Dynamical and scale invariance of charged particles slipping on a rough surface with periodic excitation

This letter deals with the dynamical and scaling invariance of charged particles slipping on a rough surface with periodic excitation. A variant of the Fermi–Ulam model (FUM) is proposed to describe the transport behavior of the particles when the electric field force F_e is smaller or larger than the friction force F_f, i.e., A<0 or A>0. For these two cases, the stability of fixed points is analyzed with the help of the eigenvalue analysis method, and further the invariant manifolds are constructed to investigate the dynamical invariance such as energy diffusion for some initial conditions in the case A>0 and decay process in the case A<0. Moreover, the scaling invariance analysis is performed to demonstrate the power law of the statistical behavior. It follows that both the FA phenomenon for A>0 and the velocity decay process for A<0 satisfy scaling invariance with respect to the nondimensional acceleration A. Besides, for A<0, the transient number n_x is proposed to evaluate the speed of the velocity decay process. More importantly, n_x is found to possess the attribute of scaling invariance with respect to both the initial velocity V₀ and the nondimensional acceleration A. These results are very useful for the in-depth understanding of the energy transport properties of charged particle systems.