  It has recently been shown that the computing abilities of Boltzmann
machines, or Ising spin-glass models, can be implemented by chaotic billiard
dynamics without any use of random numbers. In this paper, we further
numerically investigate the capabilities of the chaotic billiard dynamics as a
deterministic alternative to random Monte Carlo methods by applying it to
classical spin models in statistical physics. First, we verify that the
billiard dynamics can yield samples that converge to the true distribution of
the Ising model on a small lattice, and we show that it appears to have the
same convergence rate as random Monte Carlo sampling. Second, we apply the
billiard dynamics to finite-size scaling analysis of the critical behavior of
the Ising model and show that the phase transition point and the critical
exponents are correctly obtained. Third, we extend the billiard dynamics to
spins that take more than two states and show that it can be applied
successfully to the Potts model. We also discuss the possibility of extensions
to continuous-valued models such as the XY model.
