Dynamics of phase transitions and metastability in stochastic particle systems

Phase transitions are ubiquitous in systems consisting of a large number of interacting components, which can be as simple as a gas or as complex as human society. Often one observes intriguing associated dynamic phenomena such as a separation of time scales and metastability. A classical example is the liquid-gas transition of water, which exhibits metastability of supersaturated vapour over relatively long time scales, followed by a rapid transition to the liquid phase. A phenomenological description of such metastable states goes back to van der Waals theory of non-ideal gases. Besides a proper description of the states, the relevant dynamic aspects are the lifetimes of the states and how transitions occur between them. In reality transitions are often triggered by small impurities (causing e.g. droplet nucleation in vapour), and in mathematical models this is often achieved by adding randomness to dynamics. Stochastic particle systems, where idealized particles move and interact in a discrete (lattice) geometry, provide therefore a very natural class of models to study and understand the dynamics of such transitions and the concept of metastability.A mathematically rigorous approach poses very challenging research questions, and is an active area of modern probability theory where significant recent progress has been achieved. The proposed research builds on these developments and aims towards a full rigorous understanding of a condensation transition in zero-range processes, a particular class of stochastic particle systems which has attracted recent research interest in theoretical physics. Condensation here means, that with increasing density the system switches from a homogeneous distribution of particles to a state where a macroscopic fraction of all the particles condenses on a single lattice site. Zero-range processes are therefore used as generic models of condensation phenomena with applications ranging from clustering of granular materials to the formation of giant hubs in complex network dynamics. They are also of theoretical interest as effective models of domain wall dynamics separating different phases in more general systems, explaining phenomena like the formation of traffic jams on highways.Zero-range processes show a very rich critical behavior with interesting dynamic phenomena on several time scales including metastability, which have been understood on a heuristic level in statistical physics, inspiring many ideas in this proposal. The aim of the project is to underpin these findings with rigorous probabilistic results by proving scaling limits for the dynamics of effective observables on several different time scales. These concrete outcomes will be put into a wider context and will lead to methodological advances, by improving and generalizing recent mathematical techniques and understanding the exact conditions of validity of heuristic arguments used for predictions. Also conceptual insights are invisaged, by exploring new approaches for the mathematical characterization of metastability phenomena in stochastic particle systems.

The proposed research will have a direct impact in probability and stochastic processes, contributing methodological as well as conceptual advances to the theory of stochastic particle systems. Further impact is achieved in Statistical Physics in the area of driven diffusive systems, where the zero-range process has attracted significant research activity over recent years. As detailed in the Academic Beneficiaries Section, impact within Mathematics and Statistical Physics will be assured by journal publications, seminar talks and regular attendance at relevant meetings and conferences. The anticipated results lead also to a deeper understanding of dynamic aspects of critical phenomena such as the separation of time scales and metastability, where the rigorous approach provides a precise control of approximation errors, which is important for modeling, model reduction and to connect approaches on different scales. Therefore the proposed research can have indirect impact in a wider community of applied scientists and non-academic beneficiaries. Applications where dynamic critical phenomena such as metastability play a crucial role are numerous, including e.g. the spread of diseases and epidemics, the formation of jams in traffic flow or the dynamics of polymers or granular materials, which are important in policy making and manufacturing. The PI is a dedicated staff member of the Doctoral Training Centre (DTC) in Complexity Science, which provides various opportunities to establish new contacts and discuss concrete applications of the conceptual insight from the proposal. The DTC has connections to non-academic and industrial partners (such as the NHS Institute for Innovation and Improvement or IBM) and runs a weekly seminar and several other research events such as short workshops and schools on a regular basis. Warwick also has a strong Interdisciplinary Mathematics programme which hosted the European Study Group with Industry this year. Applicable outcomes of the proposal can be exploited in joint projects involving also Complexity MSc and PhD students. These opportunities will be followed up mostly in the second half of the proposal when first results are available. In this context a 2-day workshop is planned as part of this proposal within the DTC, which is devoted to separation of time scales and dynamics of critical phenomena in complex systems and applications, where non-academic participants will be invited. To summarize, the proposed research provides indirect impact opportunities in several applied areas and efforts will be made to identify and exploit these connections via the DTC in Complexity Science. Nevertheless, it should be stressed that the proposal uses a single-disciplinary approach: The methods, collaborations and the formulation of results will be entirely mathematical within the area of probability and stochastic processes. The planned research outputs as explained in the proposal are not claimed to have any direct impact in applied areas and for non-academic beneficiaries. This is the best strategy to deliver the prime objectives of the proposed research, enabling a coherent and efficient approach within the limits of this proposal.