Dynamical and statistical properties of time-dependent one-dimensional nonlinear Hamilton systemsDimitrios Andresas
, 2015, doktorska disertacija
Opis: We study the one-dimensional time-dependent Hamiltonian systems and their statistical behaviour, assuming the microcanonical ensemble of initial conditions and describing the evolution of the energy distribution in three characteristic cases: 1) parametric kick, which by definition means a discontinuous jump of a control parameter of the system, 2) linear driving, and 3) periodic driving. For the first case we specifically analyze the change of the adiabatic invariant (the canonical action) of the system under a parametric kick: A conjecture has been put forward by Papamikos and Robnik (2011) that the action at the mean energy always increases, which means, for the given statistical ensemble, that the Gibbs entropy in the mean increases (PR property). By means of a detailed rigorous analysis of a great number of case studies we show that the conjecture largely is satisfied, except if either the potential is not smooth enough (e.g. has discontinuous first derivative), or if the energy is too close to a stationary point of the potential (separatrix in the phase space). We formulate the conjecture in full generality, and perform the local theoretical analysis by introducing the ABR property. For the linear driving we study first 1D Hamilton systems with homogeneous power law potential and their statistical behaviour under monotonically increasing time-dependent function A(t) (prefactor of the potential). We used the nonlinear WKB-like method by Papamikos and Robnik J. Phys. A: Math. Theor., 44:315102, (2012) and following a previous work by Papamikos G and Robnik M J. Phys. A: Math. Theor., 45:015206, (2011) we specifically analyze the mean energy, the variance and the
adiabatic invariant (action) of the system for large time t→∞. We also show analytically that the mean energy and the variance increase as powers of A(t), while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is correct. For the periodic driving cases we study the 1D periodic quartic oscillator and its statistical behaviour under periodic time-dependent function A(t) (prefactor of the potential). We compare the results for three different drivings, the periodic parametrically kicked case (discontinuous jumps of $A(t)$), the piecewise linear case (sawtooth), and the smooth case (harmonic). Considering the Floquet map and the energy distribution we perform careful numerical analysis using the 8th order symplectic integrator and present the phase portraits for each case, the evolution of the average energy and the distribution function of the final energies. In the case where we see a large region of chaos connected to infinity, we indeed find escape orbits going to infinity, meaning that the energy growth can be unbounded, and is typically exponential in time.
The main results are published in two papers:
Andresas, Batistić and Robnik Phys. Rev. E, 89:062927, (2014) and
Andresas and Robnik J. Phys. A: Math. Theor., 47:355102, (2014).
Ključne besede: one-dimensional nonlinear Hamiltonian systems, adiabatic invariant, parametric kick, periodic driving, linear driving, energy distribution, WKB method, action
Objavljeno: 02.03.2015; Ogledov: 812; Prenosov: 15
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