Quantum statistics describe the behavior of indistinguishable particles through the Fermi–Dirac (FD) and Bose–Einstein (BE) distributions. This study compares both distributions directly to emphasize their fundamental statistical differences. A Python-based computational model was developed to visualize the contrasting behaviors of FD and BE distributions in a unified 3D representation. The simulation uses NumPy and Matplotlib with an energy interval of 0.01–1.00, temperatures ranging from 100 K to 600 K, and a chemical potential of The results show that the FD distribution forms a sharp profile near the Fermi energy and becomes step-like at low temperatures, while the BE distribution rises sharply as the chemical potential is approached, reflecting the characteristic mathematical divergence of Bose–Einstein statistics rather than indicating actual Bose–Einstein condensation. As temperature increases, the differences between the Fermi–Dirac and Bose–Einstein distributions become less pronounced, consistent with the expected high-temperature limit. The model yields an average relative deviation below 0.5%, confirming numerical stability and accuracy. This work provides an open-source Python framework for the visualization and comparative analysis of ideal quantum statistical distributions, intended primarily for educational and computational purposes rather than for predicting new physical phenomena.
Quantum statistics describe the behavior of indistinguishable particles through the Fermi–Dirac (FD) and Bose–Einstein (BE) distributions. This study compares both distributions directly to emphasize their fundamental statistical differences. A Python-based computational model was developed to visualize the contrasting behaviors of FD and BE distributions in a unified 3D representation. The simulation uses NumPy and Matplotlib with an energy interval of 0.01–1.00, temperatures ranging from 100 K to 600 K, and a chemical potential of The results show that the FD distribution forms a sharp profile near the Fermi energy and becomes step-like at low temperatures, while the BE distribution rises sharply as the chemical potential is approached, reflecting the characteristic mathematical divergence of Bose–Einstein statistics rather than indicating actual Bose–Einstein condensation. As temperature increases, the differences between the Fermi–Dirac and Bose–Einstein distributions become less pronounced, consistent with the expected high-temperature limit. The model yields an average relative deviation below 0.5%, confirming numerical stability and accuracy. This work provides an open-source Python framework for the visualization and comparative analysis of ideal quantum statistical distributions, intended primarily for educational and computational purposes rather than for predicting new physical phenomena.
Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
Abstract Knowledge of the chemical potential is essential in application of the Fermi–Dirac and the Bose–Einstein distribution functions for the calculation of properties of quantum gases. We give expressions for the chemical potential of ideal Fermi and Bose gases in 1, 2 and 3 dimensions in terms of inverse polylogarithm functions. We provide Mathematica functions for these chemical potentials together with low- and high-temperature series expansions. In the 3d Bose case we give also expansions about TB T B . The Mathematica routines for the series allow calculation to arbitrary order.
Bose gases and Fermi gases represent two fundamental particle systems in statistical mechanics, governed by Bose-Einstein and Fermi-Dirac distributions, respectively. This paper provides a comprehensive analysis of the thermodynamic quantities, phase transitions, chemical potentials, and energy level relationships of these gases. The study utilizes Laurent series expansions and group theory analysis to explore the statistical properties and behavior of Bose and Fermi gases under various conditions. Additionally, the paper derives a new energy-mass equation based on metric and curvature R.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
We study the semidefinite programs (SDPs) of quantum information whose variables are density operators, measurement operators, or unbounded positive semidefinite operators, together with their thermodynamic regularizations by the von Neumann, Fermi-Dirac, and Bose-Einstein entropies, which have been associated with Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, respectively. We find that this correspondence is exact: the Fermi-Dirac and Bose-Einstein regularized SDPs coincide with the constrained free-energy minimization of an ideal Fermi or Bose gas whose Hamiltonian and non-commuting charges are the second quantizations of the SDP data and whose one-body density matrix is the SDP variable, while the Boltzmann framework corresponds to the single-particle case. We show that the three frameworks can be unified under one framework, in which each entropy-regularized dual is, up to a term linear in the chemical potentials, the grand potential of the corresponding system; the duals are unconstrained and concave in the chemical-potential vector, with gradient and Hessian given by thermal expectation values that hybrid quantum-classical algorithms can estimate. We further identify the Fermi-Dirac and Bose-Einstein relative entropies with the Umegaki relative entropy between quasi-free states, which yields their monotonicity under gauge-covariant Gaussian channels. The result places a wide class of SDPs and their solvers within the thermodynamics of ideal quantum gases and points to extensions to interacting gases.
In this work we present a classical kinetic model of intermediate statistics. In the case of Brownian particles we show that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions can be obtained, just as the Maxwell-Boltzmann (MB) distribution, as steady states of a classical kinetic equation that intrinsically takes into account an exclusion-inclusion principle. In our model the intermediate statistics are obtained as steady states of a system of coupled nonlinear kinetic equations, where the coupling constants are the transmutational potentials ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\kappa}}\mathrm{\ensuremath{\kappa}}\ensuremath{'}}$. We show that, besides the FD-BE intermediate statistics extensively studied from the quantum point of view, we can also study the MB-FD and MB-BE ones. Moreover, our model allows us to treat the three-state mixing FD-MB-BE intermediate statistics. For boson and fermion mixing in a D-dimensional space, we obtain a family of FD-BE intermediate statistics by varying the transmutational potential ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. This family contains, as a particular case, when ${\mathrm{\ensuremath{\eta}}}_{\mathrm{B}\mathrm{F}=0}$, the quantum statistics recently proposed by L. Wu, Z. Wu, and J. Sun [Phys. Lett. A 170, 280 (1992)]. When we consider the two-dimensional FD-BE statistics, we derive an analytic expression of the fraction of fermions. When the temperature T\ensuremath{\rightarrow}\ensuremath{\infty}, the system is composed by an equal number of bosons and fermions, regardless of the value of ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. On the contrary, when T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$ becomes important and, according to its value, the system can be completely bosonic or fermionic, or composed both by bosons and fermions.