Nonequilibrium steady state of trapped active particles

We consider an overdamped particle with a general physical mechanism that creates noisy active movement (e.g., a run-and-tumble particle or active Brownian particle etc.), that is confined by an external potential. Focusing on the limit in which the correlation time $\tau$ of the active noise is small, we find the nonequilibrium steady-state distribution $P_{\text{st}}\left(\mathbf{X}\right)$ of the particle's position $\mathbf{X}$. While typical fluctuations of $\mathbf{X}$ follow a Boltzmann distribution with an effective temperature that is not difficult to find, the tails of $P_{\text{st}}\left(\mathbf{X}\right)$ deviate from a Boltzmann behavior: In the limit $\tau \to 0$, they scale as $P_{\text{st}}\left(\mathbf{X}\right)\sim e^{-s\left(\mathbf{X}\right)/\tau}$. We calculate the large-deviation function $s\left(\mathbf{X}\right)$ exactly for arbitrary trapping potential and active noise in dimension $d=1$, by relating it to the rate function that describes large deviations of the position of the same active particle in absence of an external potential at long times. We then extend our results to $d>1$ assuming rotational symmetry.