Spatio-temporal fluctuations in the passive and active Riesz gas on the circle

We consider a periodic Riesz gas consisting of $N$ classical particles on a circle, interacting via a two-body repulsive potential which behaves locally as a power law of the distance, $\sim g/|x|^s$ for $s>-1$. Long range (LR) interactions correspond to $s<1$, short range (SR) interactions to $s>1$, while the cases $s=0$ and $s=2$ describe the well-known log-gas and the Calogero-Moser (CM) model respectively. We study the fluctuations of the positions around the equally spaced crystal configuration, both for Brownian and run-and-tumble particles (RTP). Focusing on the regime of weak noise, we obtain exact expressions for the space-time correlations, both at the macroscopic and microscopic scale, for $N \gg 1$ and at fixed mean density $\rho$. They are characterized by a dynamical exponent $z_s=\min(1+s,2)$. We also obtain the gap statistics, described by a roughness exponent $\zeta_s=\frac{1}{2} \min(s,1)$. For $s>0$ in the Brownian case, we find that in a broad time window, the mean square displacement of a particle is sub-diffusive as $t^{1/2}$ for SR as in single-file diffusion, and $t^{s/(1+s)}$ for LR interactions. Remarkably, this coincides, including the amplitude, with a recent prediction obtained using macroscopic fluctuation theory. These results also apply to RTPs beyond a time-scale $1/\gamma$, with $\gamma$ the tumbling rate, and a characteristic length-scale. Instead, for either shorter times or shorter distances, the active noise leads to a rich variety of static and dynamical regimes, with distinct exponents. For $-1<s<0$, the displacements are bounded, leading to true crystalline order at weak noise. The melting transition, recently observed numerically, is discussed in light of our calculation. Finally, we extend our method to the active Dyson Brownian motion and active CM model in a harmonic trap, generalizing to finite $\gamma$ the results of our earlier work.