Mathematical conception of the gas theory
  In this paper, using of the rigorous statement and rigorous proof the Maxwell distribution as an example, we establish estimates of the distribution depending on the parameter $N$, the number of particles. Further, we consider the problem of the occurrence of dimers in a classical gas as an analog of Bose condensation and establish estimates of the lower level of the analog of Bose condensation. Using of the dequantization principles we find the relationship of this level to "capture" theory in the scattering problem corresponding to an interaction of the form of the Lennard-Jones potential. This also solves the problem of the Gibbs paradox.   We derive the equation of state for a nonideal gas as a result of pair interactions of particles in Lennard-Jones models and, for classical gases, discuss the $\lambda$-transition to the condensed state (the state in which $V_{\text{sp}}$ does not vary with increasing pressure; for heat capacity, this is the $\lambda$-point). 
V.P. Maslov