  We propose a replica-exchange method (REM) which does not use pseudo random
numbers. For this purpose, we first give a conditional probability for Gibbs
sampling replica-exchange method (GSREM) based on the heat bath method. In
GSREM, replica exchange is performed by conditional probability based on the
weight of states using pseudo random numbers. From the conditional probability,
we propose a new method called deterministic replica-exchange method (DETREM)
that produces thermal equilibrium distribution based on a differential equation
instead of using pseudo random numbers. This method satisfies the detailed
balance condition using a conditional probability of Gibbs heat bath method and
thus results can reproduce the Boltzmann distribution within the condition of
the probability. We confirmed that the equivalent results were obtained by REM
and DETREM with two-dimensional Ising model. DETREM can avoid problems of
choice of seeds in pseudo random numbers for REM using a differential equation.
