### Abstract

Variational principles for lower bounds to the energy, or free energy for T > 0°, of many-body systems are obtained in a form requiring density matrix minimization subject to certain model restrictions. The latter restrict the domain in which the density matrices can vary, and only utilize the energy - or free energy - for the model Hamiltonian H_{M}. Increasingly accurate bounds are obtained as the model system begins to resemble the system of interest, and the behavior of the error as H - H_{M} approaches zero is shown by two examples based upon the Ising model. Coupling the lower bound principle for the free energy with the standard Gibbs-Bogoliubov upper bound principle results in bounds on generalized susceptibility as well.

Original language | English (US) |
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Pages (from-to) | 2184-2193 |

Number of pages | 10 |

Journal | Journal of Mathematical Physics |

Volume | 8 |

Issue number | 11 |

DOIs | |

State | Published - Jan 1 1967 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Mathematical Physics*,

*8*(11), 2184-2193. https://doi.org/10.1063/1.1705140