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In the relaxation evolution of finite weakly-coupled star clusters, starsundergo stochastic acceleration due to the `discreteness' of the clusters (thefiniteness of the total stellar number), in addition to fundamental two-bodyrelaxation processes. The acceleration is the essential non-collectivemany-body relaxation process. However, existing works have never detailed the`discreteness' stochastic acceleration and the corresponding mathematicalmodel, i.e., the generalized-Landau (g-Landau) kinetic equation for the stellardistribution function. The present paper shows the kinetic formulation of anorbit-averaged g-Landau equation in action-angle coordinates, beginning withBogoliubox-Born-Green-Kirkwood-Yvon hierarchy. We show that only the g-Landauequation can satisfy the anti-normalization condition among existingapproximated star-cluster kinetic equations if gravitational polarization isneglected. Accordingly, only the equation can correctly define the total energyand the total number of stars in phase space. It also holds the conservationlaws and the H-theorem. We further show that stars undergoing the discretenessstochastic acceleration can hold these physical properties independently ofstars experiencing the two-body relaxation. Lastly, we suggest that thestochastic acceleration tends to isotropize the stellar DF and to drive starclusters into a stationary state in the early relaxation-evolution stage beforethe two-body relaxation thermalizes stars.

An orbit-averaged generalized-Landau kinetic equation for the relaxation  evolution of finite weakly-coupled star clusters: `Discreteness' stochastic  acceleration and anti-normalization