  Stochastic systems that undergo random restarts to their initial state have
been widely investigated in recent years, both theoretically and in
experiments. Oftentimes, however, resetting to a fixed state is impossible due
to thermal noise or other limitations. As a result, the system configuration
after a resetting event is random. Here, we consider such a resetting protocol
for an overdamped Brownian particle in a confining potential $V(x)$. We assume
that the position of the particle is reset at a constant rate to a random
location $x$, drawn from a distribution $p_R(x)$. To investigate the
thermodynamic cost of resetting, we study the stochastic entropy production
$S_{\rm Total}$. We derive a general expression for the average entropy
production for any $V(x)$, and the full distribution $P(S_{\rm Total}|t)$ of
the entropy production for $V(x)=0$. At late times, we show that this
distribution assumes the large-deviation form $P(S_{\rm Total}|t)\sim
\exp\left[-t^{2\alpha-1}\phi\left(\left(S_{\rm Total}-\langle S_{\rm
Total}\rangle\right)/t^{\alpha}\right)\right]$, with $1/2<\alpha\leq 1$. We
compute the rate function $\phi(z)$ and the exponent $\alpha$ for exponential
and Gaussian resetting distributions. In the latter case, we find the anomalous
exponent $\alpha=2/3$ and show that $\phi(z)$ has a first-order singularity at
a critical value of $z$, corresponding to a real-space condensation transition.

Statistical Mechanics