01 / The original paper · 2012
Explosive Condensation in a Mass Transport Model
Bartłomiej Wacław & Martin R. Evans
Physical Review Letters 108, 070601 (2012)
DOI: 10.1103/PhysRevLett.108.070601
An interactive field guide · 2012 → 2015
Small clusters.
Explosive
consequences.
Can a larger system organise itself faster? Follow a few simple hopping rules to a surprising answer in the physics of condensation.
A ring. Many particles. One condensate.
Order, at an accelerating pace.
Particles move both ways.γ > 3
Explosion is possible.
01 / The idea
When the background
can’t hold any more.
Place particles on a ring of L sites. They hop to neighbours; the total number N stays fixed. Above a critical density, the background settles at ρc and a single site holds the excess. That is condensation.
A place for the excess
Equilibrium schematic Background Excess on one site
An idealised mass budget at L = 32, γ = 4, d = 0.3. Equal background bars show its mean; real configurations fluctuate.
SparseDense
≈ 0.30CRITICAL DENSITY · ρc
—MASS IN CONDENSATE
The background is saturated.
Increasing the density puts more mass into a single condensate.
Mcond ≈ (ρ − ρc)L
For ρ > ρc, in the large-system limit. The condensate contains a finite fraction of all particles.
Why is there a critical density? Open the stationary-state argument.
The grand-canonical stationary measure factorises into single-site distributions νφ(n) = w(n)φn/z(φ). At fixed N, the product is conditioned on ∑ηx = N; occupations are therefore constrained, not independent.
w(n) = ∏k=1…n [(k − 1)γ + d] / kγ ∼ n−γ
ρc = ∑n≥0 n w(n) / ∑n≥0 w(n)
ρc = ∑n≥0 n w(n) / ∑n≥0 w(n)
The largest allowed fugacity is φ = 1. The first moment of its power-law tail is finite only for γ > 2. For γ ≤ 2 there is no finite critical density in this fixed-parameter model. The displayed ρc is numerically estimated from the weight recurrence (50,000 terms plus a power-law tail correction).
Chau, Connaughton & Grosskinsky · §2.2, equations (5)–(10) ↗
Next: make the clusters move →02 / Wacław & Evans · 2012
The more it gathers,
the faster it goes.
In the original model, every particle hops to the right. The jump rate grows with the mass at both ends of a bond. Clusters travel, exchange particles in collisions, and accelerate as they grow.
One bond, under the microscope
Exact local rates4
DEPARTURE · m
2
ARRIVAL · n
u(m,n) = [(m + ε)γ − εγ](n + ε)γ
Forward rate · c(m → n)638.28
γ = 3, ε = 0.1. Reverse hops are forbidden. The formula is the total transition rate for one particle to move, not a per-particle rate.
The original insight
A stationary distribution doesn’t tell you
how fast you will get there.
Zero-range models can have similar power-law stationary weights, yet their condensates typically move very slowly. Here, the large clusters become the fastest objects in the system.
v(m) ∼ mγ
Larger clusters → faster travel → more mass exchange.
Larger clusters → faster travel → more mass exchange.
Collisions exchange particles; they are not simply irreversible mergers. In the original model, a small mean transfer towards larger clusters fuels growth.
A race with more contestants
There are order L candidate clusters. The first to run away sets the condensation time. An extreme-value argument gives:
⟨TSS⟩ ∼ (ln L)1−γ → 0
For γ > 2 and ρ > ρc. “Explosive” means the formation time vanishes as L grows at fixed density. Every finite system still takes a positive time.
i. Nucleate
Fluctuations seed clusters.
Uneven local occupation grows into small, concentrated packets of mass.
ii. Accelerate
The leaders get faster.
Clusters travel in one direction and exchange mass when they encounter one another.
iii. Condense
One cluster dominates.
The excess mass concentrates, and the condensate keeps moving in the stationary state.
Wacław & Evans · Physical Review Letters 108, 070601 (2012), equations (1), (2), (5), (19) ↗
Next: what if particles can go both ways? →03 / Chau, Connaughton & Grosskinsky · 2015
Take away the bias.
Keep the feedback.
The next question: allow left and right hops equally in the spatial kernel. With no preferred direction, clusters revisit old ground. Does that slow exploration prevent an explosion?
c(η, x, y) = p(x,y) ηxγ(d + ηyγ) with p(x,x±1) = ½
The 2015 paper uses a simpler rate function with the same large-mass asymptotics as the original. Spatial symmetry does not make every local forward and reverse rate equal: occupancies still matter.
A tiny, real particle system
Gillespie simulation · live in your browserMODEL TIME · t0
LARGEST SITE—
PARTICLES · N64
SECOND MOMENT—
Ready. Press Run or advance one hop.
ρ = 2; symmetric d = 0.1; original ε = 0.1. Model, γ or L changes restart the run. New seed randomises the initial configuration.
Exact continuous-time event sampling, within floating-point precision. Playback batches a chosen number of events per frame, so screen speed is not physical time. Follow the clock. The tallest bar is the largest site, including during a cluster’s transfer. This small finite run illustrates microscopic dynamics; it is not evidence for a thermodynamic-limit scaling law.
What is the simulator actually doing?
At each event it computes every allowed directed nearest-neighbour rate, adds them to a total R, samples a waiting time Δt = −ln(U)/R and selects a bond with probability c/R. Exactly one particle moves. The original mode uses [(m + ε)γ − εγ](n + ε)γ to the right; the symmetric mode uses ½mγ(d + nγ) on each of the two outgoing bonds. The sites form a ring, so the two ends of the bar chart are neighbours.
The “second moment” readout is the single-run quantity L−1∑ηx2. The paper’s σ2 is its ensemble expectation, not the variance. Runs pause when this panel leaves the viewport, the browser tab is hidden, or 2 million events are reached; press Run to continue.
Chau et al. · §2.1, equations (1)–(3); §4, simulation discussion ↗
Next: find the tipping point →04 / The central result
One extra threshold.
Two different futures.
Condensation needs γ > 2. In a symmetric, one-dimensional system, explosive condensation needs something stronger: γ > 3. Change γ below to see why the distinction matters.
100 sites1,000,000 sites
Logarithmic size slider. All comparisons assume a fixed supercritical density.
Can the critical cluster fit?
mc / L decreases as the system grows.
mc ∼ L2/(γ−1) Extensive mass scale L
Unit-prefactor scaling guides. The actual available excess is (ρ − ρc)L. mc is a crossover scale; rapid coverage requires m ≫ mc.
Does a larger system take longer?
Relative time compared with L = 100.
Symmetric · 2015 One-way · 2012
Asymptotic laws normalised separately to T(100) = 1, without finite-size corrections. Curves compare size dependence, not absolute times between models or different γ.
i. Initiate a move
rate ∼ mγ
A large cluster sends a first particle to a neighbour very quickly.
ii. Complete the move
D(m) ∼ mγ−1
During nucleation, a transfer succeeds with probability of order 1/m because mass can return.
iii. Explore the ring
tcover ∼ L² / D(m)
Symmetric random motion revisits sites. Covering a 1D ring costs order L² steps.
iv. Compare scales
mc ∼ L2/(γ−1)
For mc ≪ L, the exponent must be less than one. That happens precisely at γ > 3.
Different mechanisms.
The same effective growth equation.
dm / dt = A mγ−2
Above γ = 3 the growth rate is superlinear. The effective solution blows up in finite time; a finite system instead saturates when its excess mass is collected.
Here m means typical cluster size in coarsening, and dominant-cluster size during explosion. A > 0 absorbs density-dependent factors.
Show the growth law and its time scale.
m(t) = [m₀3−γ + (3 − γ)At]1/(3−γ)
At γ = 3, m(t) = m₀eAt. For γ > 3 the formal blow-up time is tbu = m₀3−γ/[(γ − 3)A]. Using the paper’s heuristic early-cluster scale m₀ ∼ ln L gives T ∼ (ln L)3−γ. For 2 < γ < 3, reaching m = O(L) costs order L3−γ. At the boundary it costs order ln L.
During coarsening the equilibrated, truncated background enhances cluster motion to D(m) ∼ mγ, unlike the nucleation estimate. With cluster spacing, mass exchange and merger probability accounted for, the same effective growth exponent emerges. The dynamical arguments are heuristic and are supported by the paper’s simulations.
Anatomy of the growth curveγ = 4.50
Analytic illustration, not simulation data: m₀ = 4, final mass 128, A = 1. Time is divided by the time to reach 128 for each γ. The cap represents finite available mass.
Chau et al. · §3, equations (18), (20)–(22), (24), (28), (30), (33)–(39) ↗
Next: how an average can hide an explosion →05 / A subtle observation
The average can
miss the event.
Individual runs explode at different times because nucleation is random. Average them at the same clock time, and a sharp event looks gradual. Compare the same runs aligned at a common mass level.
Individual trajectories Clock-time average Aligned average
Each trajectory rises sharply, but different onset times spread out the ordinary average.
Illustrative family of 15 shifted, capped γ = 5 growth curves from the effective equation. Synthetic curves explain Figure 8; they are not digitised paper data.
A better way to average
The paper averages the first time each run reaches a chosen second-moment level, then inverts that relation. This preserves the shape of a typical explosion. Aligning at 50% above shows the intuition; the paper’s method uses every level.
t(q) = ⟨inf {s > 0 : S₂(s) ≥ q}⟩
where S₂(s) = L−1 ∑x ηx(s)²
where S₂(s) = L−1 ∑x ηx(s)²
Concentration has a measurable signature.
Squaring occupations makes large clusters dominate the second moment. In the phase-separated scaling picture, σ² ≈ (ρ − ρc)m. At stationarity it saturates near (ρ − ρc)²L.
The random wait matters. After the critical size is reached, explosive growth is nearly deterministic up to a time shift. Coarsening instead exhibits self-averaging as the system grows.
Chau et al. · §3.3, Figure 8 and equation (37) ↗
Next: bring the picture together →06 / The bigger picture
An explosion doesn’t need
a preferred direction.
It needs strong enough feedback.
The balance is between mass-dependent acceleration and the time needed to explore space. In one dimension, symmetric motion creates a new coarsening window. Above it, even unbiased motion can collect the excess in a time that shrinks with system size.
| Transport | Nonlinearity | Condensation time | As L → ∞ |
|---|---|---|---|
| One-way · Wacław & Evans | γ > 2 | (ln L)1−γ | → 0 · Explosive |
| Symmetric · 1D | 2 < γ < 3 | L3−γ | → ∞ · Coarsening |
| Symmetric · 1D | γ = 3 | ln L | → ∞ · Boundary |
| Symmetric · 1D | γ > 3 | (ln L)3−γ | → 0 · Explosive |
What changes in higher dimensions?
Section 4 predicts explosive condensation for symmetric transport for all γ > 2 in dimensions ≥ 2. To leading power, the critical cluster scale grows more slowly than the volume: mc ∼ LD/(γ−1) ≪ LD, where D is spatial dimension and L is side length. Logarithmic cover-time corrections are suppressed in this argument.
The paper expects a nucleation-time scaling of order (ln L)2−γ. These are heuristic extensions, not higher-dimensional numerical results established in the paper. The verified numerical study is one-dimensional.
Zero net current does not mean no motion.
In the symmetric stationary system, left and right currents cancel on average. Yet the activity, the amount of hopping, can be very large. The paper predicts that condensate-dominated activity grows with system size as order Lγ for supercritical density. A stationary distribution can support rapidly moving clusters.
A compact glossary
ηx: occupation of site x. N: total conserved particles. L: number of sites on the 1D ring. ρ = N/L: density. ρc: critical density. γ: strength of the nonlinear rate. d: positive rate parameter in the 2015 model, enabling jumps into empty sites. ε: the original model’s positive offset. m: cluster mass. mc: critical cluster scale. D(m): effective cluster jump rate, proportional to a diffusivity. TSS: first time a site reaches the excess mass (ρ − ρc)L. ∼: scaling up to multiplicative factors in this guide. ⟨…⟩: ensemble average.
02 / The symmetric model · 2015
Explosive condensation in symmetric mass transport models
Yu-Xi Chau, Colm Connaughton & Stefan Grosskinsky
arXiv:1508.07516v1 · 30 August 2015
This guide follows the supplied arXiv version.
About this guide: equations and claims are attributed to the source papers. The mass-budget schematic, analytic scaling plots and time-shift illustration are explanatory constructions. Only the live lattice panel samples the microscopic stochastic process. Everything runs locally in your browser, with no external libraries or data requests.