From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation

Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the $2D$ conformal algebra. In the case without external forces, the symmetry of the conformally invariant transport equation is first generalised by considering the particle momentum as an independent variables. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined.Comment: Latex2e, 18 pages, no figure

Dynamical symmetries in the non-equilibrium dynamics of the directed spherical model

The dynamical scaling and ageing in the relaxational dynamics of the quenched directed spherical model is analysed. The exact two-time correlation and resposne functions display new regimes of ballistic or anisotropic ballistic scaling, at larger distance than probed in the usual regime of diffusive scaling. The r\^ole of long-ranged initial correlations on the existence of these scaling regimes is clarified. Their dynamical symmetries are described in terms of recent biased extensions of the Schr\"odinger algebra in that the anisotropic ballistic scaling regime can be interpreted in terms of meta-Schr\"odinger invariance while the regime of isotropic ballistic scaling is meta-conformally invariant.Comment: 36 pages, 2 figure