Broken Ergodicity in a Stochastic Model with Condensation

We introduce a variant of the asymmetric random average process with continuous state variables where the maximal mass transport is restricted by a cutoff. For periodic boundary conditions, we show the existence of a phase transition between a pure high flow phase and a mixed phase, whereby the latter consists of a homogeneous high flow and a condensed low flow substate without translation invariance. The finite system alternates between these substates which both have diverging lifetimes in the thermodynamic limit, so ergodicity is broken in the infinite system. However, the scaling behavior of the lifetimes in dependence of the system size is different due to different underlying flipping mechanisms.

Figure 1

Current-cutoff diagram for system size $L=100$ and different densities $\rho$ ($\frac{2}{5}=♦$, $1=○$, $2=\square$).

Figure 2

Current-time diagram of the mixed flow phase. Parameters used: $L=100$ and $\tilde{\Delta }=0.846$. Each current value is averaged over ${10}^3$ time steps.

Figure 3

Linear-logarithmic plot of the lifetime ${\tau }_H$ of the high flow state in dependence of the system size $L$ for several rescaled cutoffs $\tilde{\Delta }=0.76$ ($○$), 0.80 ($\square$), 0.84 ($♦$), and 0.88 ($▵$).

Figure 4

Log-log plot of the lifetime ${\tau }_L$ of the low flow state in dependence of the system size $L$ for the rescaled cutoffs $\tilde{\Delta }=0.92$ ($○$) and 0.88 ($\square$).

Figure 5

Low flow state current in dependence of the rescaled cutoff. Comparison of analytical (–) and numerical [$L=100$$○$), 200 ($\square$), 500 ($♦$)] data.