By John Earman

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As anticipated, the relevance of these processes depends on the temperature regime of the plasma considered. In the models discussed in the following paragraphs, the magnitude of the various non-linear terms is expressed by symbolic coef®cients (for instance a, c, s; see list of symbols) that must be appropriately evaluated in terms of realistic plasma parameters in each case. The interest here is primarily that of studying the formal properties of these solutions. In the context of fusion research, the most important collision processes are, of course, those occurring at high temperature between light ions (typically deuterium and tritium) resulting in thermonuclear reaction with production of particles with high kinetic energy (typically 14 MeV neutrons) and helium nuclei (alpha particles): D T 3 He4 3X5 MeV n 14 MeVX Copyright © 2001 IOP Publishing Ltd 3X1X4 Exact Solutions of Reaction±Diffusion Equations 33 The fusion power density released by a D±T reaction is Pa PDT 5X6 Â 10ÿ13 nD nT hsviDT erg sÿ1 3X1X5 with hsviDT 3X68 Â 10ÿ12 Tÿ2a3 eÿ19X94T cm3 sÿ1 .

Furthermore, the type of the evolution differential equations is still formidable as they are non-linear and partial, PDE. There is scope for further simpli®cations without destroying essential information. ). In some cases, it is indeed possible to extend to con®guration space the procedure used to construct moments in velocity space. In principle, consider the simplest example of a ®rst-order ¯uid evolution equation, presently supposed with linear sources ¶Ya ÿr Á uYa S Ya X ¶t 3X1 Taking some suitable complete basis of orthonormal functions fck xY tg, one may expand the ®eld variable and the source function Ya xY t k k Copyright © 2001 IOP Publishing Ltd cak ck xY tY v xY t bk ck xY tY S xY t k sk ck xY t 3X2 30 Modelling a System with a Finite Number of Variables where the coef®cients are de®ned in terms of a suitable scalar product in the relevant functional space, cak ua Y ck Y bk uY ck Y sk SY ck and obtain a set of in®nite rate equations for the ®eld amplitudes, necessarily coupled with the momentum balance equation (not written here for brevity).

The cross-section is a fast growing function of energy, just below the excitation threshold, with a maximum close to the threshold, followed by a decaying behaviour, due to the decrease of the interaction time of the electron with the atom. The inelastic collisions lead to the formation of atoms in excited states, which last typically 10ÿ8 s before relaxation through emission of radiation. 2]. Of course, disexcitation of an excited atom leads to the inverse process e a0 6 a j e of formation of electrons with energy equal to the difference between impact energy and excited energy level.