On The Linear Delaying Maryland Model

This thesis introduce our work on the Linear Delaying Maryland Model.The Linear Delaying Maryland Model is a classical chaos model.But it is different from the classical Maryland model.The disturbing potential is not periodical spread but Linear Delaying.The energy increasing is interesting.when the interval is near the Resonance condition,the energy increase will have a obvious change.If the Resonance condition is not satisfied,then the energy increase is limited.We also show more about the resonance phenomenon.We find the energy increase of the linear delaying Maryland model system is very sensitive to the system parameter.If the system parameter increase a very small term,the increasing pattern of the energy of the Linear Delaying Maryland Model will dramatic differences.we explain it as a resonance raised by a series of kicks.After a certain series of kicks,the momentum and position of the kicked rotor will return to its initial value.The periodic energy evolution appear.it is a special case for the Linear Delaying Maryland Model.When the system parameter satisfied a certain condition,then the periodic energy evolution will appear.If the special resonance condition is not satisfied,the energy increase will show a ballistic increase.Classical chaos study the phenomenon of classical system.The evolution of those classical chaos system are very sensitive to the initial conditions and system parameters.A very similar initial condition may have a complete different track in the phase space.Different tracks show exponential expansion.In the first chapter,we introduce the classical kicked rotor as a well-known classical chaos model.We introduce the Chirikov standard map to analyse the evolution of the classical kicked rotor.We also introduce the energy increase of the classical kicked rotor.The energy increase of the classical kicked rotor shows a quasi-linear increase.The increasing rate of energy is a function of the period of kicks.It is also shows the period as a important parameter of the classical kicked rotor system.in the second chapter,we introduce the quantum kicked rotor system.It is one of the representative model in quantum chaos field.People find the quantum kicked rotor model share a similar localization mechanism with the Anderson localization in the condensed matter physics area.It raised people’ interest on the quantum simulation to study the corresponding physical phenomenon.In the third chapter,we introduce the Maryland model and the quantum relativistic kicked rotor model.The Maryland model is first introduced by D.R.Grempel,Shmuel Fishman and R.E.Prange.M.V.Berry studied this model in his paper.He shows the Maryland model is sensitive to the system parameter α.We also introduce the quantum relativistic kicked rotor model(QRKR)in this chapter.The quantum relativistic kicked rotor model(QRKR)study a kicked rotor described by the Dirac equation.It is easy to find the similarity between the Maryland model and the quantum relativistic kicked rotor model(QRKR).In the fourth chapter,we introduce the linear delaying Maryland model,its kick interval is not equal,but linear increasing.We find some interesting phenomenon in this model,we study a mathematic problem to explain those interesting phenomenon.We will also show our study on a non-periodic quantum relativistic kicked rotor model(QRKR)in this chapter.In the last chapter,a conclusion of our work and the future development will be given.