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Equations of motion of test particles for solving the spin-dependent Boltzmann-Vlasov equation
A consistent derivation of the equations of motion (EOMs) of test particles
for solving the spin-dependent Boltzmann-Vlasov equation is presented. The
resulting EOMs in phase space are similar to the canonical equations in
Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg
picture of quantum mechanics. Considering further the quantum nature of spin
and choosing the direction of total angular momentum in heavy-ion reactions as
a reference of measuring nucleon spin, the EOMs of spin-up and spin-down
nucleons are given separately. The key elements affecting the spin dynamics in
heavy-ion collisions are identified. The resulting EOMs provide a solid
foundation for using the test-particle approach in studying spin dynamics in
heavy-ion collisions at intermediate energies. Future comparisons of model
simulations with experimental data will help constrain the poorly known
in-medium nucleon spin-orbit coupling relevant for understanding properties of
rare isotopes and their astrophysical impacts.Comment: 5 page
Stability analysis of the five-dimensional energy demand-supply system
summary:In this paper, a five-dimensional energy demand-supply system has been considered. On the one hand, we analyze the stability for all of the equilibrium points of the system. For each of equilibrium point, by analyzing the characteristic equation, we show the conditions for the stability or instability using Routh-Hurwitz criterion. Then numerical simulations have been given to illustrate all of cases for the theoretical results. On the other hand, by introducing the phenomenon of time delay, we establish the five-dimensional energy demand-supply model with time delay. Then we analyze the stability of the equilibrium points for the delayed system by the stability switching theory. Especially, Hopf bifurcation has been considered by showing the explicit formulae using the central manifold theorem and Poincare normalization method. For each cases of the theorems including the Hopf bifurcation, numerical simulations have been given to illustrate the effectiveness of the main results
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