Stationary nonequilibrium bound state of a pair of run and tumble particles

We study two interacting identical run-and-tumble particles (RTPs) in one dimension. Each particle is driven by a telegraphic noise and, in some cases, also subjected to a thermal white noise with a corresponding diffusion constant D. We are interested in the stationary bound state formed by the two RTPs in the presence of a mutual attractive interaction. The distribution of the relative coordinate y indeed reaches a steady state that we characterize in terms of the solution of a second-order differential equation. We obtain the explicit formula for the stationary probability P(y) of y for two examples of interaction potential V (y). The first one corresponds to V (y) similar to |y|. In this case, for D = 0 we find that P(y) contains a delta function part at y = 0, signaling a strong clustering effect, together with a smooth exponential component. For D > 0, the delta function part broadens, leading instead to weak clustering. The second example is the harmonic attraction V (y) similar to y2 in which case, for D = 0, P(y) is supported on a finite interval. We unveil an interesting relation between this two-RTP model with harmonic attraction and a three-state single-RTP model in one dimension, as well as with a four-state single-RTP model in two dimensions. We also provide a general discussion of the stationary bound state, including examples where it is not unique, e.g., when the particles cannot cross due to an additional short-range repulsion.

Automated summary

This study focuses on the stationary bound state formed by two interacting identical run-and-tumble particles (RTPs) in one dimension. The distribution of the relative coordinate y reaches a steady state characterized by the solution of a second-order differential equation. Explicit formulas for the stationary probability of y under different interaction potentials are obtained.