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We study the dynamics of the separation (gap) between a pair of interactingrun and tumble particles (RTPs) moving in one dimension in the presence ofadditional thermal noise. On a ring geometry the distribution of the gapapproaches a steady state. We analytically compute this distribution and findthat this is exponentially localised in space, in contrast to the `jammed'configuration, seen earlier in the absence of thermal noise. We also study therelaxation which is an exponential, characterised by a time scale $\tau_r$. Weobserve that this time scale undergoes a crossover from a size independentvalue to a size dependent form with increasing size $l$ of the ring. We studythe full eigenvalue spectrum of the evolution operator $\mathcal{L}$ and findthat the spectrum can be classified into four sectors depending on thesymmetries of $\mathcal{L}$. For large $l$, we find explicit expressions forthe low lying eigenvalues in each of the sectors. On infinite line theseparation does not reach a steady state. In the long times we find that theparticles behave as interacting Brownian particles, except for the presence ofa peak in the distribution at small separation which is a remnant of activity.

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