  It is well-known that the semi-circle law, which is the limiting distribution
in the Wigner theorem, is the minimizer of the logarithmic energy penalized by
the second moment. A very similar fact holds for the Girko and
Marchenko--Pastur theorems. In this work, we shed the light on an intriguing
phenomenon suggesting that this functional is monotonic along the mean
empirical spectral distribution in terms of the matrix dimension. This is
reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann
equation, the monotonicity of the free energy along ergodic Markov processes,
and the Shannon monotonicity of entropy or free entropy along the classical or
free central limit theorem. While we only verify this monotonicity phenomenon
for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square
Laguerre unitary ensemble, numerical simulations suggest that it is actually
more universal. We obtain along the way explicit formulas of the logarithmic
energy of the mentioned models which can be of independent interest.
