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By D. Ya. Petrina

The monograph is dedicated to at least one of an important traits in modern mathematical physics, the research of evolution equations of many-particle platforms of statistical mechanics. The e-book systematizes rigorous effects acquired during this box in recent times, and it offers modern equipment for the research of evolution equations of infinite-particle platforms. The ebook is meant for specialists in statistical physics, mathematical physics, and chance idea and for college students of universities really good in arithmetic and physics

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10) for arbitrary t1 ; t2 . , qi qj D a. q1 ; p1 ; : : : ; qi ; pi ; : : : ; qi a; pj ; : : : ; xN / will be considered as identical in the sense that they determine the same initial state for the system. We will choose this point as initial that particles after the collision should depart. In this case, it is necessary to distinguish between positive or negative time. , one chooses x as the initial point. pi pj /  0, then one chooses x  as the initial point. , one chooses x as the initial point.

T / as in the usual Hamilton equations. For systems of N particles, one can also use equations with ı-functions but without boundary conditions or equivalent equations without ı-functions but with boundary conditions. Denote by WN the set of forbidden configurations in the phase space of N particles. p1 ; : : : ; pN /. This means that WN consists of phase points such that the distance between the centers of at least two hard spheres is less than their diameter a. qi ; pi / at time t D 0. t; x/ for t  R1 and fixed x defines a trajectory in the phase space.

Integration with respect to qsC1 ; : : : ; qsCn can be carried out by using ı-functions; therefore, all qsC1 ; : : : ; qsCn are expressed via q1 ; : : : ; qs . 15). x/sCi /. 11). 5). In this respect, the stochastic Boltzmann hierarchy has the same chaos property as the ordinary Boltzmann hierarchy. However, the proper stochastic Boltzmann hierarchy has two advantages. x/s / only outside all hyperplanes qi pi  D qj pj ; i; j 2 ¹1; : : : ; sº; 0    t . 5 Justification of the Boltzmann–Grad limit 43 can be derived from the Hamiltonian dynamics.

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