Abstract
The authors propose an effective evolution equation for a coarse-grained distribution function of a long-range-interacting system preserving the symplectic structure of the Vlasov equation valid at the fine-grained level. Symmetry considerations suggest a general form of the evolution equation, that is then explicitly derived in the case of one-dimensional systems. The equation predicts how the damping times depend on the coarse-graining scale.
Highlights
Long-range interactions, whose potential energy decays with the distance r between interacting bodies slower than r−d, where d is the dimension of space [1,2], are relevant to astrophysics and plasma physics [3,4], since gravitational and unscreened Coulomb forces are long-ranged, as well as to condensed matter, given that dipolar interactions in d = 3 or effective interactions between cold atoms in an electromagnetic cavity [5,6] are long-ranged and occur in two-dimensional fluids [7]
We present an effective evolution equation for a coarse-grained distribution function of a long-rangeinteracting system preserving the symplectic structure of the noncollisional Boltzmann, or Vlasov, equation
To get a truly coarse-grained distribution function one should average over an interval of angles θ, but it is more convenient to consider such an average as carried over a time interval t, that is, to assume that we are blind to changes of the coordinates of the particles occurring on time scales smaller than t
Summary
The Boltzmann entropy is a Casimir, corresponding to C( f ) = − f ln f , so that it is a constant of motion and no H theorem holds All these properties seem to suggest that no relaxational dynamics is possible: any time dependence of f should survive forever in the N → ∞ limit and at least up to t ≈ τR when collisional effects set in for a large but finite system. It is widely believed that the mechanism of violent relaxation is similar to Landau damping [13,22,23] This means that the Vlasov dynamics never stops: rather it trickles down towards smaller and smaller scales until it no longer affects the behavior of any coarse-grained observable. Some proofs and some further details on the numerics are reported in Appendixes A–C
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