Transport theory for an interacting fermionic system is reviewed and applied to the chiral Lagrangian of the Nambu-Jona-Lasinio model. Two expansions must be applied: an expansion in the inverse number of colors, 1/N c , due to the nature of the strong coupling theory, and a semiclassical expansion, in powers ofh. The quasiparticle approximation is implemented at an early stage, and spin effects are omitted. The self-energy is evaluated, self-consistently only in the Hartree approximation, and semi-perturbatively in the collision integral. In the Hartree approximation, O((1/N c ) 0 ), the Vlasov equation is recovered to O(h 1 ), together with an on-mass shell constraint equation, that is automatically fulfilled by the quasiparticle ansatz. The expressions for the self-energy to order O((1/N c )) lead to the collision term. Here one sees explicitly that particle-antiparticle creation and annihilation processes are suppressed that would otherwise be present, should an off-shell energy spectral function be admitted. A clear identification of the s, t and u channel scattering processes in connection with the self-energy graphs is made and the origin of the mixed terms is made evident. Finally, after ordering according to powers inh, a Boltzmannlike form for the collision integral is obtained.
We discuss a nonlocal generalization of the nonlinear Schrödinger equation and study propagation of solitary waves in a nonlinear nonlocal medium at its critical state, the response of which obeys the power law with the exponent k. Using the time-dependent variational principle, we derive a set of dynamical equations and develop the fixed-point analysis. A critical behavior is found in the k dependence of the width of the wave packet. We also present a proof of the stability of the system and discuss an oscillatory phenomenon in the self-focusing process. ͓S1063-651X͑98͒11104-2͔
We show the value of mass-momentum diagrams for analyzing collision problems in classical mechanics in one dimension. Collisions are characterized by the coefficient of restitution and the momentum of the interacting particles both before and after the collision. All those quantities are presented in the massmomentum diagrams without the need to do any calculations. We also show that the same diagrams can be used to investigate collisions with respect to the center-of-mass system. In this case, also, we do not need to do any calculations to obtain the momentum. Since we give an alternative way of looking at collisions in classical mechanics, this article is aimed at undergraduate-level students.
We present the usefulness of the diagrammatic approach for analyzing two dimensional elastic collision in momentum space. In the mechanics course, we have two major purposes of studying the collision problems. One is that we have to obtain velocities of the two particles after the collision from initial velocities by using conservation laws of momentum and energy. The other is that we have to study two ways of looking collisions, i.e. laboratory system and center-of-mass system. For those two major purposes, we propose the diagrammatic technique. We draw two circles. One is for the center-of-mass system and the other is for the laboratory system. Drawing these two circles accomplish two major purposes. This diagrammatic technique can help us understand the collision problems quantitatively and qualitatively.
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