Asymptotic expansions of the hypergeometric function with two large parameters $-$ application to the partition function of a lattice gas in a field of traps

Abstract

The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function ${}_2{F}_1(a,b;c;z)$ in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of ${}_2{F}_1(a,b;c;z)$ have been provided in the literature, they are all limited by a narrow domain of validity, either in the complex plane of the variable or in the parameter space. Overcoming this restriction, we provide new asymptotic expansions for the hypergeometric function with two large parameters, which are valid for the entire complex plane of $z$ except for a few specific points. We show that these expansions work well even when we approach the possible singularity of ${}_2{F}_1(a,b;c;z)$, $|z|=1$, where the current expansions typically fail. Using our results we determine asymptotically the partition function of a lattice gas in a field of traps in the different possible physical limits of few/many particles and few/many traps, illustrating the applicability of the derived asymptotic expansions of the HGF in physics.

32 pages (SI included), 8 figures (+6 in SI), published in J. Phys. A: Math. Theor. Several references added and some notation and formal issues fixed. Expanded by a number of physical applications and applied to solve the problem of partition function of lattice gas (new structure and one section added). Abstract and conclusion improved

References in corpus (3)

• Coherent Excitation of a Two-Level Atom driven by a far off-resonant Classical Field: Analytical Solutions
• Statistics of black hole radiance and the horizon area spectrum
• The hypergeometric series for the partition function of the 2-D Ising model

Cited by in corpus (10)

• An even lighter QCD axion
• Entropy and modular Hamiltonian for a free chiral scalar in two intervals
• Finite Features of Quantum De Sitter Space
• Policies for elementary links in a quantum network
• Complex geodesics in de Sitter space
• Correlation functions in scalar field theory at large charge
• Exploring entropy by counting microstates of the p-state paramagnet
• Proper time to singularity and thermal correlators
• On the Density arising from the Domain of Attraction between Sum and Supremum: the $\alpha$-Sun operator
• Thermodynamic Constraints on the Emergence of Intersubjectivity in Quantum Systems