### Abstract

The method of functional differentiation, used in classical statistical mechanics to obtain approximate integral equations for the pair distribution function, is extended to quantum systems obeying Maxwell-Boltzmann statistics. The grand partition function is written as a path integral, and space-temperature distributions are generated from it by successive functional differentiation. Distributions can be expanded in powers of an external potential via a functional Taylor series. When the series is truncated so that no distributions higher than second order enter and the external potential is specialized to one arising from a fixed particle in the system, coupled integral equations result for the two kinds of pair distributions. The first-order Ursell function calculated from these equations yields the Montroll-Ward ring summation for the grand potential. This approximate Ursell function determines also the Fourier transform of the momentum density (useful in calculating the p = 0 occupation number). Although divergences appear in the low-temperature limit, they can be removed by a simple modification of the independent functional. An extension to Fermi-Dirac and Böse-Einstein statistics is indicated at appropriate places in the text.

Original language | English (US) |
---|---|

Pages (from-to) | 1029-1036 |

Number of pages | 8 |

Journal | Journal of Mathematical Physics |

Volume | 11 |

Issue number | 3 |

State | Published - 1970 |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*11*(3), 1029-1036.

**Space-temperature correlations in quantum-statistical mechanics.** / Kunkin, William; Percus, Jerome.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 11, no. 3, pp. 1029-1036.

}

TY - JOUR

T1 - Space-temperature correlations in quantum-statistical mechanics

AU - Kunkin, William

AU - Percus, Jerome

PY - 1970

Y1 - 1970

N2 - The method of functional differentiation, used in classical statistical mechanics to obtain approximate integral equations for the pair distribution function, is extended to quantum systems obeying Maxwell-Boltzmann statistics. The grand partition function is written as a path integral, and space-temperature distributions are generated from it by successive functional differentiation. Distributions can be expanded in powers of an external potential via a functional Taylor series. When the series is truncated so that no distributions higher than second order enter and the external potential is specialized to one arising from a fixed particle in the system, coupled integral equations result for the two kinds of pair distributions. The first-order Ursell function calculated from these equations yields the Montroll-Ward ring summation for the grand potential. This approximate Ursell function determines also the Fourier transform of the momentum density (useful in calculating the p = 0 occupation number). Although divergences appear in the low-temperature limit, they can be removed by a simple modification of the independent functional. An extension to Fermi-Dirac and Böse-Einstein statistics is indicated at appropriate places in the text.

AB - The method of functional differentiation, used in classical statistical mechanics to obtain approximate integral equations for the pair distribution function, is extended to quantum systems obeying Maxwell-Boltzmann statistics. The grand partition function is written as a path integral, and space-temperature distributions are generated from it by successive functional differentiation. Distributions can be expanded in powers of an external potential via a functional Taylor series. When the series is truncated so that no distributions higher than second order enter and the external potential is specialized to one arising from a fixed particle in the system, coupled integral equations result for the two kinds of pair distributions. The first-order Ursell function calculated from these equations yields the Montroll-Ward ring summation for the grand potential. This approximate Ursell function determines also the Fourier transform of the momentum density (useful in calculating the p = 0 occupation number). Although divergences appear in the low-temperature limit, they can be removed by a simple modification of the independent functional. An extension to Fermi-Dirac and Böse-Einstein statistics is indicated at appropriate places in the text.

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M3 - Article

AN - SCOPUS:36849096784

VL - 11

SP - 1029

EP - 1036

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

ER -