Equilibrium statistical mechanics of self-consistent wave-particle system

The equilibrium distribution of $N$ particles and $M$ waves (e.g. Langmuir) is analysed in the weak-coupling limit for the self-consistent hamiltonian model $H = \sum_r p_r^2 /(2m) + \sum_j \omega_j I_j + \varepsilon \sum_{r,j} (\beta_j / k_j) \cos (k_j x_r - \theta_j)$ [1]. In the canonical ensemble, with temperature $T$ and reservoir velocity $v < \inf_j {\omega_j/k_j}$, the wave intensities are almost independent and exponentially distributed, with expectation $\langle I_j \rangle = k_{\rm B} T / (\omega_j - k_j v)$. These equilibrium predictions are in agreement with Monte Carlo samplings [2] and with direct simulations of the dynamics, indicating equivalence between canonical and microcanonical ensembles. \par\noindent [1] Y. Elskens and D.F. Escande, Microscopic dynamics of plasmas and chaos (IoP publishing, Bristol, 2003). \par\noindent [2] M-C. Firpo and F. Leyvraz, 30th EPS conf. contr. fusion and plasma phys., P-2.8 (2003).