Explosive Condensation

An interactive demo of a mass-transport model I studied years ago • arXiv:1508.07516

Note from the author: This is a paper I wrote many years ago. Since then I have left my academic career entirely to work in industry, so this page is more nostalgic than a reflection of up-to-date research in the field. The models and results here still hold, but the story is now a personal snapshot from a different chapter.
Low mass
Medium mass
High mass
Condensate peak
Bias β 1.5
β = 0: unbiased random walk • 0 < β < 1: weak attraction, normal coarsening • β ≥ 1: explosive condensation — the condensate forms in vanishing time as system size grows
Max mass
1
1 / 100 = 1.0%
Particles present
100
lattice sites occupied
Gini coefficient
0.000
0 = uniform, 1 = all in one site
Steps elapsed
0
0 steps/sec

What is explosive condensation?

Imagine a ring of boxes, each holding some number of balls. At each tick, you pick one ball at random and move it to a neighbouring box. If the move is completely unbiased, every box ends up with roughly the same number of balls. That is normal diffusive mixing.

Now add a twist: the probability of a ball moving to a box depends on how many balls are already there. The richer the box, the more likely it is to attract new arrivals. This is the misanthrope process with nonlinear jump rates.

With strong enough nonlinearity, a condensate appears — one box hoovers up a finite fraction of all the balls. In the explosive regime, this condensate forms so fast that the time to reach the steady state actually vanishes as the system grows. It is the spatial analogue of "instantaneous gelation" in mean-field coagulation.

Our result

Waclaw & Evans (PRL 2012) had proven explosive condensation for asymmetric one-dimensional dynamics. In our paper we showed the same explosion happens for symmetric dynamics too, provided the nonlinearity is strong enough (β ≥ 1). Below that threshold, the system enters ordinary coarsening where the relaxation time grows with system size. In dimensions d ≥ 2 we argued explosive condensation should be generic for all parameter values.

Paper

arXiv:1508.07516 • J. Stat. Mech. (2015) P11031

Explosive condensation in symmetric mass transport models
Yu-Xi Chau, Colm Connaughton, Stefan Grosskinsky

Read on arXiv →   PDF →

Further reading

  • B. Waclaw & M. R. Evans, Explosive Condensation in a Mass Transport Model, Phys. Rev. Lett. 108, 070601 (2012)
  • C. Godrèche, Dynamics of condensation in zero-range processes, J. Phys. A 36, 6313 (2003)
  • M. R. Evans & T. Hanney, Nonequilibrium statistical mechanics of the zero-range process, J. Phys. A 38, R195 (2005)

About this demo

Sites are picked uniformly from the non-empty ones and each sends out one particle per unit time. The destination (left or right neighbour) is chosen with probability proportional to m^β, where m is the mass already there. In the misanthrope / zero-range process the departure rate from a site is independent of its occupation, which is what creates the positive feedback: a crowded site receives more than it loses. The Gini coefficient measures mass inequality. When β ≥ 1, watch how quickly the white peak swallows the system.