2019/09/30 by Sabine Jansen
Engineering · Materials Science · Mathematics · Physics and Astronomy · #Compressibility #Constant (computer programming) #Elasticity and Material Modeling #Hypercube #Navier-Stokes equation solutions #Phase (matter) #Phase transition #Set (abstract data type) #Solidification and crystal growth phenomena #math-ph #math.MP #math.PR #msc:82B20 #msc:82B26
paper · pdf · doi:10.1007/s10955-020-02531-1
published as J. Stat. Phys. (2020) · 31 pages
openalex created_date 2019/09/26 · openalex publication_date 2020/03/24 · arxiv created 2020/03/25 · arxiv updated 2020/03/26 · openalex updated_date 2026/08/05
Abstract We investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in \mathbb Zd <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> of sidelengths 2j <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> </mml:msup> </mml:math> , j∈ \mathbb N0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> . Cubes belong to an admissible set \mathbb B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> such that if two cubes overlap, then one is contained in the other. Cubes of sidelength 2j <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> </mml:msup> </mml:math> have activity zj <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> and density ρ j <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> . We prove explicit formulas for the pressure and entropy, prove a van-der-Waals type equation of state, and invert the density-activity relations. In addition we explore phase transitions for parameter-dependent activities zj(μ ) = exp ( 2dj μ - Ej) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>exp</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>dj</mml:mi> </mml:mrow> </mml:msup> <mml:mi>μ</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We prove a sufficient criterion for absence of phase transition, show that constant energies Ej≡ λ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>≡</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.