Memory as activity: pattern formation in a conserved scalar field

TL;DR

The paper addresses how temporal memory and delayed feedback can drive pattern formation in scalar active matter. It formulates a continuum Cahn-Hilliard–type model with a history-dependent active chemical potential $\mu_{ac}$ controlled by a memory kernel $\gamma$ and an activity strength $\alpha$, and analyzes linear stability, dispersion, and nonlinear states, including traveling waves and spirals. Key contributions include a formal memory-driven CH dynamics with frequency-dependent mobility, a Lambert W–based analysis of stability revealing non-equilibrium oscillatory modes, and generalizations to multi-delay and nonlinear active terms that enrich the pattern landscape. The results establish a minimal framework for memory-induced pattern formation in conserved fields, with potential implications for active solids and aging phenomena that involve temporal feedback and non-Markovian dynamics.

Abstract

We explore the concept of memory in scalar active matter systems, focusing on the collective dynamics of particles whose interactions depend on their evolutionary history rather than on their present configuration. We do so by introducing the idea of an active particle whose velocity acquires an active contribution that depends on its past trajectory suitably weighted by a memory kernel. The memory kernel is unrelated to the thermal noise acting on the particle, meaning that the particle breaks detailed balance at the microscopic level. The number density of these active particles is described by a Cahn-Hilliard equation, which typically describes passive phase separation, suitably modified to account for this particular non-equilibrium effect. Through theory and simulations we establish the novel emergent features of the model and use the example of time delayed interactions to highlight the novel pattern-forming abilities of the model.

Figures (7)

• Figure 1: Memory as activity: (a) We consider an active particle that has access to information about its own past evolution. The memory persists for a finite duration, meaning the particle stores and utilizes information from a limited time in the past. In this paper, we investigate the collective behavior of a mixture of such particles, which are active due to the violation of fluctuation-dissipation relation at the level of a single particle. (b) The feedback received by the particles from the past is associated with a strength $\alpha$ (see Sec. \ref{['sec:scalarMemory']}), and the inter-particle interaction strength effective interaction ($a$) of these particles determines broadly the collective behaviour observed in the system. Strikingly, when the two effect compete in a particular way (bottom-right sector), spontaneously moving patterns, including travelling waves, spirals, or in more complex cases irregular chaotic, emerge in the scalar system.
• Figure 2: Dynamical steady states in Cahn-Hilliard with delay: As an example of the general framework we consider a system where the particles receives feedback from a time $\tau_d$ in the past with strength $\alpha$, a parameter that controls the level of activity in the system (see Eq. \ref{['eq:activeJ']}, and Sec. \ref{['sec:scalarMemory']}). An exploration of the phase space in the $\alpha - \tau_d$ plane, in one and two space dimensions reveals four distinct stable steady states: uniform state, phase separation, traveling waves (with or without long-lived defects), and spiral patterns. (a) As summarised in Fig. \ref{['fig:MemorySchematic']} (b) system-spanning bulk phases appear when the passive and the active are both attractive, or when the first is repulsive and the second is attractive. (b) Travelling patterns arise when physical interactions are attractive while the time-delayed interactions are repulsive. (c) Travelling waves are accompanied with long lived dislocations. (d) Typically at large enough $\alpha$ multiple stable spirals are found. (e) Qualitatively similar features are also observed in one dimension where stable, often positionally ordered defects persist in the steady state. (f) As a fixed position in space, the value of the field at the current time and the lagged time traces a limit cycle in phase space.
• Figure 3: State diagram for Cahn-Hilliard with delay: The phase space diagram of 2D steady states across varying values of $\alpha$ and $\tau_d$ reveals four distinct phase-separated regimes: homogeneous phase separation, traveling waves, traveling waves with defects, and spiral patterns (listed in order of increasing complexity).
• Figure 4: Linear stability of the uniform system, $\alpha = 2.0, \tau_d = 5.0$ (a) The real part of the eigenmodes are shown. Stability analysis reveals multiple modes in the system, shown in different colours. Distinct colors denote different branches (k), reflecting the multivalued character of the equation, with branch labeling following the convention of the Lambert $W$ function. In panel (a), the principal branch $\lambda_{k=0}$ (green) approaches $0$ as $q \to 0$, while the branch $\lambda_{k=-1}$ (orange) diverges to $-\infty$. Each branch becomes unstable at a finite value of $q$, with the corresponding critical $q_c$ shown as a function of $\tau_d$ in the Appendix. (b) The imaginary part is shown here. For the conserved mode the exceptional point appears at finite $q$. Unlike $\operatorname{Re}(\lambda)$, the branches with $k<0$ remain stable and distinct from their unstable conjugates. Roots are computed using the Lambert $W$ function (solid line) and the Newton–Raphson method (dashed line). In both panels points of Hopf-bifurcation are denoted by black circle (a pair of complex eigenvalues change stability) and the exceptional points are denoted by stars (two roots coalesce and transition from real-complex or the reverse).
• Figure 5: Dispersion relation Roots of \ref{['eq:one_mode']} obtained for a fixed $\tau_d = 10$, $q_0 = 0.7$. Green indicates physical solutions that satisfy the amplitude condition and black indicates roots that don't. Hence, as seen in the graph, the trivial solution ($\Omega = 0$) is not viable beyond $\alpha_c = 0.51$, marking the point at which the phase separated state is unstable. Additionally, note that as $\alpha$ increases, the range of $\Omega$ values also increases. The bifurcation points(indicated by red dotted circles), at which the number of solutions of the equation changes, are discussed and provided in the appendix.