We study the effects of the gain and the loss of polaritons on the wave propagation in polariton condensates. This system is described by a modified Gross–Pitaevskii equation. In the case of small damping, we use the reductive perturbation method to transform this equation; we get a modified Burgers equation in the dispersionless limit and a damped Korteweg – de Vries equation in a more general case. We demonstrate that the shock wave occurrence depends on the gain and the loss of polaritons in the dispersionless polariton condensate. The resolution of the damped Korteweg – de Vries equation shows that the soliton behaves like a damped wave in the case of a constant damping. Based on an asymptotic solution, the survival time and the distance traveled by this soliton are evaluated. We solve the damped Korteweg – de Vries equation and the modified Gross–Pitaevskii numerically to validate the analytical calculations and discuss especially the soliton propagation in the system.
The time-independent mean-field theory of collisions is applied to the collision of four one-dimensional particles. Their two-body interactions are taken as separable, with Lorentzian form factors. This allows completely analytical solutions. A unique and satisfactory physical branch emerges out of 29 candidate solutions. It is very stable when the strength of the interaction is modified.
A proper treatment of the Pauli principle is usually impossible in Jacobi coordinate representation when heavy nuclei are involved; hence a shell-model representation of the collision becomes mandatory. This raises the traditional problems of shell-model spurious states and recoil corrections. We show how recoil parameters can actually be used as dynamical parameters in collision theories.
For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble. The second ensemble is then used. The results are supposed to be the same, at least in the thermodynamic limit. Is this actually the case? In this work, we consider a system of N noninteracting bosons distributed among few energy levels. We can calculate the canonical partition function in this case and deduce the canonical mean energy. We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles. We consider the case of a large number of particles.
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