analytical solutions of the eikonal equation, which are exact on the beam axis and provide ͑i͒ a first-principles determination of the self-focusing position, thus replacing the widely used empirical Marburger formula, ͑ii͒ a mathematical condition for obtaining the filament intensity, ͑iii͒ a benchmark solution for numerical simulations, and ͑iv͒ a tool for the experimental determination of the high-order nonlinear susceptibility. Successful comparison with experiment is presented.Nonlinear light self-focusing is a self-induced modification of the optical properties of a material which leads to beam collapse at a certain point z sf in the media. This effect, first observed in the 1960s, plays nowadays a key role in all scientific and technological applications related to the propagation of intense light beams ͓1͔, such as material processing ͓2͔, environmental sciences ͓3͔, femtochemistry in solutions ͓4͔, macromolecule chromatography ͓5͔, medicine ͓6͔, etc.Usually, z sf is estimated using the empirical Marburger formula ͓1,7,8͔, which has been constructed via fitting the results of extensive numerical simulations obtained for the case when the refractive index n is a linear function of the electric field intensity n = n͑I͒ = n 0 + n 2 I ͑n 2 Ͼ 0͒ ͓9͔. Under the geometrical optics approximation, and for the same form of refractive index, exact analytical expressions for z sf have been obtained in Refs. ͓10-13͔. In most modern experiments, however, high beam intensities are used for which the linear approximation breaks down, and further contributions to n͑I͒ must be considered ͓7,8,14͔. For these cases no general mathematical condition for the behavior of z sf and the filament intensity has been derived so far. Most theoretical results are based on numerical studies, or on variational calculations assuming a fixed beam profile inside the medium ͑see, e.g., ͓7͔, and references therein͒. An analytical theory, able to accurately describe beam collapse in highly nonlinear optics, is still missing. Moreover, it is widely believed that the exact treatment of beam propagation in a highly nonlinear medium can only be done numerically ͓1͔.In this paper, we construct analytical solutions for the eikonal equations with highly nonlinear forms of the refractive index avoiding any a priori assumptions on the form of the beam during propagation. The results obtained are exact on the beam axis within the geometrical optics approximation, which we demonstrate to be accurate for many of the situations taking place in modern experiments. Our approach permits not only to obtain exact expressions for z sf for different nonlinear functions n͑I͒ in ͑1+1͒ and ͑1+2͒ dimensions, but also to find a general mathematical framework which corrects traditionally used formulas for the filament inten...