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Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction | | Jean-Baptiste Fouvry
; Pierre-Henri Chavanis
; Christophe Pichon
; | | Date: |
29 Jul 2020 | | Abstract: | Finite-$N$ effects unavoidably drive the long-term evolution of long-range
interacting $N$-body systems. The Balescu-Lenard kinetic equation generically
describes this process sourced by ${1/N}$ effects but this kinetic operator
exactly vanishes by symmetry for one-dimensional homogeneous systems: such
systems undergo a kinetic blocking and cannot relax as a whole at this order in
${1/N}$. It is therefore only through the much weaker ${1/N^{2}}$ effects,
sourced by three-body correlations, that these systems can relax, leading to a
much slower evolution. In the limit where collective effects can be neglected,
but for an arbitrary pairwise interaction potential, we derive a closed and
explicit kinetic equation describing this very long-term evolution. We show how
this kinetic equation satisfies an $H$-theorem while conserving particle number
and energy, ensuring the unavoidable relaxation of the system towards the
Boltzmann equilibrium distribution. Provided that the interaction is
long-range, we also show how this equation cannot suffer from further kinetic
blocking, i.e., the ${1/N^{2}}$ dynamics is always effective. Finally, we
illustrate how this equation quantitatively matches measurements from direct
$N$-body simulations. | | Source: | arXiv, 2007.14685 | | Services: | Forum | Review | PDF | Favorites | |
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