Dynamics of a Disordered, Driven Zero-Range Process in One Dimension

We study a disordered, driven zero range process which models a closed system of attractive particles that hop with site-dependent rates and whose steady state shows a condensation transition with increasing density. We characterize the dynamical properties of the mass fluctuations in the steady state in one dimension both analytically and numerically and show that there is a dynamic phase transition in the density-disorder plane. We also determine the form of the scaling function which describes the growth of the condensate as a function of time, starting from a uniform density distribution.

Figure 1

Plot of the tagged particle correlation $C(k,t)$ vs $t$ at the critical point showing the existence of kinematic waves for $n>1$. For $n=2$, the solid curve passing through the minima is a power law with exponent $2/3$ and the initial tangential straight line has a slope equal to $c=1/2$. For $n=1/2$, the $y$ axis has been scaled down by a factor of 15. The data have been averaged over all $k$ for both values of $n$.

Figure 2

Scaling collapse of the cumulative distribution in Eq. (8) for (a) $n=1/3$ and (b) $n=5/4$ at $t={10}^4$ (△), $2\times {10}^4$ (□), $4\times {10}^4$ (+), and $8\times {10}^4$ (×) in the condensate phase. The data are obtained from the two slowest sites in $150$ disorder configurations, using $c=1/2$. The broken line is a guide to the eye. The inset shows the temporal growth of average mass as a power law with exponent $\beta$ given in the text.