The thesis so far has treated the relative feedback between nonlinearity and disorder in the radiation-matter interaction processes by using a perturbative approach for what concerns the theoretical analysis. Furthermore, we dealt with a limited number of localizations. In the previous chapter, for example, we studied the interaction of a single localized wave-form (just one soliton) with the surface Anderson localizations. We have treated both nonlinearity that disorder as a perturbation. One can argue if is it possible to study the emerging phenomena by treating nonlinearity and disorder on the same level. What happens if we simultaneously consider the light-matter interaction when a large number of localizations are taken into account? Is it possible to employ some method dealing with physical systems with multiple bodies? In order to apply the mean-field theories for many bodies systems, we treat with a standard Random Laser (RL). Later we discuss with details the physical features of this kind of laser, for now it is enough to know that a RL is an optical device sustaining laser action in a disordered medium. This system presents a large number of electromagnetic modes with overlapping resonances. So the RL displays all the features we are looking for: many disorderly distributed states interacting in a nonlinear manner. We are dealing with a complex system and we need an analytical framework able to treat the multi bodies problem. Through some approximations, it is possible to express the interacting light in the disordered resonant system via very general equations, relating to a mean-field spin-glass model . The Spin-Glass theory is an approach to obtain the dynamical and thermodynamical behaviors of a complex system. We solve our model with the replica method, a subtle trick for which the physical system is replicated n-times in order to calculate the partition function and all the physical observables. By operating on the degree of disorder and nonlinearity (through the energy furnished to the system), we are able to obtain the phase-diagram of the RL, describing the very interesting complex landscape of behavior of the nonlinear waves in random systems
Partition Function Electromagnetic Mode Replica Method Random Laser Replica Symmetry Breaking
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