  We analyze a token-based Brownian circuit in which Brownian particles, coined
`tokens,' move randomly by exploiting thermal fluctuations, searching for a
path in multi-token state space corresponding to the solution of a given
problem. The circuit can evaluate a Boolean function with a unique solution.
However, its computation time varies with each run. We numerically calculate
the probability distributions of Brownian adders' computation time, given by
the first-passage time, and analyze the thermodynamic uncertainty relation and
the thermodynamic cost based on stochastic thermodynamics. The computation can
be completed in finite time without environment entropy production, i.e.,
without wasting heat to the environment. The thermodynamics cost is paid
through error-free output detection and the resets of computation cycles. The
signal-to-noise ratio quantifies the computation time's predictability, and it
is well estimated by the mixed bound, which is approximated by the square root
of the number of token detections. The thermodynamic cost tends to play a minor
role in token-based Brownian circuits in computation cycles. This contrasts
with the logically reversible Brownian Turing machine, in which the entropy
production increases logarithmically with the size of the state space, and thus
worsens the mixed bound.
