Gravitational Fugacity as the seed of self-organized violent relaxaton process toward Local Virial relation

We propose the self-organized relaxation process which drives a collisionless self-gravitating system (SGS) to the equilibrium state satisfying local virial (LV) relation. During the violent relaxation process, particles can move widely within the time interval as short as a few free fall times, because of the effective potential oscillations. Since such particle movement causes further potential oscillations, it is expected that the system approaches the critical state where such particle activities, which we call gravitational fugacity, is independent of the local position as much as possible. Here we demonstrate that gravitational fugacity can be described as the functional of the LV ratio, which means that the LV ratio is a key ingredient estimating the particle activities against gravitational potential. We also demonstrate that LV relation is attained if the LV ratio exceeds the crticaial value $b=1$ everywhere in the bound region during the violent relaxation process. The local region which does not meet this criterion can be trapped into the pre-saturated state. However, small phase-space perturbation can bring the inactive part into the LV critical state.