Abstract:
In the framework of statistical physics, the energy distribution functions for classical (Maxwell–Boltzmann distribution) and quantum (Fermi–Dirac and Bose–Einstein distributions) particles were established. The development of nanotechnology has led to the necessity to use the Tsallis energy distribution function for the ensemble of fractal particles. Distinctive features of the listed associations of particles are: the distinguishability of classical particles; the presence of spin (half-integer – fermions, whole – bosons) in quantum particles; geometrical differences of fractal particles. On the other hand, the interrelation of organizational levels of matter raises the question about the existence of a unified distribution function on energies of the mentioned objects. The type of distribution function is found by using the Boltzmann cell method, by calculating the large statistical sum, by using the variational method, etc. In this paper, the representation of the known distribution functions in the form of solutions of the corresponding Cauchy problems allowed us to establish the form of a unified expression to describe the average numbers of particles in quantum, classical and fractal ensembles. It is shown that at the “deformation” index $q$ = 0.5 the fractal ensemble is described by a function similar to the energy noise in the system. In systems with $q <$ 0, fractal ensembles originate at a certain threshold negative value (the internal energy of a fractal particle is less than its chemical potential) of the dimensionless energy.
Keywords:energy state, particle ensemble, temperature, chemical potential, fractal dimensionality.
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