On the Boltzmann Equation applied in various sectors of String Theory and the Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory

In this paper we have showed the various applications of the Boltzmann equation in string theory and related topics. In the Section 1, we have described some equations concerning the time dependent multi-term solution of Boltzmann’s equation for charged particles in gases under the influence of electric and magnetic fields, the Planck’s blackbody radiation law, the Boltzmann’s thermodynamic derivation and the connections with the superstring theory. In the Section 2, we have described some equations concerning the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string cosmologies. In the Section 3, we have described some equations concerning the entropy of an eternal Schwarzschild black hole in the limit of infinite black hole mass, from the point of view of both canonical quantum gravity and superstring theory. We have described some equations regarding the quantum corrections to black hole entropy in string 2 theory. Furthermore, in this section, we have described some equations concerning the thesis “Can the Universe create itself?” and the adapted Rindler vacuum in Misner space. In the Section 4, we have described some equations concerning p-Adic models in HartleHawking proposal and p-Adic and Adelic wave functions of the Universe. Furthermore, we have described in the various Sections the various possible mathematical connections that we’ve obtained with some sectors of Number Theory and, in the Section 5, we have showed some mathematical connections between some equations of arguments above described and p-adic and adelic cosmology. 1. On some equations concerning the time dependent multi-term solution of Boltzmann’s equation for charged particles in gases under the influence of electric and magnetic fields. [1] [2] [3] The behaviour of a uniform swarm of electrons in gases under the influence of electric and magnetic fields is described by the Boltzmann equation. This equation represents the time t evolution of the distribution function   t c f , in velocity space c . The distribution function is defined such that  dc t c f , is the probability of finding a particle within dc of c at time t . The explicit form of Boltzmann’s equation for charged particle of charge q and mass m under the influence of spatially homogeneous orthogonal electric E and magnetic B fields is     0 , f f J c f B c E m q t f           (1.1) Swarm conditions are assumed to apply and   0 , f f J denotes the rate of change of f due to binary, particle conserving collisions with the neutral molecules only. The original Boltzmann collision operator and its semi-classical generalisation are used for elastic and inelastic process respectively:                  jk k j dc g d g jk g c f t c f c f t c f f f J 0 0 0 0 0 0 ' ˆ ; ' , ' , ,   . (1.2) Thence, the eq. (1.1) can be rewritten also             c f B c E m q t f               jk k j dc g d g jk g c f t c f c f t c f 0 0 0 0 0 ' ˆ ; ' , ' ,   . (1.2b)