Kinetic theory of spatially inhomogeneous stellar systems without collective effects

Abstract

We review and complete the kinetic theory of spatially inhomogeneous stellar systems when collective effects (dressing of the stars by their polarization cloud) are neglected. We start from the BBGKY hierarchy issued from the Liouville equation and consider an expansion in powers of 1/N in a proper thermodynamic limit. For N → +∞, we obtain the Vlasov equation describing the evolution of collisionless stellar systems like elliptical galaxies. This corresponds to the mean field approximation. At the order 1/N, we obtain a kinetic equation describing the evolution of collisional stellar systems like globular clusters. This corresponds to the weak coupling approximation. This equation coincides with the generalized Landau equation derived from a more abstract projection operator formalism. This equation does not suffer logarithmic divergences at large scales since spatial inhomogeneity is explicitly taken into account. Making a local approximation, and introducing an upper cut-off at the Jeans length, it reduces to the VlasovLandau equation which is the standard kinetic equation of stellar systems. Our approach provides a simple and pedagogical derivation of these important equations from the BBGKY hierarchy which is more rigorous for systems with long-range interactions than the two-body encounters theory. Making an adiabatic approximation, we write the generalized Landau equation in angle-action variables and obtain a Landau-type kinetic equation that is valid for fully inhomogeneous stellar systems and is free of divergences at large scales. This equation is less general than the recently derived Lenard-Balescu-type kinetic equation since it neglects collective effects, but it is substantially simpler and could be useful as a first step. We discuss the evolution of the system as a whole and the relaxation of a test star in a bath of field stars. We derive the corresponding Fokker-Planck equation in angle-action variables and provide expressions for the diffusion coefficient and friction force.

Secular diffusion in discrete self-gravitating tepid discs

“…4 were performed with this improved approximation. Finally, in Appendix B, we detail how this same WKB formalism may be applied to the inhomogeneous Balescu-Lenard equation without collective effects (Chavanis 2013b).…”

“…where the coefficients A m 1 ,m 2 are associated with the Fourier transform in angles of the interaction potential (Pichon 1994;Chavanis 2013b) and read…”

“…(D.3) to describe the relaxation of particles of species A due to the encounters with particles of species B (but neglecting the encounters between particles of species A). This approach is further discussed in Appendix F of Chavanis (2013b). On the other hand, for a single species system, we may describe the early dynamics of the system as a whole with a good approximation by replacing F(J φ , J r , t) in the Balescu-Lenard Eq.…”

“…When collective effects are neglected, one obtains the inhomogeneous Landau equation (Chavanis 2007(Chavanis , 2013b. When collective effects are accounted for, one gets the inhomogeneous Balescu-Lenard equation (Heyvaerts 2010;Chavanis 2012a).…”

Chaos and relaxation to equilibrium in systems with long-range interactions

“…However, even for simplified models it is very difficult to arrive at any analytical results. One possibility is to explore the Lenard-Balescu equation from plasma physics [25]. However, in order to be minimally tractable, this equation requires integrable one-particle dynamics [26], ruling out the possibility of studying the effects of chaos due to broken symmetry.…”