Hard-sphere fluids with chemical self-potentials

M. K H Kiessling, Jerome Percus

Research output: Contribution to journalArticle

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 languageEnglish (US)
Article number013091JMP
JournalJournal of Mathematical Physics
Volume51
Issue number1
DOIs
StatePublished - Jan 2010

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Hard-sphere Fluid
kernel
Fixed-point Equation
newton
fluids
Fluid
elliptic differential equations
Existence of Solutions
heat
Phase Transition
Heat
gas density
Liquid
uniqueness
liquids
hydrostatics
containers
statistical mechanics
partial differential equations
Nonlocal Boundary Conditions

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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

In: Journal of Mathematical Physics, Vol. 51, No. 1, 013091JMP, 01.2010.

Research output: Contribution to journalArticle

Kiessling, M. K H ; Percus, Jerome. / Hard-sphere fluids with chemical self-potentials. In: Journal of Mathematical Physics. 2010 ; Vol. 51, No. 1.
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