Long-range correlations in a locally constrained exclusion process

This paper introduces a novel exclusion process with a local kinetic constraint that exhibits a phase transition from a homogeneous state to a clustered, translation-invariance-breaking state, characterized by glassy coarsening dynamics and a counterintuitive "faster-is-slower" effect where increased flow asymmetry reduces the stationary current.

The Gist

Imagine a long, narrow hallway with a row of lockers, each capable of holding just one person. In this hallway, people (particles) want to move around, but they have to follow a very specific set of rules. This paper introduces a new game played in this hallway that reveals some surprising secrets about how crowds behave when they are forced to be patient.

Here is the breakdown of the new rules and what happens when you play:

The New Rules of the Hallway

In the standard version of this game (known as the "Asymmetric Simple Exclusion Process"), people can usually just step forward if the locker in front of them is empty.

But in this new model, the rules are stricter and depend on who is standing behind you:

1. Moving Forward: You can only step forward if the locker immediately in front of you is empty.
2. Moving Backward: You can only step backward if two lockers behind you are empty.

The Catch: If two people are standing right next to each other (shoulder to shoulder), neither of them can move forward. The person in front is blocked by the empty space rule, and the person behind is blocked because the person in front is there. They are stuck in a "traffic jam" of their own making.

The Two Worlds: The Free Flow vs. The Traffic Jam

The researchers found that depending on how much the people "prefer" to move forward (a setting they call q), the hallway behaves in two completely different ways:

1. The "Free Flow" World (Low Asymmetry)
When the preference to move forward is low or balanced, the people move around like a calm crowd in a park. They mix well, and if you look at the hallway from a distance, it looks the same everywhere. There are no big groups, and everyone moves at a steady, predictable pace. This is the "homogeneous phase."

2. The "Traffic Jam" World (High Asymmetry)
When the preference to move forward is high (people are very eager to rush), something strange happens. Instead of rushing faster, they actually stop moving.

• The Clusters: Because everyone is trying to rush, they end up clumping together in tight groups. Once a group forms, the rules prevent them from breaking apart easily.
• The Glassy State: The system gets stuck in a "glassy" state. Imagine a crowd trying to exit a stadium; if everyone pushes at once, no one moves. The people are technically "alive" and want to move, but they are frozen in place by their own eagerness.
• Long-Range Correlations: In this jammed state, what happens at one end of the hallway affects what happens at the other end, even though they are far apart. It's as if the entire crowd is holding its breath together.

The "Faster-Is-Slower" Paradox

The most counter-intuitive discovery is what the authors call the "faster-is-slower" effect.

Usually, you think that if you tell people to move faster (increase the forward rate), the crowd will flow better. But in this model, making them move faster actually slows the whole system down.

• Why? When people are too eager to rush, they form those tight, unbreakable clusters. The more they try to push forward, the more they block each other.
• The Result: The total number of people getting from point A to point B (the current) actually decreases as you make them more aggressive. It's like a highway where everyone speeds up, causing a massive pile-up that brings traffic to a complete standstill.

The "Coarsening" Cascade

If you start with people spread out evenly and then turn on the "rush" setting, the system doesn't just jam instantly. It goes through a process called coarsening.

• Imagine a crowd of people scattered in a field. As they start to rush, small groups form.
• These small groups then merge into bigger groups.
• The bigger groups merge into even larger ones.
• Over time, the "islands" of people get larger and larger, until the whole system is dominated by a few massive, slow-moving clusters. This happens very slowly, like watching a glacier move.

Why This Matters

For a long time, scientists believed that in a one-dimensional line (like this hallway) with simple rules, you could never get a permanent "phase transition" (a sudden shift from free flow to a jam) without external walls or defects.

This paper proves that belief wrong. It shows that simple local rules (just looking at your immediate neighbors) are enough to create complex, long-range jams and spontaneous traffic patterns. It challenges our understanding of how crowds, traffic, and even biological molecules move, showing that sometimes, the best way to get things moving is to slow down and be less eager.

Technical Summary

Technical Summary: Long-range correlations in a locally constrained exclusion process

Problem Statement
The paper addresses a long-standing paradigm in statistical physics regarding one-dimensional driven diffusive systems with short-range interactions and a single conserved quantity (mass). It is widely accepted that such systems do not exhibit stationary long-range correlations or phase transitions, except when translation invariance is broken by defects or boundaries, or when multiple conservation laws or long-range interactions are present. The authors introduce a novel exclusion process to challenge this view, specifically investigating whether a simple local kinetic constraint depending on next-nearest neighbors can induce a transition to a clustered phase with long-range correlations and spontaneous breaking of translation invariance in a translation-invariant system.

Methodology
The authors define a one-dimensional lattice gas model with $L$ sites and periodic boundary conditions containing $N$ particles. The dynamics are governed by a local kinetic constraint involving next-nearest neighbors:

• A particle at site $k$ hops forward ($k \to k+1$) with rate $q \ge 0$ only if site $k-1$ is empty.
• A particle hops backward ($k \to k-1$) with rate $1$ only if both sites $k-1$ and $k-2$ are empty.

This constraint implies that if two particles occupy adjacent sites, their forward hops are mutually blocked. The study analyzes the system across different density regimes ($N < L/2$, $N = L/2$, and $N > L/2$) and asymmetry parameters ($q$).

• Analytical Approach: For the half-filled case ($N = L/2$), the authors map the configuration space to a height function $h_k$, identifying ergodic components based on the parity of minimum sites. They derive the stationary distribution using a "height" energy function $E(\eta)$ related to the area under Dyck paths, allowing for an exact solution of the partition function and steady-state probabilities.
• Numerical and Heuristic Analysis: For densities below half-filling ($N < L/2$) and $q > 1$, the authors employ numerical simulations and heuristic arguments based on metastable time scales to characterize the dynamics, including coarsening cascades and glassy relaxation.

Key Contributions and Results

1. Phase Transition and Long-Range Correlations:
   The model exhibits a transition between a homogeneous phase (short-range correlations) and a clustered phase (long-range correlations) controlled by the asymmetry parameter $q$.

   • Half-filled case ($N=L/2$): The configuration space splits into two ergodic components based on the parity of minimum sites. The steady state is analytically tractable and reversible for all $q > 0$. The system behaves similarly to the three-species ABC model, with a stationary distribution $\pi(\eta) \propto q^{E(\eta)}$.
   • Below half-filling ($N < L/2$):
     • For $q < 1$, the system relaxes rapidly ($\sim L^{3/2}$) to a steady state with a negative current, behaving similarly to the facilitated TASEP (F-TASEP).
     • For $q > 1$, the system enters a clustered phase. Typical configurations consist of blocks of particles. The system exhibits metastability, where the stationary current is characterized by abrupt jumps separated by exponentially long waiting times. This leads to a spontaneous breaking of translation invariance, as the system gets trapped in local perturbations of blocked ground states.

2. Glassy Dynamics and Coarsening:
   In the regime $q > 1$ and $N < L/2$, the system displays glassy dynamics. Starting from homogeneous initial conditions, the system undergoes a coarsening cascade where blocked clusters grow in size. The density of active (unblocked) particles, $a(t)$, decays asymptotically as $a(t) \sim (\ln(1+q)t)^{-1/3}$. This decay is universal, depending only on $q$ after a transient phase, and is independent of the particle density $\rho$.

3. Negative Differential Mobility (Faster-Is-Slower Effect):
   The paper demonstrates a counterintuitive phenomenon where increasing the forward hopping rate $q$ (increasing the driving force) leads to a decrease in the stationary current.

   • For $q > 1$, the current is positive but vanishes in the thermodynamic limit ($L, N \to \infty$ with fixed $\rho \in (0, 1/2)$).
   • The blocking mechanism becomes super-exponentially slow as $q$ increases, causing the current to drop. This "faster-is-slower" effect occurs because higher individual speeds enhance the formation of stable, blocked clusters that hinder overall transport.

4. Breakdown of Hydrodynamics:
   The authors show that the standard hydrodynamic approach, which assumes a local current-density relation $j(\rho)$, breaks down in this model. The current in the clustered phase depends on the system size and global history (metastability) rather than just the local density, invalidating the assumption of local equilibrium.

Significance and Claims
The paper claims to disprove the belief that translation-invariant driven diffusive systems with one conserved quantity and short-range interactions cannot undergo phase transitions or exhibit long-range correlations.

• Theoretical Implication: The model provides a simple counter-example to the "positive-rates conjecture" for cellular automata, which posits that phase separation in 1D systems requires certain transition probabilities to vanish. Here, phase separation occurs with continuous-time dynamics where no transition probability is zero.
• Universality: The work suggests that the nature of fluctuations near the transition point may belong to new universality classes (potentially related to the Edwards-Wilkinson or KPZ classes, or a Fibonacci family of classes), though this remains an open problem.
• Novelty: The introduction of a minimal kinetic constraint that leads to glassy dynamics, negative differential mobility, and long-range correlations in a single-species, translation-invariant system represents a significant departure from standard exclusion processes like ASEP or F-TASEP.

The authors conclude that these findings necessitate a reconsideration of the role of zero transition probabilities in 1D phase transitions and highlight the breakdown of regular hydrodynamics due to the buildup of long-range correlations in kinetically constrained systems.