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Kießling, Friedrich; Nefzger, Peter; Nolasco, João Felix; Kaintzyk, Ulf

doi:10.1007/978-3-642-97879-1_12

Cited by 1 publication

(6 citation statements)

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“…In this case the system must find a compromise structure: either a drop of liquid surrounded by vapor or a bubble of vapor inside liquid, depending on the mechanical boundary conditions. It is reassuring to find this scenario confirmed numerically for V = V W and neutral mechanical boundary conditions, [33,32] and in particle simulations of many hard balls with attractive −r −6 interactions. [28,34] Interestingly enough, in Ref.…”

Section: Introductionmentioning

confidence: 71%

Section: Remarkmentioning

confidence: 92%

“…Interestingly enough, our numerical studies [33,32,51] revealed that the change from quasi-uniform vapor state to droplet state in the canonical ensemble is not gradual but involves another first-order phase transition which is embedded in the (α, N)-region "jumped" by the grand canonical phase transition; see also Refs. [32,51]. While a complete analytical proof of all the interesting details revealed by our numerical studies seems futile, our next theorem does assert the existence of a petit canonical first-order transition between a quasi-uniform vaporous and a strongly non-uniform free energy minimizer with same (α, N).…”

Section: Remarkmentioning

confidence: 92%

“…Ref. [51,32], even though the algebraic fixed point problem (26) has at most three solutions, still. This shows that naïve inferences from the algebraic fixed point problem (26) onto the integral equation (8) are not to be drawn.…”

Section: P Stability and The Gas ↔ Liquid Phase Transitionmentioning

confidence: 99%

“…We also note that since V N (| · |) ∈ L 1 (R 3 ), the fixed point problem (7) is not well defined in R 3 as it stands with V N * η given by (5); however, replacing V N * η by φ N and stipulating the familiar Poisson equation ∆φ N = 4πη, solutions in R 3 for the related PDE problem ∆φ N = 4π℘ ′ • (γ − αφ) do exist; it is easy to numerically compute radial solutions, which have applications in planetary science. [51,32] To summarize, non-uniform van der Waals fluid theory furnishes an accessible model to study the structure of non-uniform density functions η(r) of a hard-sphere fluid in R 3 . Evidently, boundary effects are absent in R 3 .…”

Section: Introductionmentioning

confidence: 99%

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Hard-sphere fluids with chemical self-potentials

Kiessling

Percus

2010

Journal of Mathematical Physics

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Existence, uniqueness and stability of solutions is 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 ⊂ R 3 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 ∈ 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 V N (|r|) = −|r| −1 , the Yukawa kernel V Y (|r|) = −|r| −1 e −κ|r| , and a van der Waals kernel V W (|r|) = −(1 + κ 2 |r| 2 ) −3 . The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic 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.

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