  The key factor currently limiting the advancement of computational power of
electronic computation is no longer the manufacturing density and speed of
components, but rather their high energy consumption. While it has been widely
argued that reversible computation can escape the fundamental Landauer limit of
$k_B T\ln(2)$ Joules per irreversible computational step, there is disagreement
around whether indefinitely reusable computation can be achieved without energy
dissipation. Here we focus on the relatively simpler context of sampling
problems, which take no input, so avoids modeling the energy costs of the
observer perturbing the machine to change its input. Given an algorithm $A$ for
generating samples from a distribution, we desire a device that can perpetually
generate samples from that distribution driven entirely by Brownian motion. We
show that such a device can efficiently execute algorithm $A$ in the sense that
we must wait only $O(\text{time}(A)^2)$ between samples. We consider two output
models: Las Vegas, which samples from the exact probability distribution every
$4$ tries in expectation, and Monte Carlo, in which every try succeeds but the
distribution is only approximated. We base our model on continuous-time random
walks over the state space graph of a general computational machine, with a
space-bounded Turing machine as one instantiation. The problem of sampling a
computationally complex probability distribution with no energy dissipation
informs our understanding of the energy requirements of computation, and may
lead to more energy efficient randomized algorithms.
