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We derive a new effective macroscopic Cahn-Hilliard equation whose homogeneous free energy is represented by 4-th order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly…

Mathematical Physics · Physics 2013-10-02 Markus Schmuck , Marc Pradas , Greg A. Pavliotis , Serafim Kalliadasis

The Cahn-Hilliard equation is a fundamental model that describes the phase separation process in multi-component mixtures. It has been successfully extended to many different contexts in several scientific fields. In this survey article, we…

Analysis of PDEs · Mathematics 2022-06-22 Hao Wu

The paper presents a model of lateral phase separation in a two component material surface. The resulting fourth order nonlinear PDE can be seen as a Cahn-Hilliard equation posed on a time-dependent surface. Only elementary tangential…

Numerical Analysis · Mathematics 2020-03-18 Vladimir Yushutin , Annalisa Quaini , Maxim Olshanskii

It has been shown that self-assembled chains of active colloidal particles can present sustained oscillations. These oscillations are possible because of the effective diffusiophoretic forces that mediate the interactions of colloids do not…

Soft Condensed Matter · Physics 2018-03-23 S. Gonzalez , R. Soto

The Cahn-Hilliard equation with an externally-prescribed chaotic shear flow is studied in two and three dimensions. The main goal is to compare and contrast the phase separation in two and three dimensions, using high-resolution numerical…

Fluid Dynamics · Physics 2014-12-23 Lennon O'Naraigh , Selma Shun , Aurore Naso

Phase separation driven by nonequilibrium fluctuations is a hallmark of both living and synthetic active matter. Unlike equilibrium systems, where ordered states arise from the minimization of free energy, active systems are fueled by a…

The Cahn-Hilliard equation is one of the most common models to describe phase separation processes in mixtures of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic…

Numerical Analysis · Mathematics 2023-01-23 Stefan Metzger

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure…

Analysis of PDEs · Mathematics 2026-04-24 Maria Deliyianni , Boris Muha , Andrej Novak

Many systems exhibit a phase where the order parameter is spatially modulated. These patterns can be the result of a frustration caused by the competition between interaction forces with opposite effects. In all models with local…

Statistical Mechanics · Physics 2014-01-21 Simon Villain-Guillot

Motility and nonreciprocity are two primary mechanisms for self-organization in active matter. In a recent study [Phys. Rev. Lett. 131, 148301 (2023)], we explored their joint influence in a minimal model of two-species quorum-sensing…

Soft Condensed Matter · Physics 2025-04-15 Yu Duan , Jaime Agudo-Canalejo , Ramin Golestanian , Benoît Mahault

We formulate a time-dependent density functional theory for the coupled dynamics of electrons and nuclei that goes beyond the Born-Oppenheimer (BO) approximation. We prove that the time-dependent marginal nuclear probability density…

Chemical Physics · Physics 2025-11-14 Chen Li , Ryan Requist , E. K. U. Gross

Nonequilibrium phase transitions are routinely observed in both natural and synthetic systems. The ubiquity of these transitions highlights the conspicuous absence of a general theory of phase coexistence that is broadly applicable to both…

Statistical Mechanics · Physics 2023-05-30 Ahmad K. Omar , Hyeongjoo Row , Stewart A. Mallory , John F. Brady

We explore phase separation and kinetic arrest in a model active colloidal system consisting of self-propelled, hard-core particles with nonconvex shapes. The passive limit of the model, namely cross-shaped particles on a square lattice,…

Soft Condensed Matter · Physics 2020-05-11 Carl Merrigan , Kabir Ramola , Rakesh Chatterjee , Nimrod Segall , Yair Shokef , Bulbul Chakraborty

We consider a continuum model for motility-induced phase separation (MIPS) of active Brownian particles [J. Chem. Phys. 142, 224149 (2015)]. Using a recently introduced perturbative analysis [Phys. Rev. E 98, 020604(R) (2018)], we show that…

Soft Condensed Matter · Physics 2019-07-02 Lisa Rapp , Fabian Bergmann , Walter Zimmermann

Hyperuniformity emerges generically in the coarsening regime of phase-separating fluids. Numerical studies of active and passive systems have shown that the structure factor $S(q)$ behaves as $q^\varsigma$ for $q\to 0$, with hyperuniformity…

Soft Condensed Matter · Physics 2024-09-11 Filippo De Luca , Xiao Ma , Cesare Nardini , Michael E. Cates

Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatio-temporal pattern formation. Starting from a linear large-scale oscillatory instability -- a conserved-Hopf instability -- that naturally occurs in many active…

Pattern Formation and Solitons · Physics 2025-08-27 Tobias Frohoff-Hülsmann , Uwe Thiele

The Cahn--Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic…

Analysis of PDEs · Mathematics 2021-04-27 Patrik Knopf , Kei Fong Lam , Chun Liu , Stefan Metzger

Because of consuming energy to drive their motion, systems of active colloids are intrinsically out of equilibrium. In the past decade, a variety of intriguing dynamic patterns have been observed in systems of active colloids, and they…

Soft Condensed Matter · Physics 2020-04-16 Zhan Ma , Mingcheng Yang , Ran Ni

A Cahn-Hilliard-type theory for hydrodynamic fluctuations is proposed that gives a quantitative description of the slowly evolving spatial correlations and structures in density and flow fields in the early stages of evolution of freely…

Statistical Mechanics · Physics 2009-10-31 T. P. C. van Noije , M. H. Ernst

Non-equilibrium phase separating systems with reactions can break time-reversal symmetry (TRS) in two distinct ways. Firstly, the conservative and non-conservative sectors of the dynamics can be governed by incompatible free energies; when…

Soft Condensed Matter · Physics 2021-05-31 Yuting I. Li , Michael E. Cates