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Maximum entropy states of collisionless systems with long-range interaction and different degrees of mixing

Dynamics of many-particle systems with long-range interaction is collisionless and governed by the Vlasov equation. This dynamics is a flow of a six-dimensional incompressible liquid with uncountable integrals of motion. If the flow possesses the statistical property of mixing, each liquid element spreads over the entire accessible space. I derive the equilibrium microcanonical maximum entropy states of this liquid for different degrees of mixing $M$. This $M$ is the number of liquid elements which are statistically independent. To count microstates of a liquid, I develop analog of the discrete combinatorics for continuous systems by introducing the ensemble of phase subspaces and making contact with the Shannon-McMillan-Breiman theorem from the ergodic theory. If $M$ is much larger than the total number of particles $N$, then the equilibrium distribution function (DF) is found to be exactly of the Fermi-Dirac form. If the system is ergodic but without mixing, $M=0$, the DF is a formal expression which coincides with the famous DF obtained by Lynden-Bell. If the mixing is incomplete and $M\sim N$, the exponentials, which are present in Lynden-Bell's DF, appear with certain weight given by the entropy of mixing. For certainty, the \ long-range interaction is taken in the form of the Newton and Coulomb potential in three dimensional space, but the applicability of the method developed in the paper is not restricted to this case. The analogy of the obtained statistics to the Fermi-Dirac statistics allows for expressing the entropy of the system via its total energy and chemical potentials of liquid's elements. The effect of the long-range interaction to the basic thermodynamic relations is demonstrated.