Zeroth order equations particles

Together with Eq. (66), this equation describes exactly the linear response of the system to an external field, with arbitrary initial conditions. Its physical meaning is very simple and may be explained precisely as for Eq. (66) 32 the evolution of the velocity distribution results in two effects (1) the dissipative collisions between the particles which are described by the same non-Markoffian collision operator G0o(T) 35 1 the field-free case and (2) the acceleration of the particles due to the external field. As we are interested in a linear theory, this acceleration only affects the zeroth-order distribution function It is... 

The kinetic equation for homogeneous systems is given by Eq. (7.47). The evolution equation for the zeroth-order moment of the NDF is null, which is due to the fact that the collision integral does not change the number of particles, or, more explicitly, f Cdf = 0. If the rate of change of the particle velocity (i.e. particle acceleration) is a linear function of the particle velocity (i.e. f = a + b ), then the evolution equation for the first-order moments are... 

All the dynamics of the energy diffusion process are included in the probability kernel. The energy loss and energy fluctuations of the particle are determined with the aid of perturbation theory. The zeroth-order (in u,) equation of motion for the unstable mode is... 

This equation is the main result of the present considerations. In order to define the two-particle self energy (w) and for establishing the connection to the familiar form of Dyson s equation we adopt a perturbation theoretical view where a convenient single-particle description (e. g. the Hartree-Fock approximation) defines the zeroth order. We will see later that the coupling blocks and vanish in a single-particle approximation. Consequently the extended Green s function is the proper resolvent of the zeroth order primary block which can be understood as an operator in the physical two-particle space ... 

Details of the method outlined below can be found i n [5], A stochastic system of several extensive variables X. is supposed tc be described by a master equation which can be explicitly written when the transition probabilities per unit time W( Xj Xj ) are known. In a reaction diffusion system, X. may be the number of chemical species a in a cell located by the vector r and is denoted by X. = X. Introducing the toaka tic, pot ntiai U defined by P = exp(-S - N U), where P is the probability, N is proportional to the total volume of the system and S stands for the normalization factor, we switch to the quasicontinu-ous intensive variables x = X /N, where N may be the mean number of particles in one cell of a reaction-diffusion system. If we assume that for all states for which liJ ( X j -> X1 ) are nonnegligible, x - xj is much smaller than 1, the equation for U can be expressed, at the zeroth order in 1/N, in terms of xj and 3U/9xj. liie thus obtain a Hamilton Jacobi type of equation ... 

Equation (7.19) was also obtained by Hounslow et al. (1988) after correcting the original equation proposed by Batterham et al. (1981). However, diis equation is nodiing odier than the fixed-pivot technique applied for a geometric grid with ratio equal to two and after superimposing the conservation of the zeroth- and first-order moments (with particle mass as internal coordinate). 

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