Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Pfadintegrale und Cluster

Peter Borrmann

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

Recursion formulas are derived for Fermi-Dirac and Bose-Einstein statistical partition functions for systems where the energy can be written as a sum of one particle energies. Thus their exact partition functions can be calculated for any N. We consider a system of N particles for which the energy can be written as a sum of one-particle energies such as ideal gases or simple shell models. We aim at a recursion of Z(N)/ in terms of Z for subsystems. The idea may be seen as a vague analogy to cluster expansion methods.

Related papers

Quantum Statistics of Nonideal Systems

Frank H. Stillinger, John G. Kirkwood

A new cluster development for the logarithm of the grand partition function of a system of interacting particles is derived. The leading term in this expansion is the pressure exerted by an ideal Bose or Fermi gas at the same temperature and absolute activity $Z$ as the actual system. Succeeding terms involve quantum cluster integrals which themselves depend upon $Z$, unlike their classical analogs. The definition of these cluster integrals follows in a natural fashion using techniques illustrated by construction (in closed form) of the successive $Z$ derivatives of the Bose and Fermi ideal gas grand partition functions. It is not possible (except in the classical limit) to eliminate $Z$ explicitly between the pressure and density series, so that the equation of state must remain in parametric form.

Physical Review•Apr 15, 1960•2 citations

Investigations on finite ideal quantum gases

Heinz–Jürgen Schmidt, Jürgen Schnack

Recursion formulae of the N-particle partition function, the occupation numbers and its fluctuations are given using the single-particle partition function. Exact results are presented for fermions and bosons in a common one-dimensional harmonic oscillator potential, for the three-dimensional harmonic oscillator approximations are tested. Applications to excited nuclei and Bose–Einstein condensation are discussed.

•Jan 1, 1998•17 citations

Calculation of thermodynamic properties of finite Bose-Einstein systems

Peter Borrmann, Jens Harting, Oliver Mülken +1 more

We derive an exact recursion formula for the calculation of thermodynamic functions of finite systems obeying Bose-Einstein statistics. The formula is applicable for canonical systems where the particles can be treated as noninteracting in some approximation, e.g., like Bose-Einstein condensates in magnetic traps. The numerical effort of our computation scheme grows only linearly with the number of particles. As an example, we calculate the relative ground-state fluctuations and specific heats for ideal Bose gases with a finite number of particles enclosed in containers of different shapes.

arXiv:cond-mat/0008071•Physical Review A•Aug 1, 1999•51 citations

A Note on the Partition Function for Systems of Independent Particles

Douglas Ike Ford

The N-particle Bose-Einstein and Fermi-Dirac canonical partition functions are expressed explicitly in terms of the single-particle partition function. The nature of the approximations made in using corrected Maxwell-Boltzmann statistics is thereby made more obvious. A recurrence relation is developed to facilitate calculations using the exact partition function.

American Journal of Physics•Feb 1, 1971•15 citations

Cluster Expansion for Quantum Gases. II

Hiroshi Ichimura

A modified cluster expansion formula of the reduced density matrix for quantum gases is derived following the procedure analogous to the method which has been introduced in a previous paper (Prog. Theor. Phys. 31 (1964), 538) in connection with the cluster expansion of the partition function. This new cluster expansion formula is useful in highly degenerate cases. As an example, the effect of the hard-sphere interaction on the Bose-Einstein condensation is discussed.

Progress of Theoretical Physics•Mar 1, 1967•0 citations

Cluster Expansion Technique in the Many-Fermion Systems. I

Ki-ichi Nakamura

A new cluster expansion technique is developed for the calculation of the energy expectation value of the many-fermion systems with singular potentials.The wave function, which is studied in this paper, is of the formwhich, in a special case, coincides with Jastrow wave function.The formula for the energy expectation value is written in terms of the complex cluster integrals.The method introduced in the derivation of this formula has an analogy to the classical cluster expansion method in the theory 0f an imperfect classical gas of multicomponents.

Progress of Theoretical Physics•Dec 1, 1960•5 citations
Recursion formulas are derived for Fermi-Dirac and Bose-Einstein statistical partition functions for systems where the energy can be written as a sum of one particle energies. Thus their exact partition functions can be calculated for any N. We consider a system of N particles for which the energy can be written as a sum of one-particle energies such as ideal gases or simple shell models. We aim at a recursion of Z(N)/ in terms of Z for subsystems. The idea may be seen as a vague analogy to cluster expansion methods.
A new cluster development for the logarithm of the grand partition function of a system of interacting particles is derived. The leading term in this expansion is the pressure exerted by an ideal Bose or Fermi gas at the same temperature and absolute activity ZZZ as the actual system. Succeeding terms involve quantum cluster integrals which themselves depend upon ZZZ, unlike their classical analogs. The definition of these cluster integrals follows in a natural fashion using techniques illustrated by construction (in closed form) of the successive ZZZ derivatives of the Bose and Fermi ideal gas grand partition functions. It is not possible (except in the classical limit) to eliminate ZZZ explicitly between the pressure and density series, so that the equation of state must remain in parametric form.
We derive an exact recursion formula for the calculation of thermodynamic functions of finite systems obeying Bose-Einstein statistics. The formula is applicable for canonical systems where the particles can be treated as noninteracting in some approximation, e.g., like Bose-Einstein condensates in magnetic traps. The numerical effort of our computation scheme grows only linearly with the number of particles. As an example, we calculate the relative ground-state fluctuations and specific heats for ideal Bose gases with a finite number of particles enclosed in containers of different shapes.
A new cluster expansion technique is developed for the calculation of the energy expectation value of the many-fermion systems with singular potentials.The wave function, which is studied in this paper, is of the formwhich, in a special case, coincides with Jastrow wave function.The formula for the energy expectation value is written in terms of the complex cluster integrals.The method introduced in the derivation of this formula has an analogy to the classical cluster expansion method in the theory 0f an imperfect classical gas of multicomponents.