### Abstract

The mean spherical model for Ising spin systems of Lewis and Wannier replaces the condition that each spin variable σi=±12 by the weaker condition that <σi2>=14Ω, where Ω=number of lattice sites. This model has the same properties, in the thermodynamic limitΩ→, as the spherical model of Berlin and Kac, and is immediately applicable, by a well-known isomorphism, to lattice gases with an interparticle potential v(r) of the form v(r)= for r=0 (no multiple occupancy of the same lattice site), v(r) finite for r0. We have now extended this model to more general lattice gases where v(r)= for r in some domain D, i.e., lattice gases of particles with extended hard cores. This permits extension of the model to continuum systems. We find, for this model, that the direct correlation function of Ornstein and Zernike is equal to -βv(r)(β the reciprocal temperature) for r not in D, and is determined for r in D by the requirement that the two-particle distribution functions n2(r1,r2) vanish for r1,2 in D. All higher order (modified) Ursell functions (spin semi-invariants) vanish for the model. The model thus yields the same pair distribution function as the Percus-Yevick integral equation for the case when v(r)=0 for r not in D, giving also, incidentally, an upper bound to the density for which solutions of this equation exist. The thermodynamic properties of this model are also discussed and it is shown that the partition function becomes singular in the continuum limit.

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

Pages (from-to) | 251-258 |

Number of pages | 8 |

Journal | Physical Review |

Volume | 144 |

Issue number | 1 |

DOIs | |

State | Published - 1966 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physical Review*,

*144*(1), 251-258. https://doi.org/10.1103/PhysRev.144.251

**Mean spherical model for lattice gases with extended hard cores and continuum fluids.** / Lebowitz, J. L.; Percus, Jerome.

Research output: Contribution to journal › Article

*Physical Review*, vol. 144, no. 1, pp. 251-258. https://doi.org/10.1103/PhysRev.144.251

}

TY - JOUR

T1 - Mean spherical model for lattice gases with extended hard cores and continuum fluids

AU - Lebowitz, J. L.

AU - Percus, Jerome

PY - 1966

Y1 - 1966

N2 - The mean spherical model for Ising spin systems of Lewis and Wannier replaces the condition that each spin variable σi=±12 by the weaker condition that <σi2>=14Ω, where Ω=number of lattice sites. This model has the same properties, in the thermodynamic limitΩ→, as the spherical model of Berlin and Kac, and is immediately applicable, by a well-known isomorphism, to lattice gases with an interparticle potential v(r) of the form v(r)= for r=0 (no multiple occupancy of the same lattice site), v(r) finite for r0. We have now extended this model to more general lattice gases where v(r)= for r in some domain D, i.e., lattice gases of particles with extended hard cores. This permits extension of the model to continuum systems. We find, for this model, that the direct correlation function of Ornstein and Zernike is equal to -βv(r)(β the reciprocal temperature) for r not in D, and is determined for r in D by the requirement that the two-particle distribution functions n2(r1,r2) vanish for r1,2 in D. All higher order (modified) Ursell functions (spin semi-invariants) vanish for the model. The model thus yields the same pair distribution function as the Percus-Yevick integral equation for the case when v(r)=0 for r not in D, giving also, incidentally, an upper bound to the density for which solutions of this equation exist. The thermodynamic properties of this model are also discussed and it is shown that the partition function becomes singular in the continuum limit.

AB - The mean spherical model for Ising spin systems of Lewis and Wannier replaces the condition that each spin variable σi=±12 by the weaker condition that <σi2>=14Ω, where Ω=number of lattice sites. This model has the same properties, in the thermodynamic limitΩ→, as the spherical model of Berlin and Kac, and is immediately applicable, by a well-known isomorphism, to lattice gases with an interparticle potential v(r) of the form v(r)= for r=0 (no multiple occupancy of the same lattice site), v(r) finite for r0. We have now extended this model to more general lattice gases where v(r)= for r in some domain D, i.e., lattice gases of particles with extended hard cores. This permits extension of the model to continuum systems. We find, for this model, that the direct correlation function of Ornstein and Zernike is equal to -βv(r)(β the reciprocal temperature) for r not in D, and is determined for r in D by the requirement that the two-particle distribution functions n2(r1,r2) vanish for r1,2 in D. All higher order (modified) Ursell functions (spin semi-invariants) vanish for the model. The model thus yields the same pair distribution function as the Percus-Yevick integral equation for the case when v(r)=0 for r not in D, giving also, incidentally, an upper bound to the density for which solutions of this equation exist. The thermodynamic properties of this model are also discussed and it is shown that the partition function becomes singular in the continuum limit.

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U2 - 10.1103/PhysRev.144.251

DO - 10.1103/PhysRev.144.251

M3 - Article

VL - 144

SP - 251

EP - 258

JO - Physical Review

JF - Physical Review

SN - 0031-899X

IS - 1

ER -