Gap statistics of two interacting run and tumble particles in one dimension

We study the dynamics of the separation (gap) between a pair of interacting run and tumble particles moving in one dimension in the presence of additional thermal noise. On a ring geometry the distribution of the gap approaches a steady state. We analytically compute this distribution and find that this is exponentially localised in space, in contrast to the 'jammed' configuration, seen earlier in the absence of thermal noise. We also study the relaxation which is an exponential in time, characterised by a time scale τ _r. We observe that this time scale undergoes a crossover from a size independent value to a size dependent form with increasing size l of the ring. We study the full eigenvalue spectrum of the evolution operator $L$ and find that the spectrum can be classified into four sectors depending on the symmetries of $L$. For large l, we find explicit expressions for a set of the eigenvalues in each of the sectors, many of which are exact, and the rest can be structured into three groups. On the infinite line the separation does not reach a steady state. At long times we find that the particles behave as interacting Brownian particles, except for the presence of a peak in the distribution at small separation which is a remnant of activity. (paper)