  We employ a stochastic approach to probing the origin of the log-normal
distributions of halo spin in N-body simulations. After analyzing spin
evolution in halo merging trees, it was found that a spin change can be
characterized by a stochastic random walk of angular momentum. Also, spin
distributions generated by random walks are fairly consistent with those
directly obtained from N-body simulations. We derived a stochastic differential
equation from a widely used spin definition and measured the probability
distributions of the derived angular momentum change from a massive set of halo
merging trees. The roles of major merging and accretion are also statistically
analyzed in evolving spin distributions. Several factors (local environment,
halo mass, merging mass ratio, and redshift) are found to influence the angular
momentum change. The spin distributions generated in the mean-field or void
regions tend to shift slightly to a higher spin value compared with simulated
spin distributions, which seems to be caused by the correlated random walks. We
verified the assumption of randomness in the angular momentum change observed
in the N-body simulation and detected several degrees of correlation between
walks, which may provide a clue for the discrepancies between the simulated and
generated spin distributions in the voids. However, the generated spin
distributions in the group and cluster regions successfully match the simulated
spin distribution. We also demonstrated that the log-normality of the spin
distribution is a natural consequence of the stochastic differential equation
of the halo spin, which is well described by the Geometric Brownian Motion
model.
