Statistical mechanics based on fractional classical and quantum mechanics

Abstract

The purpose of this work is to study some problems in statistical mechanics based on the fractional classical and quantum mechanics. At first stage we have presented the thermodynamical properties of the classical ideal gas and the system of N classical oscillators. In both cases, the Hamiltonian contains fractional exponents of the phase space (position and momentum). At the second stage, in the context of the fractional quantum mechanics, we have calculated the thermodynamical properties for the black body radiation, studied the Bose-Einstein statistics with the related problem of the condensation and the Fermi-Dirac statistics.

Изучение Квантовых Статистических Систем В $D$-Мерном Пространстве С Использованием Дробной Производной

“…-обобщенная термическая длина волны [30]. Сравнение системы уравнений (29) с уравнением состояния свободного идеального бозе-газа [24] P…”

“…Результаты квантовой механики, полученные из понятия дробной производной, обсуждались некоторыми авторами [19], [21], [22]. В последние годы авторы работ [24], [25] обсуждали возможности введения этого понятия в квантовую статистическую физику и его физическую интерпретацию. Они также изучали некоторые проблемы теории квантовых статистических систем с энергетическим спектром дробно-степенного типа.…”

The effect of non-local derivative on Bose-Einstein condensation

“…In this work, we investigate Bose-Einstein condensation through the Caputo-Fabrizio derivative [9] and compare our findings with those presented in [6]. Unlike Riesz and Caputo FD, the Caputo-Fabrizio derivative is improved, so that the kernel becomes non-singular.…”

“…Moreover, using the Green's function approach, they showed that these systems can emulate real systems with physical inter-particle interactions. In this context, [6] studied thermodynamic properties of 3D ideal Fermi and Bose gases, 3D independent harmonic oscillators, and black-body radiation. Interestingly, [5,6] found that the critical condensation temperatures of bosons with this spectrum are larger than in the standard ideal case.…”

“…In this context, [6] studied thermodynamic properties of 3D ideal Fermi and Bose gases, 3D independent harmonic oscillators, and black-body radiation. Interestingly, [5,6] found that the critical condensation temperatures of bosons with this spectrum are larger than in the standard ideal case. More recently, [7,8] extended this approach to explore d-dimensional statistical systems.…”