  Through the Monte Carlo simulation of the three-dimensional, three-state
Potts model, which is a paradigm of finite-temperature pure gauge QCD, we study
the fluctuations of generalized susceptibilities near the temperatures of
external fields of first-, second-order phase transitions and crossover.
Similar peak-like fluctuation appears in the second order susceptibility at
three given external fields. Oscillation-like fluctuation appears in the third
and fourth order susceptibilities. We find that these non-monotonic
fluctuations are not only associated with the second-order phase transition,
but also the first-order one and crossover in a system of finite-size. We
further present the finite-size scaling analysis of the second and fourth order
susceptibilities, respectively. The exponent of the scaling characterizes the
order of the transitions, or the crossover.
