Fractional integral representation in statistical thermodynamics of confined systems

Confined systems are usually treated as integer dimensional systems, like two dimensional (2D), 1D, and 0D, by considering extreme confinement conditions in one or more directions. This approach costs piecewise representations, some limitations in confinement interval, and the deviations from the true behaviors, especially when the confinement is neither strong nor weak. In this study, fractional integral representation (FIR) is proposed as a methodology to calculate the infinite summations in statistical thermodynamics for any dimension and confinement values. FIR directly incorporates the dimension as a control variable into calculation procedures and allows us to get solutions valid for the whole confinement and dimension scales, including the fractional ones. We define the dimension of a summation and used it in the proposed FIR to calculate the partition function. The first and the higher-order FIR are introduced and high accuracy results are achieved. FIR is then extended for a generalized function to calculate thermodynamic properties directly from their fundamental expressions based on infinite sums. By using the proposed FIR approach, the thermodynamic properties of a noninteracting Maxwell-Boltzmann gas confined in an elongated rectangular domain are determined. The excess quantities induced by confinement are examined for different confinement scenarios. FIR successfully predicts the true behavior of thermodynamic properties for the whole range of confinement and dimension scales. Defining and controlling the dimension allows designing new types of thermodynamic cycles. Besides the infinite-well potential for the confinement of particles with quadratic and linear dispersion relations, quadratic and quartic confining potentials are also considered to show the success of FIR. The proposed method not only incorporates the dimension into the calculation procedures but also constitutes an application of fractional calculus in statistical thermodynamics. FIR has many potential applications especially for Bose-Einstein condensation phenomenon which inherently contains dimensional transitions.