Abstract We propose a generalized quantum statistical framework, termed Max–Min statistics, which imposes simultaneous lower ( N min ) and upper ( N max ) bounds on the occupation numbers of single-particle quantum levels. The formalism is developed using two complementary but equivalent approaches: a combinatorial microstate-counting scheme based on modified statistical weights, and an operator-algebraic construction employing a generalized number operator with constrained ladder operations. The resulting distribution function provides a unified description that continuously interpolates between Bose–Einstein, Fermi–Dirac, Gentile, and Ewkon statistics as limiting cases. Thermodynamic consequences of the occupation constraints are examined within the grand canonical ensemble, and the geometric properties of the corresponding thermodynamic state space are analyzed using Ruppeiner geometry. The scalar curvature is shown to vary systematically with the allowable occupation range Δ N = N max − N min , reflecting changes in statistical correlations induced by the imposed bounds. In particular, non-zero lower occupancy constraints lead to distinct thermodynamic behaviors that require regularization via an energy cutoff. These results establish the Max–Min framework as a flexible and mathematically consistent tool for exploring bounded and intermediate quantum statistics from a formal and phenomenological perspective.
Abstract We propose a generalized quantum statistical framework, termed Max–Min statistics, which imposes simultaneous lower ( N min ) and upper ( N max ) bounds on the occupation numbers of single-particle quantum levels. The formalism is developed using two complementary but equivalent approaches: a combinatorial microstate-counting scheme based on modified statistical weights, and an operator-algebraic construction employing a generalized number operator with constrained ladder operations. The resulting distribution function provides a unified description that continuously interpolates between Bose–Einstein, Fermi–Dirac, Gentile, and Ewkon statistics as limiting cases. Thermodynamic consequences of the occupation constraints are examined within the grand canonical ensemble, and the geometric properties of the corresponding thermodynamic state space are analyzed using Ruppeiner geometry. The scalar curvature is shown to vary systematically with the allowable occupation range Δ N = N max − N min , reflecting changes in statistical correlations induced by the imposed bounds. In particular, non-zero lower occupancy constraints lead to distinct thermodynamic behaviors that require regularization via an energy cutoff. These results establish the Max–Min framework as a flexible and mathematically consistent tool for exploring bounded and intermediate quantum statistics from a formal and phenomenological perspective.
We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for α<0. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter α. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature T∗, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature Tc, the ratio T∗/Tc depends universally on α. For negative α, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.
This chapter follows on from the previous chapter in quantum statistical mechanics but specialising on systems with identical particles. Using Gibbs prescritpion on generic states from Chapter 3, the occupation number representation is introduced. Constraints imposed on statistics by irreducible representations of the permutation group are discussed. These group-theoretic considerations are used to justify the use of Gentile’s parastatistics. Fermions and bosons are introduced as special cases of Gentile’s statistics, corresponding to the trivial representation of the permutation group for bosons, and the sign representation of the permutation group for fermions. Basic applications to fermions and bosons is given, including the Fermi–Dirac and Bose–Einstein statistics. A detailed expose of why photons are said to have zero chemical potential is also proposed.
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 compare three formulations of thermodynamic entropy for a non-interacting bosonic gas: (i) the grand-canonical Bose-Einstein entropy, (ii) the finite-N canonical entropy obtained from the exact partition function (Ziff, Uhlenbeck, and Kac construction), and (iii) the entropy of a multinomial distribution with Boltzmann categorical probabilities and temperature determined from Clausius' equation. It is well-known that the grand-canonical Bose-Einstein systematically overestimates entropy of canonical systems in regimes where particle-number fluctuations are significant. The exact canonical entropy correctly enforces the particle-number constraint, but recent experimental results suggest that it also overestimates particle-number fluctuations below the crossover temperature. The multinomial distribution is less common in thermodynamics. It addresses in a mathematically exact way the puzzle of the famous -log(N!) term introduced by Gibbs as a deus ex machina in discussions of thermodynamic entropy. One remarkable consequence is that the entropy of the multinomial distribution overcomes the issue of negative entropy at low temperature that affects the Gibbs and the Sackur-Tetrode entropies at low temperature. The analysis presented in the paper shows that the multinomial distribution, equipped with a categorical distribution calibrated in such a way that the resulting multinomial entropy fits Clausius' equation, provides accurate approximations to the canonical entropy in the classical regime, while it is smaller than the canonical entropy below the crossover temperature. One feature of the multinomial distribution is that it predicts lower peak variance of the number of particles in the ground state than the canonical distribution. This is in agreement with recent experimental results; hence, this paper identifies the thermodynamically calibrated multinomial distribution as a candidate alternative to the canonical distribution for thermodynamic bosonic entropy in finite systems.
An interesting result arises in information theory, namely that if one considers a microstate with large N trials, each representing a state i with probability p(i), then even though this is only one of many possible microstates (each being a permutation of each other) it carries information about all of the permutations based on the average occupancy number. In particular, the probability of a microstate may be written as: Product over i p(ei) [power Np(ei) ] for large N. This holds for any microstate and so all have the same probability. Given that all of the microstates are permutations in the large N case, this probability is equivalent to 1/ {number of permutations} where Number of permutations = N! / { Product over i n(ei) }. We point out the above example, because it seems there are at least two main strategies in statistical mechanics. One is to begin with a set of average occupancy numbers n(ei/T) whose specific form is unknown. The idea then seems to be to seek degeneracy in the system, possibly among the arrangement of the n(ei)’s or degeneracy within single levels ei and to maximize the number of degenerate arrangements in order to account for as little information as possible. This approach may be used to calculate the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions. In the case of the latter two, degeneracy in the single energy levels ei is key. One then maximizes the ln of the degenerate arrangements subject to constraints such as Sum over i ei n(ei) and Sum over i n(ei) = N. The degeneracies themselves are associated with momentum degeneracies yielding the same energy. A completely different approach, it seems, is to consider exp(-(E-uN)/T) as representing the probability of a state with energy E and N particles and to use either the grand canonical ensemble i.e. sum the various exp(-(E-uN)/T)’s. These may then be broken into single particle sums using E(N) = sum over i ei n(ei) for n(ei) variable or sum over i ei for N fixed. Even though momentum degeneracy still exists, it is not part of the n sum of exp(-(E-uN)/T) factors which gives a single energy level probability distribution with no maximization necessary. Thus we point out that exp(-(E-uN)/T) for microstates seems to incorporate information about degeneracies without them needing to be explicitly added. On the other hand, if one starts with average n(ei) (without knowing their form), one must explicitly consider degeneracies in order to maximize so that there is no unnecessary information in the system. Both approaches should yield the same result. We consider the Bose-Einstein example of a photon gas as an example.
We develop a statistical mean-field theory of finite quantum systems in thermal equilibrium. Our formulation employs the canonical ensemble of statistical mechanics, and it enables us to analytically determine the occupation number distributions of interacting particles obeying Bose–Einstein or Fermi–Dirac statistics. We have also developed a numerical procedure that enables us to obtain a universal scaled occupation number distribution that, for a given total number of interacting particles in a finite system, makes it possible to determine the occupation number distribution for any temperature. The developed mean-field theory is applicable to a wide range of atomic, nuclear and condensed matter systems for which finite-size effects can play an important role. In particular, the present approach makes it possible to formulate a finite temperature mean-field theory for a specific ion in a dense plasma.