The Vlasov equation (1938) was called collisionless Boltzman equation and was used for plasma physics research. Since Vlasov's approach to study plasma had aroused strong reaction by physicist-theorists, now we know well Landau damping and collisionless Boltzman equation. Whole things including Nonlocal Statistical Mechanics (NSM), Vlasov did after 1944, are still unknown [1950, 1966, 1978] for physicists and mathematicians. Vlasov's NSM does not perform any collective of particles (electrons or atomic nuclei, planets or Galaxies - any collective of particles) like two interacting objects. NSM performs collective of particles interacting by nonlocal means - "all objects of a system interact with all objects of this and other systems". Nonlocal means without interaction. Vlasov applied energy of interaction instead of interaction force.
There is miscomprehension between Vlasov and collisionless concept. The latter is used to be as rarefied (low-density) substance, where particles are so far from each other that particles cannot interact by collision. Vlasov's approach says that it does not matter how tough the volume is packed by the particles, interaction goes between all particles. The Vlasov equation has another form where interaction energy K(r-r') is under the integral sign. The interaction energy is between test corpuscle and the rest of particles of the collective (open system). That is the point. That is the difference between 19th-century approach and 20th-century one.
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Find the references to the main Vlasov's works.
1. Vlasov A.A. On the vibration properties of the electron gas / / Journal of Experimental and Theoretical Physics, 1938, Vol.8, N3. P.291. (In Rissian)
2. Vlasov A.A. The generalization of the concept of the electron plasma / / Bulletin of the USSR Academy of Sciences. Physics. 1944. Vol.8, N5. P.248. (In Rissian)
3. Vlasov A.A. Many-Particle Theory. Moscow-Leningrad, State publishing technical and theoretical literature (GITTL), 1950. 348pp. (A. A. Vlasov, Many-particle theory and its application to plasma, Trans, Russian Gordon and Breach Science Publishers. Inc., New York, 1961.).
4. Vlasov A.A. Statistical Distribution Function. Moscow: Nauka, 1966. 355pp. (In Rissian)
5. Vlasov A.A. Nonlocal statistical mechanics. Moscow: Nauka, 1978. 264pp. (In Rissian)