### Abstract

The existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container Λ- ^{R3} and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at r{small element of}Λ is the sum of the matter reservoir's contribution and a self-contribution -(V*ρ)(r), where ρ is the fluid density function and V a non-negative linear combination of the Newton kernel ^{VN}({pipe}r{pipe})=- ^{{pipe}r{pipe}-1}, the Yukawa kernel ^{VY}({pipe}r{pipe})=- ^{{pipe}r{pipe}-1e-κ{p ipe}r{pipe}}, and a van der Waals kernel ^{VW}({pipe}r{pipe})=- ^{(1+κ{script}2{p ipe}r{pipe}2)-3}. The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic partial differential equations (PDEs) of second order with a nonlinear, nonlocal boundary condition. We prove the existence of a grand canonical phase transition and of a petit canonical phase transition which is embedded in the former. The proofs suggest that, except for boundary layers, the grand canonical transition is of the type "all gas↔all liquid" while the petit canonical one is of the type "all vapor↔liquid drop with vapor atmosphere." The latter proof, in particular, suggests the existence of solutions with interface structure which compromise between the all-liquid and all-gas density solutions.

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

Article number | 013091JMP |

Journal | Journal of Mathematical Physics |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2010 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*51*(1), [013091JMP]. https://doi.org/10.1063/1.3279598

**Hard-sphere fluids with chemical self-potentials.** / Kiessling, M. K H; Percus, Jerome.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 51, no. 1, 013091JMP. https://doi.org/10.1063/1.3279598

}

TY - JOUR

T1 - Hard-sphere fluids with chemical self-potentials

AU - Kiessling, M. K H

AU - Percus, Jerome

PY - 2010/1

Y1 - 2010/1

N2 - The existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container Λ- R3 and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at r{small element of}Λ is the sum of the matter reservoir's contribution and a self-contribution -(V*ρ)(r), where ρ is the fluid density function and V a non-negative linear combination of the Newton kernel VN({pipe}r{pipe})=- {pipe}r{pipe}-1, the Yukawa kernel VY({pipe}r{pipe})=- {pipe}r{pipe}-1e-κ{p ipe}r{pipe}, and a van der Waals kernel VW({pipe}r{pipe})=- (1+κ{script}2{p ipe}r{pipe}2)-3. The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic partial differential equations (PDEs) of second order with a nonlinear, nonlocal boundary condition. We prove the existence of a grand canonical phase transition and of a petit canonical phase transition which is embedded in the former. The proofs suggest that, except for boundary layers, the grand canonical transition is of the type "all gas↔all liquid" while the petit canonical one is of the type "all vapor↔liquid drop with vapor atmosphere." The latter proof, in particular, suggests the existence of solutions with interface structure which compromise between the all-liquid and all-gas density solutions.

AB - The existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container Λ- R3 and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at r{small element of}Λ is the sum of the matter reservoir's contribution and a self-contribution -(V*ρ)(r), where ρ is the fluid density function and V a non-negative linear combination of the Newton kernel VN({pipe}r{pipe})=- {pipe}r{pipe}-1, the Yukawa kernel VY({pipe}r{pipe})=- {pipe}r{pipe}-1e-κ{p ipe}r{pipe}, and a van der Waals kernel VW({pipe}r{pipe})=- (1+κ{script}2{p ipe}r{pipe}2)-3. The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic partial differential equations (PDEs) of second order with a nonlinear, nonlocal boundary condition. We prove the existence of a grand canonical phase transition and of a petit canonical phase transition which is embedded in the former. The proofs suggest that, except for boundary layers, the grand canonical transition is of the type "all gas↔all liquid" while the petit canonical one is of the type "all vapor↔liquid drop with vapor atmosphere." The latter proof, in particular, suggests the existence of solutions with interface structure which compromise between the all-liquid and all-gas density solutions.

UR - http://www.scopus.com/inward/record.url?scp=77953244464&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953244464&partnerID=8YFLogxK

U2 - 10.1063/1.3279598

DO - 10.1063/1.3279598

M3 - Article

AN - SCOPUS:77953244464

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 1

M1 - 013091JMP

ER -