### Abstract

Within the framework for equilibrium statistical mechanics of Fermion lattice systems formulated in our preceding work, we study the local thermodynamical stability (LTS) as an alternative characterization of equilibrium states, which works without the translation invariance assumption for the states. We propose two versions, called LTS-M (mathematical) and LTS-P (physical) according to the choice of the algebra of the outside system for a local region I, LTS-M for the commutant of the local subalgebra A(I) and LTS-P for the subalgebra A(I^{c}) for the complementary region I^{c} of I. We show that the two conditions are equivalent for even states, evenness referring to Fermion numbers. By applying known methods of proof by Sewell and Araki, the following results are obtained: (1) The LTS-M condition implies the dKMS condition for a general state φ for an arbitrary general potential (in our technical sense). The same statement holds for the LTS-P condition if φ is even. (2) The LTS-M or LTS-P condition for a translation invariant state implies that the state is a solution of the variational principle for any translation covariant standard potential.

Original language | English |
---|---|

Pages (from-to) | 33-45 |

Number of pages | 13 |

Journal | Letters in Mathematical Physics |

Volume | 62 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 Oct |

Externally published | Yes |

### Fingerprint

### Keywords

- Equilibrium statistical mechanics
- Fermion lattice systems
- Local stability

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Letters in Mathematical Physics*,

*62*(1), 33-45. https://doi.org/10.1023/A:1021614924630

**Local thermodynamical stability of Fermion lattice systems.** / Araki, Huzihiro; Moriya, Hajime.

Research output: Contribution to journal › Article

*Letters in Mathematical Physics*, vol. 62, no. 1, pp. 33-45. https://doi.org/10.1023/A:1021614924630

}

TY - JOUR

T1 - Local thermodynamical stability of Fermion lattice systems

AU - Araki, Huzihiro

AU - Moriya, Hajime

PY - 2002/10

Y1 - 2002/10

N2 - Within the framework for equilibrium statistical mechanics of Fermion lattice systems formulated in our preceding work, we study the local thermodynamical stability (LTS) as an alternative characterization of equilibrium states, which works without the translation invariance assumption for the states. We propose two versions, called LTS-M (mathematical) and LTS-P (physical) according to the choice of the algebra of the outside system for a local region I, LTS-M for the commutant of the local subalgebra A(I) and LTS-P for the subalgebra A(Ic) for the complementary region Ic of I. We show that the two conditions are equivalent for even states, evenness referring to Fermion numbers. By applying known methods of proof by Sewell and Araki, the following results are obtained: (1) The LTS-M condition implies the dKMS condition for a general state φ for an arbitrary general potential (in our technical sense). The same statement holds for the LTS-P condition if φ is even. (2) The LTS-M or LTS-P condition for a translation invariant state implies that the state is a solution of the variational principle for any translation covariant standard potential.

AB - Within the framework for equilibrium statistical mechanics of Fermion lattice systems formulated in our preceding work, we study the local thermodynamical stability (LTS) as an alternative characterization of equilibrium states, which works without the translation invariance assumption for the states. We propose two versions, called LTS-M (mathematical) and LTS-P (physical) according to the choice of the algebra of the outside system for a local region I, LTS-M for the commutant of the local subalgebra A(I) and LTS-P for the subalgebra A(Ic) for the complementary region Ic of I. We show that the two conditions are equivalent for even states, evenness referring to Fermion numbers. By applying known methods of proof by Sewell and Araki, the following results are obtained: (1) The LTS-M condition implies the dKMS condition for a general state φ for an arbitrary general potential (in our technical sense). The same statement holds for the LTS-P condition if φ is even. (2) The LTS-M or LTS-P condition for a translation invariant state implies that the state is a solution of the variational principle for any translation covariant standard potential.

KW - Equilibrium statistical mechanics

KW - Fermion lattice systems

KW - Local stability

UR - http://www.scopus.com/inward/record.url?scp=0038569439&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038569439&partnerID=8YFLogxK

U2 - 10.1023/A:1021614924630

DO - 10.1023/A:1021614924630

M3 - Article

AN - SCOPUS:0038569439

VL - 62

SP - 33

EP - 45

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 1

ER -