Phase transitions in nonequilibrium systems

Traffic jam on a highway is most widely known example of a phase transition produced in nonequilibrium system. Typically caused by a blockage of some sort, it demonstrates the key difference from an equilibrium phase transition: in equilibrium, the local disturbance is unable to trigger a phase transition unless the system is already in the critical state. The main feature of nonequilibrium system is the flux, transporting the information from the local obstacle over the whole system.

The transport flux singles out the preferred direction and therefore can be modelled by one-dimensional models. The features of traffic are captured quite well by simplified models of hopping particles, which hop unidirectionally along the chain, with only hard core interaction between them (i.e. two particles cannot occupy the same place simultaneously).

The key ingredients of road traffic with an obstacle are present in a toy model [ 14 ]. Due to the fact that the stationary probability distribution of all possible configurations of the system can be found exactly, one can obtain different chacteristics of a traffic jam formation, including the n-point correlation functions. One learns that a) traffic jam formation is truly first order phase transition b) the system segregates into two regions - high(jam) and low (free traffic) regions separated by a shock.

Nonequilibrium phase transitions can also be induced in the open systems via boundary effects (in this case, the system left and right end act like a local disturbances).

The constraints imposed by the ends spead (due to the flux) to the whole system. As a result, the system stationary state (the state a nonequilibrium system reaches after long time; stationarity means the relevant physical properties e.g. flux, correlations become time-independent ) gets dominated by the boundary processes.

We study the role of boundary processes in driven diffusive systems. In systems with only one specie of particle, we obtained a generic extremal principle for the flux. Using it, one can predict the stationary flux (and density) in open system, if current versus average density relation (the so-called fundamental diagram) for the system is known [ 15 ]. One can apply the principle directly to popular cellular models of vehicular traffic [ 18 ]. In all cases, one can characterize diferent regimes (phases) by an order parameter, an average density of particles, and the corresponding current.

If there are more than two different particle species in the system, the situation is more complicated, and more intriguing. One finds e.g. spontaneous symmetry breaking, either to the boundary [Evans M. R., Foster D. P., Godréche C., and Mukamel D., Phys. Rev. Lett. 74, 208 (1995)], or bulk [P. F. Arndt, T. Heinzel and V. Rittenberg, J. Stat. Phys. 90, 783, (1998)] effects.

We study driven systems with particles hopping along two or more parallel chains. Interaction between the particle on different chains is included via the hopping rates. We found that already in a simple two-chain model we have studied, there are regions in the phase space where the stationary state cannot be described via the average particle chain densities, as they strongly fluctuate (see a Fig. below).

Caption: Modelling of two-chain driven system. Time evolution of the average particle densities for different choices of boundary conditions. For details see [ 20 ].

In one of the new phases (Mixed), quite unusual dynamical coexistence between sharply separated symmetric and totally asymmetric regions were established (see Fig.)

Caption:Typical snapshot of average density profiles in the Mixed Phase (see previous Fig.), where one sees coexistence of symmetric states (the left segment) and asymmetric states (the right segment).

In the other (Seesaw), we find correlated densities fluctuations with preference to strongly different densities (`weak symmetry breaking').
We aim at better understanding of the properties of multichannel nonequilibrium systems. In hydrodynamic limit, the multichannel driven systems transform into systems of conservation laws, as the number of particles in the bulk is conserved. The Causchy problem for systems of conservation laws is a long-standing problem in mathematics [Serre, D.: Systemes de lois de conservation, Cambridge University Press 2000]. This fact makes the investigation of underlying particle systems even more challenging.