Two-point correlation function of an exclusion process with hole-dependent rates

Abstract

We consider an exclusion process on a ring in which a particle hops to an empty neighbouring site with a rate that depends on the number of vacancies n in front of it. In the steady state, using the well known mapping of this model to the zero range process, we write down an exact formula for the partition function and the particle-particle correlation function in the canonical ensemble. In the thermodynamic limit, we find a simple analytical expression for the generating function of the correlation function. This result is applied to the hop rate u(n) = 1 + (b/n) for which a phase transition between high-density laminar phase and low-density jammed phase occurs for b > 2. For these rates, we find that at the critical density, the correlation function decays algebraically with a continuously varying exponent b − 2. We also calculate the two-point correlation function above the critical density, and find that the correlation length diverges with a critical exponent ν = 1/(b − 2) for b < 3 and 1 for b > 3. These results are compared with those obtained using an exact series expansion for finite systems.

Critical dynamics of the jamming transition in one-dimensional nonequilibrium lattice-gas models

“…In the former case, at the critical point, the single site mass distribution p(m) ∝ exp −b(1 − λ) −1 m 1−λ which decays faster than a power law as in the fluid phase. As a consequence, the roughness exponent remains one half [21] and the dynamical behavior is the same as in the TASEP for all 0 < λ < 1, b > 0. We have also studied a bidirectional version of the model in Sec.…”

“…Using the last expression in (21), we immediately obtain the result (32) in [21] for the generating function of the correlation function:…”

“…The large distance behavior of the correlation function can be found by expanding G(y) about y = 1, and at the critical point, we have [21] …”

“…In a recent work [21], this correlation function was calculated in the fluid phase and at the critical point using an exact expression for it. It was shown that at the critical point, the equal time correlation function decays as a power law with a continuously varying exponent, see (25) below.…”

“…Recently, we studied the equal time density-density correlation function in the stationary state [21], and it was shown that at the critical point, it decays with distance as a power law with a continuously varying exponent. Using this result and numerical simulations, we find that at the critical point, the autocorrelation function decays as a power law in time with an exponent that changes continuously with a parameter in the model when the static correlation function decays slower than inverse distance, but the TASEP behavior for the autocorrelation function, viz., t −2/3 decay holds otherwise.…”