  This is the second in a series of two papers studying $\mu$-cscK metrics and
$\mu$K-stability from a new perspective, inspired by observations on
$\mu$-character in arXiv:2004.06393 and on Perelman's $W$-entropy in the first
paper arXiv:2101.11197.
  This second paper is devoted to studying a non-archimedean counterpart of
Perelman's $\mu$-entropy. The concept originally appeared as $\mu$-character of
polarized family in the previous research arXiv:2004.06393, where we used it to
introduce an analogue of CM line bundle adapted to $\mu$K-stability.
  We firstly show some differential of the characteristic $\mu$-entropy
$\mathbf{\check{\mu}}^\lambda$ is the minus of $\mu^\lambda$-Futaki invariant,
which connects $\mu^\lambda$K-semistability to the maximization of
characteristic $\mu^\lambda$-entropy. It in particular provides us a criterion
for $\mu^\lambda$K-semistability working without detecting the vector $\xi$
involved in the $\mu^\lambda_\xi$-Futaki invariant.
  In the latter part, we propose a non-archimedean pluripotential approach to
the maximization problem. In order to adjust the characteristic $\mu$-entropy
$\mathbf{\check{\mu}}^\lambda$ to Boucksom--Jonsson's non-archimedean
framework, we introduce a natural modification
$\mathbf{\check{\mu}}^\lambda_{\mathrm{NA}}$ which we call non-archimedean
$\mu$-entropy. We extend the non-archimedean $\mu$-entropy from the set of test
configurations to a space $\mathcal{E}^{\exp} (X, L)$ of non-archimedean psh
metrics on the Berkovich space $X^{\mathrm{NA}}$, which is endowed with a
complete metric structure. We introduce a measure $\int \chi
\mathcal{D}_\varphi$ on Berkovich space called moment measure for this sake,
which can be considered as a hybrid of Monge--Amp\`ere measure and
Duistermaat--Heckman measure.
