1063-780X/04/3010- $26.00 © 2004 MAIK “Nauka/Interperiodica” 0816 Plasma Physics Reports, Vol. 30, No. 10, 2004, pp. 816–864. From Fizika Plazmy, Vol. 30, No. 10, 2004, pp. 877–929. Original English Text Copyright © 2004 by Tsytovich, Morfill, Thomas. 1 1. THEORETICAL APPROACHES IN COMPLEX PLASMAS 1.1. Introduction In the previous parts of our review, we described in detail the processes of dust charging in complex plas- mas [1]; the elementary processes in complex plasmas [2], including the external forces that act on dust grains and dust–dust interactions; and state-of-the-art experi- ments [3]. We recall here the notation used in the previous parts of our review and used in this part as well. The quantity P = n d Z d /n i is the so-called Havnes parameter, showing the relative number of charges on the dust grains (here, n d is the dust density, n i is the ion density, and Z d is the dust charge in units of elementary charge). In state-of-the-art experiments, the parameter P (which is always less than unity) is on the order of unity (the minimum value of P in state-of-the-art experiments is 0.5 × 10 –2 . The dimensionless ion and electron densities are denoted as n = n i /n i, 0 and n e n e /n i, 0 (n i, 0 is a cer- tain characteristic ion density, either the density far from the structures or the density corresponding the basic state, the parameters of which are determined by 1 This article was submitted by the authors in English. both the charge and power balance (see [2])). The small parameter τ = T i /T e is on the order of 10 –2 in state-of- the-art experiments. The sharpness of a dust structure boundary is determined by another (much smaller) parameter τ d = T d e 2 / (where a is the dust size and T d is the dust temperature). In state-of-the-art experi- ments, τ d is no larger than 2 × 10 –3 , but it is usually on the order of 10 –6 . The ion Debye radius is determined by the expression λ Di = . To estimate the contributions from the various pro- cesses, we used in [3] expressions for the elementary processes in complex plasmas given in [2]. For simplic- ity, these processes were described by some average quantities, such as the electron, ion, and dust densities; the temperatures; and the ion drift velocity. The description of elementary processes in this form is very useful in applications. However, even in such a descrip- tion, some processes are determined by the particle velocity distributions. For example, this is true for the charging currents [1], the ion drag force, and the dust– dust interactions [2], which depend strongly on the ion velocity distribution. For simplicity and to be able to perform some estimates, we have already given (see [2]) the results for the case of thermal distributions of electrons and ions. More general is a kinetic approach T e 2 za T i /4 π n i 0 , e 2 DUSTY PLASMA Complex Plasmas IV: Theoretical Approaches to Complex Plasmas and Their Application 1 V. N. Tsytovich*, G. E. Morfill**, and H. Thomas** *Prokhorov Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia e-mail: tsytov@tp.lpi.ac.ru ** Max Planck Institut für Extraterrestrische Physik, 85740 Garching, Postfach 1312, Germany e-mail: gem@mpe.mpg.de, thomas@mpe.mpg.de Received September 26, 2002; in final form, December 23, 2003 Abstract—This paper completes a series of reviews devoted to the physics of complex plasmas, in which one of the components (dust) is in a crystalline or liquid state, while the others (electron, ions, and neutral atoms) are in a gaseous state. This review is devoted to the theoretical approaches used to describe complex plasmas so far. The main theoretical developments have been concentrated in the gaseous and weakly nonideal states of complex plasmas. Here, we describe the achievements in the new kinetic and new hydrodynamic approaches to complex plasmas. At present, only generalizations of the van der Waals approach for complex plasmas have been used to describe phase transitions and plasma condensation in complex plasmas. Here, criteria for transi- tions are described and compared with the existing experimental observations. Theoretical and numerical results for nonlinear structures, such as dust layers, dust voids, dust sheaths, and dust convective vortices, obtained by solving the stationary balance equations, are also discussed and compared with state-of-the-art experiments. At present, experiments in this field are progressing very fast, while theory is not advancing at the same rate of development. To further develop new theoretical models, one can use the elementary physical pro- cesses in complex plasmas described in the previous parts of the review. However, the detailed comparison of theory and experiments also needs more detailed experimental diagnostics of the phenomena observed. In the concluding part of our review, the trends in experiment and theory, as well as some existing applications, includ- ing industrial, environmental, and astrophysical ones, are described. © 2004 MAIK “Nauka/Interperiodica”.