Emergence of condensation in stochastic systems

A complex system undergoes condensation if a positive fraction of an observed quantity concentrates in a single state, asymptotically as time (or some other model parameter) goes to infinity. Condensation is a characteristic feature of many complex systems including wealth condensation in macroeconomic systems, gelation of molecules, the physical phenomenon of Bose-Einstein condensation, and the formation of traffic jams.

In this project we investigate how condensation can build up, or emerge, in complex systems. This will be done by investigating relatively simple mathematical toy models. In these models we zoom into a small neighbourhood of the condensation state at a finite time and rescale the distribution of the observable around this site. If the scaled distribution converges as time goes to infinity and scale goes to zero we call the limit a condensing wave. In other words, the condensing wave describes the shape of the distribution of the observable before it collapses to form the condensate. A recent pilot study by the investigator and his collaborators has shown that for several models from very different contexts the shape of the condensing wave is universal, i.e., it does not depend on the specific model details. In all the examples the shape was that of a well-known probability distribution, the gamma distribution.

The pressing questions resulting from these observations are:

(1) How large is the universality class of complex systems exhibiting condensation in this form?
(2) Are there other universality classes with different wave shapes?
(3) How can we characterize systems in different universality classes in terms of accessible parameters?
(4) What features other than the shape of condensing wave are characteristic of a given class?

The project will address these questions for a variety of models of different complexity.

If the objectives of this project are fully achieved, it will provide a full classification of different types of condensation behaviour in terms of basic and in many cases observable information. This classification will be rigorously verified for relatively simple toy models, but can be extrapolated to realistic models of a wide range of condensation phenomena, including wealth condensation, the physical phenomenon of Bose-Einstein condensation, or the formation of traffic jams. Our theory will give scientists in these areas a handle to classify systems far too complex to study mathematically rigorously or numerically. Information contained in this classification, which will go well beyond the shape of the condensing wave, will be exploited in a range of interdisciplinary follow-on projects and will have numerous benefits, including earlier detection of the onset condensation or new approaches to enforce or stop the formation of condensation.