Optical Analogue Of Self-Gravitating Systems

Many interactions in nature are long range. As long range interactions, we refer to those in which parts of the system far away from each other interact considerably. An historical example of such systems are stars inside a galaxy or globular clusters. Typically, in such a system, a star is subjected to a force dominated by the ensemble of the other stars (long range) rather than, for example, the neighboring ones (short range). As a consequence, one expects a very different behavior (and often counter-intuitive) compared to what happens in the short-range interacting regime usually encountered in statistical physics textbooks. For example, in the thermodynamic equilibrium, there is nonequivalence of ensembles or a possible apparition of negative specific heat in the micro-canonical ensemble. The macroscopic dynamics is also very different: starting from arbitrary initial conditions, the system forms rapidly a quasi-stationary state (like a galaxy) and then relaxes towards thermodynamic equilibrium. Recently, it has been observed that a laser beam propagating through a nonlinear, nonlocal medium presents a behavior very similar to the formation of a quasi-stationary state of self-gravitating bosons in the non-relativistic regime, which is a serious candidate for dark matter in the halos of galaxies. The purpose of this internship, is studying theoretically and numerically such an optical analogue of a self-gravitating system. The targeted objectives are: - Full description of the analogy between self-gravitating systems and nonlinear optical propagation, together with a variational approach in order to derive an analytical solution of the equation. - Study the evolution of the system for various initial conditions by writing a code to solve the Schroedinger-Newton equation.- Comparisons between theory and simulations oriented to the realisation of the optimal experimental procedure in the optical framework.