Absorbing phase transitions with memory in critical scaling

This paper demonstrates that driven systems with absorbing states can exhibit history-dependent critical scaling and nonunique quasi-stationary behavior when their configuration space fractures into disconnected macroscopic classes, thereby challenging the conventional assumption that critical exponents are independent of initial conditions.

The Gist

The Big Idea: When History Matters in a Game of Chance

Imagine you are watching a game played by a crowd of people. The game has two possible endings:

1. The Active State: People are moving around, talking, and interacting.
2. The Absorbing State (The "Game Over"): Everyone stops moving and sits perfectly still. Once they hit this state, they can never get up again.

In physics, many systems behave like this. Think of a forest fire (it burns until there's no wood left) or a species in a forest (it survives until it goes extinct). Usually, scientists assume that if you wait long enough, the "active" part of the game settles into a predictable, unique pattern, regardless of how the game started. They believe the system "forgets" its past.

This paper says: "Not always."

The authors show that under specific conditions, a system can get stuck in a "memory loop." If you start the game with a slightly different setup, the system might settle into a completely different long-term pattern, and the rules that describe how it behaves near the edge of extinction change based on where it started.

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The Analogy: The Mountain Hikers

To understand how this works, imagine a group of hikers on a mountain range.

• The Hikers: These are the particles in the system.
• The Mountain: The landscape of possible states.
• The Valley (Absorbing State): A deep pit at the bottom of the mountain. Once a hiker falls in, they are trapped forever (extinction).
• The Peaks: The active areas where hikers can roam.

Scenario A: The Connected Peaks (The Old Assumption)

Imagine all the peaks are connected by bridges. A hiker starting on the North Peak can eventually walk to the South Peak, and vice versa.

• The Result: No matter where you drop the hiker, they will eventually wander around the whole mountain range. If you wait long enough, the distribution of hikers across the mountain becomes the same, regardless of the starting point. The system has "forgotten" where it began. This is the standard behavior physicists have always expected.

Scenario B: The Fractured Peaks (The New Discovery)

Now, imagine a massive earthquake splits the mountain range. The North Peak and the South Peak are now separated by a deep chasm. There are no bridges between them.

• The Catch: Hikers can still move around within the North Peak, and they can move around within the South Peak. But they can never cross over.
• The Result:
  • If you drop a hiker on the North Peak, they will eventually settle into a pattern specific to the North.
  • If you drop a hiker on the South Peak, they will settle into a pattern specific to the South.
  • The system retains its memory. The final outcome depends entirely on which "island" you started on.

The Specific Experiment: Birth, Death, and Diffusion

The authors tested this idea using a specific mathematical model called a Birth-Death-Diffusion (BDD) model. Think of this as a simulation of bacteria in a petri dish.

1. Diffusion: Bacteria move around randomly (mixing).
2. Death: Bacteria die off.
3. Birth: New bacteria are born.

The Twist:
The authors looked at two versions of this game:

• Version 1 (Birth is ON): New bacteria are constantly being born.

  • What happens: The "bridges" between different population sizes are open. Even if the population drops low, a birth event can jump-start it again, connecting all possible population sizes. The system behaves like Scenario A (Connected Peaks). The long-term behavior is unique and predictable.

• Version 2 (Birth is OFF): No new bacteria are born; they can only die or move.

  • What happens: If you start with 10 bacteria, you can never go back to 11. You can only go down to 9, 8, 7, etc. The "bridges" are broken. The system is now trapped in a specific "population sector" (e.g., the 10-bacteria island).
  • The Surprise: Even though the bacteria are dying, the system doesn't just drift randomly toward extinction. Instead, it settles into a "quasi-stationary" state (a long-lived active state) that remembers the initial number of bacteria.

The Critical Finding: Memory at the Edge of Extinction

The most surprising part of the paper happens right at the "edge of the cliff"—the critical point. This is the precise moment where the system is balanced between surviving for a long time and dying out quickly.

In standard physics, the "critical exponents" (mathematical numbers that describe how the system behaves near this edge) are universal. They are like the laws of gravity: they shouldn't change based on how you set up the experiment.

The paper claims:
In this "No-Birth" scenario, the critical exponents change depending on the initial conditions.

• If you start with a specific distribution of bacteria, the math describing the system's fluctuations near extinction will have one set of numbers.
• If you start with a different distribution, the numbers change.

It's as if the "laws of physics" for the dying system change depending on how you introduced the bacteria to the dish.

Why Does This Happen? (The "Escape Rate" Bottleneck)

The authors explain this using the concept of escape rates.

• Imagine the hikers on the fractured peaks are trying to escape to the "Game Over" valley.
• In the "No-Birth" scenario, the rate at which a group of hikers escapes to the valley depends on how many hikers are there.
• The authors found that in these fractured systems, the "escape rates" between different population groups become so incredibly slow (exponentially slow) that the system effectively gets stuck in its starting group for a very long time.
• Because the system can't "mix" between groups fast enough to forget its start, the memory of the initial setup imprints itself on the critical scaling laws.

Summary

• The Norm: Usually, complex systems forget their past. If they survive, they settle into one unique pattern.
• The Exception: If the system's possible states are "fractured" into isolated islands (like different population sizes with no way to jump between them), the system gets stuck on its island.
• The Consequence: The system retains a "memory" of how it started. This memory is so strong that it changes the fundamental mathematical rules (critical exponents) describing how the system behaves right before it dies out.

The paper challenges the long-held belief that "universality" (the idea that details don't matter) always applies to systems with absorbing states. It shows that in certain controlled environments, history matters, even at the very edge of extinction.

Technical Summary

Technical Summary: Absorbing Phase Transitions with Memory in Critical Scaling

Problem Statement
Driven systems with absorbing states (e.g., extinction in predator-prey dynamics or inactivity in reaction-diffusion processes) are conventionally assumed to exhibit a unique long-lived quasi-stationary (QS) state conditioned on survival, independent of initial conditions. This uniqueness is typically underpinned by the ergodicity of the non-absorbing configuration space, often guaranteed by the Perron-Frobenius theorem for irreducible Markov chains. However, this assumption may fail when the configuration space fractures into multiple macroscopic open communicating classes (CCs). The paper investigates whether such structural fragmentation leads to non-unique QS states and, crucially, whether this memory of initial preparation persists at criticality, thereby altering the universality class and critical exponents of absorbing phase transitions (APTs).

Methodology
The authors employ a minimal Birth-Death-Diffusion (BDD) model on a one-dimensional periodic lattice of size $L$. The system consists of hard-core particles with occupation variables $s_i \in \{0, 1\}$. The dynamics involve:

1. Diffusion: Particles hop between sites at a unit rate, ensuring rapid mixing within a fixed particle number sector.
2. Birth-Death Events: These occur at rates $B(\rho)$ and $D(\rho)$ dependent on the instantaneous global density $\rho = N/L$. Crucially, the authors impose a strong separation of time scales: birth and death events are exponentially slow ($\sim e^{-L}$) compared to diffusion.

This separation allows the reduction of the full BDD dynamics to an effective one-dimensional random walk (RW) in particle-number space ($N$). The authors analyze two distinct regimes:

• Case 1 ($b > 0$): Birth rates are non-zero, allowing transitions between different particle-number sectors.
• Case 2 ($b = 0$): Birth rates vanish, restricting dynamics to sectors of fixed particle number $N$, effectively decoupling the non-absorbing sectors.

The analysis utilizes spectral decomposition of the Markov generator matrix, saddle-point approximations for large $L$, and quasi-stationary simulation methods to verify analytical predictions.

Key Contributions and Results

1. Uniqueness vs. Non-Uniqueness of QS States:

   • With Birth ($b > 0$): The non-absorbing sector forms a single open communicating class. The system is irreducible, and the Perron-Frobenius theorem ensures a unique QS state. The long-time behavior is independent of the initial condition, and the system exhibits standard critical scaling with fixed exponents ($\beta=1, \gamma=0, \nu=2$).
   • Without Birth ($b = 0$): The non-absorbing sector fractures into multiple open CCs, each corresponding to a fixed particle number $N$. The generator becomes reducible. The long-time QS state becomes non-unique and depends explicitly on the initial particle number (or initial density profile). The system retains a measurable memory of its preparation.

2. Memory-Dependent Critical Scaling:

   • In the no-birth limit ($b=0$), the authors demonstrate that the memory of the initial condition persists even at the critical point of the absorbing phase transition.
   • By considering initial density distributions $\pi(\rho_{in}) \propto \rho_{in}^a$, the critical exponents are shown to depend continuously on the preparation exponent $a$.
   • Specifically, while the order parameter exponent remains $\beta = 1$, the susceptibility exponent $\gamma$ becomes $\gamma = -(a+3)$. This contrasts with conventional universality, where critical exponents are determined solely by symmetry and dimensionality, not initial conditions.

3. General Conjecture for Thermodynamic Non-Uniqueness:
   The paper proposes a precise structural criterion for non-unique QS behavior in Markov processes with absorbing states:

   • The non-absorbing part must split into at least two macroscopic open CCs.
   • If these CCs form a linear directed acyclic graph terminating at an absorbing sink, the ratios of the escape rates from the slowest class ($\bar{k}$) to all competing classes ($k < \bar{k}$) must vanish in the thermodynamic limit as $1/L^u$ with $u \ge 1$.
   • This vanishing ratio prevents the system from "forgetting" its initial class, allowing the initial condition to imprint on the critical scaling.

Significance and Claims
The paper claims to challenge the conventional universality hypothesis in non-equilibrium statistical mechanics. It demonstrates that for a specific class of systems (BDD with vanishing birth rates), the critical exponents of an absorbing phase transition are not universal in the traditional sense but are history-dependent.

The authors emphasize that this phenomenon arises from the topological structure of the Markovian dynamics (fractured communicating classes) and the specific scaling of inter-class escape rates. They note that this is distinct from systems with marginal parameters where exponents vary with coupling constants; here, the variation is driven by the initial condition.

The work suggests that such memory-dependent scaling could be observable in controlled lattice or colloidal systems with tunable particle numbers. However, the authors remain modest, noting that this behavior is specific to systems where the active sector fractures and escape-rate ratios vanish sufficiently fast. They explicitly state that non-uniqueness of the steady state alone is insufficient to alter critical exponents; the specific structural criteria regarding escape rates must also be met. The findings revise the assumption that long-time behavior conditioned on survival is always unique and independent of preparation, pointing to a class of systems where "history matters" even at criticality.