Phase Transitions with memory in critical scaling

TL;DR

This work addresses whether long-time behavior conditioned on survival at absorbing transitions is always unique. By analyzing a birth–death–diffusion model with density-dependent rates and a clear separation of time scales, the authors show that the QS state can depend on the initial open CC when the active sector fractures into multiple macroscopic CCs in the thermodynamic limit. In the presence of births ($b>0$) the QS state is unique; when births are suppressed ($b=0$) the QS becomes nonunique, and critical scaling exponents can inherit memory from the preparation (e.g., $\beta=1$, $\gamma=-(a+3)$ for initial density distribution $\pi(\rho_{\rm in})\propto \rho_{\rm in}^a$), with a proposed relation $\nu=\gamma+2\beta$. A general criterion is formulated: nonuniqueness arises iff the active sector divides into multiple open CCs and the escape-rate ratios vanish with system size, signaling history-dependent critical dynamics and a challenge to universality in absorbing-state phenomena. The findings suggest history-mardened critical scaling in tunable lattice or colloidal systems and have broad implications for ecological, epidemiological, and biochemical networks where multiple metastable active communities can influence long-time outcomes.

Abstract

Many driven systems alternate between bursts of activity and quiescence and can become trapped in an absorbing state, such as complete inactivity in reaction-diffusion processes or extinction in predator-prey dynamics. It is generally assumed that, conditioned on survival, their long-lived (quasi-stationary) behavior is unique and independent of the initial condition. We show this need not hold, even for memoryless Markov dynamics. When the configuration space fractures into multiple macroscopic communicating classes, where configurations can be reach from one another within a class but not across classes, the system retains a measurable memory of its preparation, which can directly affect the critical exponents near absorbing transitions. Using a minimal birth-death-diffusion model, we demonstrate that the quasi-stationary state is unique when birth processes are present, but becomes nonunique and initial-condition dependent when they are suppressed. This mechanism, arising from vanishing of inter-class escape-rate ratios in thermodynamic limit, challenges the conventional universality hypothesis and suggests possibility of history-dependent critical scaling in controlled lattice or colloidal systems with tunable particle-number.

Figures (7)

• Figure 1: Communicating class (CC) structure controls the uniqueness of long‑time survival‑state behavior. Nodes are states $n\in S=\{1,\dots,9\}$; arrows indicate allowed transitions. Open CCs: rounded blue; closed CCs: grey. (a) One open CC $S\setminus\{9\}$ and one closed CC $\{9\}$. (b) Time evolution of the average survival state $\langle n(t)\rangle$ (conditional on not being in the closed CC), started from $n_i\in\{1,3,4,6,8\}$ and computed for the network in (a); all curves converge to the same asymptotic value, reflecting unique long‑time behavior. (c) Multiple open CCs $\{1,2,3\}$, $\{5,6\}$, $\{4,7,8\}$ and one closed CC $\{8,9\}$. (d)$\langle n(t)\rangle$ for the network in (c), initialized at the same $n_i$; here the asymptotic form depends on the initial open CC, indicating memory of preparation.
• Figure 2: Absorbing phase diagram in the birth–death–diffusion process. Quasi–stationary steady states are classified by the density $\varrho$: absorbing ($\varrho=0$), active ($0<\varrho<1$), and maximal density ($\varrho=1$). The APT occurs at $d=0$. Along the PQR line separating active and maximal-density phases, the complementary order parameter $\psi\equiv 1-\varrho$ exhibits an ordinary transition: it is continuous (second order) on QR and discontinuous (first order) on PQ, meeting at a tricritical point $Q$ where $b=d=1$.
• Figure 3: Reduction of a birth–death–diffusion process to an effective random-walk description. Time evolution of the density $\langle\rho(t)\rangle$ for the rates in Eq. (\ref{['eq:BDex']}) with $b=1.4$ and $d=0.6$. (a) Decay for $L=10$ with initial densities $\rho(0)=0.3,\,0.5,\,0.7$. (b) Decay for $L=8,16,32$ initialized with $N(0)=\lfloor 0.7L\rfloor$. In each panel, three overlapping curves show Monte Carlo simulation of the full BDD model (solid red), simulation of the effective RW (dashed yellow), and numerical integration of the RW master equation (solid teal), demonstrating the validity of the reduction. Horizontal dotted lines mark $\varrho=\langle\rho(t\to\infty)\rangle$ obtained using the quasi–stationary simulation method of Ref. Oliveira2005. Inset of (B): finite–size approach of $\varrho_L$ to the asymptotic value $\varrho_{\infty}=0.6$ in agreement with Eq. (\ref{['eq:rho_s']}); the deviation follows $\varrho_{\infty}-\varrho_L\sim L^{-0.428}$ (dashed guide line).
• Figure 4: Initial-condition–dependent scaling at the absorbing transition$\langle\rho\rangle$ and its fluctuation $\langle\rho^2\rangle-\langle\rho\rangle^2$ versus $d$ for initial distributions $\pi(\rho)=(1+a)\rho^a$ with $a=0,1,2$. For $d_c\approx 0$, $\langle\rho\rangle\sim d^{\beta}$ with $\beta=1$, and $\langle\rho^2\rangle-\langle\rho\rangle^2\sim d^{-\gamma}$ with $\gamma=-(a+3)$, showing initial-condition dependence of the fluctuation exponent.
• Figure 5: Schematic communicating-class structure in birth–death–diffusion models. Circles denote open CCs and squares closed (absorbing) CCs; arrows indicate allowed transitions. Classes are ordered by decreasing order parameter $\rho$. (a) With birth active, the nonabsorbing sector collapses into a single open CC, yielding a unique QS state. (b) No-birth limit with transitions among open CCs. (c) No-birth limit with multiple open CCs and inter-species transitions.