  Adopting the Mahler measure from number theory, we introduce it to toric
quiver gauge theories, and study some of its salient features and physical
implications. We propose that the Mahler measure is a universal measure for the
quiver, encoding its dynamics with the monotonic behaviour along a so-called
Mahler flow including two special points at isoradial and tropical limits.
Along the flow, the amoeba, from tropical geometry, provides geometric
interpretations for the dynamics of the quiver. In the isoradial limit, the
maximization of Mahler measure is shown to be equivalent to $a$-maximization.
The Mahler measure and its derivative are closely related to the master space,
leading to the property that the specular duals have the same functions as
coefficients in their expansions, hinting the emergence of a free theory in the
tropical limit. Moreover, they indicate the existence of phase transition. We
also find that the Mahler measure should be invariant under Seiberg duality.
