Extreme fluctuations of current in the symmetric simple exclusion process: a non-stationary setting

We use the macroscopic fluctuation theory (MFT) to evaluate the probability distribution P of extreme values of integrated current J at a specified time t = T in the symmetric simple exclusion process (SSEP) on an infinite line. As shown recently (Meerson and Sasorov 2014 Phys. Rev. E 89 010101), the SSEP belongs to the elliptic universality class. Here, for very large currents, the diffusion terms of the MFT equations can be neglected compared with the terms coming from the shot noise. Using the hodograph transformation and an additional change of variables, we reduce the 'inviscid' MFT equations to Laplace's equation in an extended space. This opens the way to an exact solution. Here we solve the extreme-current problem for a flat deterministic initial density profile with an arbitrary density 0 < n(0) < 1. The solution yields the most probable density history of the system conditional on the extreme current, J/root T -> infinity and leads to a super-Gaussian extreme-current statistics, lnP similar or equal to -Phi (n(0)) J(3) / T, in agreement with Derrida and Gerschenfeld (2009 J. Stat. Phys. 137 978). We calculate the function Phi(n(0)) analytically. It is symmetric with respect to the half-filling density n(0) = 1/2, diverges at n(0) -> 0 and n(0) -> 1 and exhibits a singularity Phi(n(0)) similar to vertical bar n(0)-1/2 vertical bar at the half-filling density n(0) = 1/2.