Nonlinear Kac model: Spatially homogeneous solutions and the Tjon effect

We study Kac's nonlinear model of the Boltzmann equation when the cross section() does not satisfy the special symmetry condition()=(-). We determine a differential system for the Laguerre moments of the odd and even velocity parts of the solutions. We consider the spatially homogeneous model in 1+1 dimensions (velocityv and timet) when the even velocity part of the solution is provided by the Bobylev-Krook-Wu closed solutions and study the associated odd velocity part. We find that the solutions depend on the microscopic models of(). For one class of(), which has sums of exponential terms for the Laguerre moments, we establish the relations allowing the construction of the time-dependent solutions associated with any initial distribution. We find sufficient conditions on() and on the even part such that the Laguerre series of the odd part converges. We establish a criterion for a well-defined linear combination of the moments cross section, and we check its validity for different numerical examples. We find that if the relaxation time for the even part is smaller than the corresponding one for the odd part and if the initial distribution has a narrow peak, then the Tjon effect exists for the complete B.K.W. solution (even+odd parts).