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arXiv:2512.03876 · Contributions to Plasma Physics

Generalized Beth–Uhlenbeck Entropy Formula From the Φ‐Derivable Approach

David Bernhard Blaschke, GERD RÖPKE, Gordon Baym

Published Jun 15, 2026• 1 citation
arXivHTMLPDF

Abstract

ABSTRACT We derive a generalized Beth–Uhlenbeck formula for the entropy of a dense fermion system with strong two‐particle correlations, including scattering states and bound states. We work within the Φ‐derivable approach to the thermodynamic potential. The formula takes the form of an energy‐momentum integral over a statistical distribution function times a unique spectral density. In the near mass‐shell limit, the spectral density reduces, contrary to naïve expectations, not to a Lorentzian but rather to a “squared Lorentzian” shape. The relation of the Beth–Uhlenbeck formula to the Φ‐derivable approach is exact at the two‐loop level for Φ. The formalism we develop, which extends the Beth–Uhlenbeck approach beyond the low‐density limit, includes Mott dissociation of bound states, in accordance with Levinson's theorem, and the self‐consistent back reaction of correlations in the fermion propagation. We discuss applications to further systems, such as quark matter and nuclear matter.

DOI
10.1002/ctpp.70145

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ABSTRACT We derive a generalized Beth–Uhlenbeck formula for the entropy of a dense fermion system with strong two‐particle correlations, including scattering states and bound states. We work within the Φ‐derivable approach to the thermodynamic potential. The formula takes the form of an energy‐momentum integral over a statistical distribution function times a unique spectral density. In the near mass‐shell limit, the spectral density reduces, contrary to naïve expectations, not to a Lorentzian but rather to a “squared Lorentzian” shape. The relation of the Beth–Uhlenbeck formula to the Φ‐derivable approach is exact at the two‐loop level for Φ. The formalism we develop, which extends the Beth–Uhlenbeck approach beyond the low‐density limit, includes Mott dissociation of bound states, in accordance with Levinson's theorem, and the self‐consistent back reaction of correlations in the fermion propagation. We discuss applications to further systems, such as quark matter and nuclear matter.
The Beth Uhlenbeck formula gives an exact (quantum) expression of the second virial correction to the equation of state of a (slightly degenerate) dilute gas. We show how this result can be extended to arbitrary degeneracy provided that the interaction potential has a sufficiently short range. For this purpose we develop a formalism based on the use of Ursell operators, which contain no symmetrization in themselves (they correspond to an auxiliary system of distinguishable particles) and we show how they can be used for a system of identical particles. A concise expression generalizing the Beth Uhlenbeck formula is obtained, which is equally valid for bosons and fermions Higher order corrections are also introduced. The formalism is rather general and will be applied to other cases in forthcoming articles.
A dense interacting Bose or Fermi gas is considered. Within the Kadanoff-Baym real-time Green-function technique a generalized kinetic equation is derived in the nonequilibrium ladder approximation. As a Markovian limit the Boltzmann-Uehling-Uhlenbeck equation is recovered. The relevance of retardation effects in the kinetic equation of dense systems is shown. It turns out that the stationary solution of the generalized kinetic equation reproduces the second virial coefficient of the equation of state, including the Beth-Uhlenbeck quantum corrections. We establish a generalized kinetic equation describing correlation effects in a consistent approach to nonequilibrium for Bose and Fermi systems. In this way we give a generalization of the Beth-Uhlenbeck formula that applies to nonequilibrium and to finite systems. The derivation makes use of the time dependent T-matrix equation, which establishes a generalization of the Bethe-Goldstone equation. The correlated parts of the density and energy, which can be interpreted as virial corrections, are thermodynamically consistent. In this way global energy conservation is fulfilled.
Abstract We investigate the thermodynamics at finite density of a generalized (2+1)(2 + 1)(2+1)D Gross–Neveu model of NNN fermion species with various types of four-fermion interactions. The motivation for considering such a generalized schematic model arises from taking the Fierz transformation of an effective Coulomb current–current interaction and certain symmetry-breaking interaction terms, as considered for graphene-type models in Ref. [29]. We then apply path-integral bosonization techniques, based on the large-NNN limit, to derive the thermodynamic potential. This includes the leading-order mean-field (saddle point) contribution as well as the next-order contribution of Gaussian fluctuations of exciton fields. The main focus of the paper is then the investigation of the thermodynamic properties of the resulting fermion–exciton plasma. In particular, we derive an extended Beth–Uhlenbeck form of the thermodynamic potential, and discuss the Levinson theorem and the decomposition of the phase of the exciton correlation into resonant and scattering parts.
Abstract The equation of state for a dense system of interacting Fermions is derived using functional integral technique. To investigate the formation of bound states, a Hubbard‐Stratonovich transformation is applied which expresses the action as functional of pair fields. The evaluation of the partition function in lowest orders with respect to the pair fields leads to a result which can be interpreted as the contribution of two‐particle states, accounting for density corrections as Pauli blocking and self‐energy shift. Comparison is performed with results for the equation of state of hot dense matter (plasmas, nuclear matter) obtained within a Green's function approach.
The two-particle spectrum in a dense Fermion system is treated using a thermodynamic Green function approach. A self-consistent description of possible bound states and a superfluid condensate in a correlated medium is given. A detailed understanding of superfluidity and superconductivity in correlated Fermion systems, especially the transition from the Cooper-paired state (weak-coupling limit) to the Bose condensed state of tightly bound pairs of Fermions (strong coupling limit) [1], is of great interest for very different physical systems. The problem of a unified treatment of Bose–Einstein condensation (BEC) and the Bardeen–Cooper–Schrieffer (BCS) phase arises not only in describing the electron structure of strongly correlated electron superfluids such as superconductors [2], the electronhole system in semiconductors [3], spin-polarized hydrogen and liquid He [4], but also in the theory of nuclear matter [5] and quark–gluon systems [6]. Recently, there have been several new approaches to this stimulating problem. A Monte-Carlo simulation of a finite He system at zero temperature has been performed in Ref. [7]. The microscopic theory of strongly coupled quantum fluids has been treated within the Jastrow approximation (cf. Ref. [8]) to obtain the ground state and low-lying excited states of strongly correlated boson quantum fluids. The crossover from weak to strong coupling superconductivity has been considered using a functional integral representation [2] (see also Ref. [7]).
We consider expectation values of local operators in (continuum) integrable models in a situation when the mean value is calculated in a single Bethe state with a large number of particles. We develop a form factor expansion for the thermodynamic limit of the mean value, which applies whenever the distribution of Bethe roots is given by smooth density functions. We present three applications of our general result: (i) in the framework of integrable quantum field theory (IQFT) we present a derivation of the LeClair–Mussardo formula for finite temperature one-point functions. We also extend the results to boundary operators in boundary field theories. (ii) We establish the LeClair–Mussardo formula for the non-relativistic 1D Bose gas in the framework of the algebraic Bethe ansatz (ABA). This way we obtain an alternative derivation of the results of Kormos et al for the (temperature dependent) local correlations using only the concepts of the ABA. (iii) In IQFT we consider the long-time limit of one-point functions after a certain type of global quench. It is shown that our general results imply the integral series found by Fioretti and Mussardo. We also discuss the generalized eigenstate thermalization hypothesis in the context of quantum quenches in integrable models. It is shown that a single mean value always takes the form of a thermodynamic average in a generalized Gibbs ensemble, although the relation to the conserved charges is rather indirect.