Dynamical phase transitions: an Emerging Talents Lab Talk with Alexandre Lazarescu

By Phil Brown on August 18, 2017

In a newly published Emerging Talents paper in Journal of Physics A: Mathematical and Theoretical Alexandre Lazarescu investigates a Generic dynamical phase transition in one-dimensional bulk-driven lattice gases with exclusion. Alexandre spoke to JPhys+ to explain more about dynamical phase transitions.

An isolated system of particles hopping on a lattice and interacting together is known to have large scale dynamics which follow a hydrodynamic equation, where the evolution of the local average density of particles $\rho(x)$ satisfies a continuity equation $\dot\rho=-\nabla j(\rho)$ , involving noisy currents $j$ whose distribution only depend on $\rho$ , and not on any higher moment of the particle distribution. This happens when the spatial correlations present at the microscopic scale disappear at the macroscopic scale, making the system much simpler to analyse.

Fig.1: Standard totally asymmetric simple exclusion process ; particles jump from site to site with a uniform rate 1 to the right. Jumping to an occupied site is forbidden. Particles can also enter and exit the system at its boundaries with different rates.

Figure 1. Standard totally asymmetric simple exclusion process ; particles jump from site to site with a uniform rate 1 to the right. Jumping to an occupied site is forbidden. Particles can also enter and exit the system at its boundaries with different rates.

One such system, named the totally asymmetric simple exclusion process (TASEP – figure 1), is particularly interesting in that context: it is hydrodynamic as long as the current of particles fluctuates below a certain threshold, but long-range spatial correlations appear when it goes above it, through a so-called `dynamical phase transition’. Figure 2(a) shows the dynamical phase diagram of the TASEP in terms of the two boundary rates and the fluctuation of the current $j$ with respect to the typical current $\overline j$ . That transition can be seen in the scaling of the probability to observe a certain current: it is extensive in the system size $L$ in the correlated phase, and is independent of $L$ in the hydrodynamic phase.

$Fig.2: a) Dynamical phase diagram of the TASEP, in terms of the two boundary rates and the fluctuation of the current $j$ with respect to the typical current $\overline j$. The hydrodynamic phase (blue) and the correlated phase (red) are separated by a dynamical phase transition (purple). b) Density-current diagram of a TASEP with extra nearest-neighbour interactions. The coloured curve indicates the density in the system as a function of the current fluctuation.$

Figure 2. a) Dynamical phase diagram of the TASEP. The hydrodynamic phase (blue) and the correlated phase (red) are separated by a dynamical phase transition (purple). b) Density-current diagram of a TASEP with extra nearest-neighbour interactions. The coloured curve indicates the density in the system as a function of the current fluctuation.

The TASEP happens to be exactly solvable, which is a blessing and a curse: we can perform a very precise analysis of said phase transition, but the methods involved are not transposable to other models. In spite of that, I was able to use non-specific methods to analyse extreme fluctuations of a large class of non-solvable generalisations of the TASEP, with arbitrary site-dependent jump rates and extra interactions. I found that the scalings of the probabilities of very low and very high currents are the same as for the regular TASEP, and I conjecture that a similar dynamical phase transition separates those behaviours. This phase transition can be understood by looking at the typical current for a given density, $J^\star(\rho)$ (figure 2(b), black curve). Because of the exclusion condition, the current vanishes at $\rho=0$ and $\rho=1$ , implying that it has a maximal value $J^\star_{max}$ somewhere in-between. The system can change its density by maintaining a localised defect, at a non-extensive probability cost, but the accessible currents are then all below $J^\star_{max}$ (blue line). The only way for the system to obtain larger currents (red line) is by producing spatial correlations, the cost of which is extensive. The next step would be to find ways to get more precise information on the nature of the transition.