  We study the mechanism of the gauge cooling technique to stabilize the
complex Langevin method in the one-dimensional periodic setting. In this case,
we find the exact solutions for the gauge transform which minimizes the
Frobenius norm of link variables. Thereby, we derive the underlying stochastic
differential equations by continuing the numerical method with gauge cooling,
and thus provide a number of insights on the effects of gauge cooling. A
specific case study is carried out for the Polyakov loop model in $SU(2)$
theory, in which we show that the gauge cooling may help form a localized
distribution to guarantee there is no excursion too far away from the real
axis.
