We consider simple mean field continuum models for first order liquid-liquid demixing and solidliquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn-Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid-liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit.However, in a system of finite size or when a finite time horizon is considered, metastable states often play an important role and even unstable states may be crucial, as transient states, for extended time periods. The full set of states and their dependence on the various control parameters is conveniently presented in the form of bifurcation diagrams, well known in the context of dynamical systems and pattern formation theory [3][4][5]. The place of thermodynamic phase diagrams is taken by 'morphological phase diagrams' or state diagrams and stability diagrams [3,4,6]. The notion of a Maxwell point is often used in the context of pattern formation in nonconserved systems [7][8][9][10][11] to indicate equal energy states because of its dynamical significance [12,13]. In this context this notion applies equally to finite and infinite systems [7,8] although for finite systems it lacks the thermodynamic relevance as the condition for phase coexistence. In the context of buckling the corresponding concept is the Maxwell load [14].In this paper we show and discuss how the discontinuities in the TL represented by the Maxwell construction arise from the bifurcation diagrams relating stable, metastable and unstable steady states in finite-size systems. We focus on two systems: (i) phase decomposition of a binary liquid mixture and (ii) the liquid to crystalline solid phase transition. We investigate the transitions that occur in the context of the most basic mean-field continuum models for these two different phase transitions, namely, the Cahn-Hilliard equation [15][16][17] and the phase field crystal (PFC) model (or conserved Swift-Hohenberg equation) [18][19][20].Some aspects related to this question have been considered previously, in particular in relation to the nature of some of the states that can arise in finite-size systems in the two-phase region. References [21-24] describe theory and computer simulation results for atomistic models exhibiting gas-liquid, liquid-hexatic and hexaticsolid phase transitions that indicate how the Maxwell construction develops as the system size increases or the temperature decreases. For example, figures 3-5 of [23] compare Monte-Carlo computer simulation results in a finite three-dimensional domain (see also figures 2 and 3 of [22]) with the results from a capillary drop type model, also in a finite domain, with a mean-field expression for the chemi...
The classical Cahn–Hilliard (CH) equation corresponds to a gradient dynamics model that describes phase decomposition in a binary mixture. In the spinodal region, an initially homogeneous state spontaneously decomposes via a large-scale instability into drop, hole or labyrinthine concentration patterns of a typical structure length followed by a continuously ongoing coarsening process. Here, we consider the coupled CH dynamics of two concentration fields and show that non-reciprocal (or active or non-variational) coupling may induce a small-scale (Turing) instability. At the corresponding primary bifurcation, a branch of periodically patterned steady states emerges. Furthermore, there exist localized states that consist of patterned patches coexisting with a homogeneous background. The branches of steady parity-symmetric and parity-asymmetric localized states form a slanted homoclinic snaking structure typical for systems with a conservation law. In contrast to snaking structures in systems with gradient dynamics, here, Hopf instabilities occur at a sufficiently large activity, which results in oscillating and travelling localized patterns.
Active field theories, such as the paradigmatic model known as ‘active model B+’, are simple yet very powerful tools for describing phenomena such as motility-induced phase separation. No comparable theory has been derived yet for the underdamped case. In this work, we introduce active model I+, an extension of active model B+ to particles with inertia. The governing equations of active model I+ are systematically derived from the microscopic Langevin equations. We show that, for underdamped active particles, thermodynamic and mechanical definitions of the velocity field no longer coincide and that the density-dependent swimming speed plays the role of an effective viscosity. Moreover, active model I+ contains an analog of the Schrödinger equation in Madelung form as a limiting case, allowing one to find analoga of the quantum-mechanical tunnel effect and of fuzzy dark matter in active fluids. We investigate the active tunnel effect analytically and via numerical continuation.
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