Exponential energy growth in adiabatically changing Hamiltonian systems

We show that the mixed phase space dynamics of a typical smooth Hamiltonian system universally leads to a sustained exponential growth of energy at a slow periodic variation of parameters. We build a model for this process in terms of geometric Brownian motion with a positive drift, and relate it to the steady entropy increase after each period of the parameters variation.

Figure 1

Quadratic energy growth in ergodic regime. We change parameters so that the frozen system remains chaotic, with no visible stability islands. The ensemble averaged energy versus time is shown. In the inset the rates $r\left(n\right)=\frac{1}{n}ln\left[E\left(n\right)/E\left(0\right)\right]$ are shown for two trajectories for a larger number of periods. Both rates tend to zero, corroborating the lack of exponential acceleration.

Figure 2

Exponential energy growth. (a) Parameter space for the quartic Hamiltonian system (4). Parameters are chosen such that the frozen Hamiltonian exhibits chaotic dynamics (along $a=0.01$), quasiperiodic motion (along $b=0$), and mixed behavior (along the connecting arc). The insets show the dynamics of the frozen Hamiltonian. Performing the cycle leads to the exponential growth of energy. In (b) we show the ensemble energy growth with the energy given in logarithmic scale. For the first 70 cycles the higher ensemble rate $r_{\text{en}}=0.0368$ holds. Further, a crossover to a lower rate starts. In (c) a prediction of the GBM model for the relation between ${\rho }^{+}$, $\rho$, and $\sigma$ is verified (the rates are varied by changing the value of the parameter $a_0$ at fixed $A=1$). The red solid line is the identity.

Figure 3

Distribution of energies is log-normal. The distribution of energies for an ensemble of $2\times {10}^4$ particles starting with initial conditions randomly distributed in the energy shell $E_0$. Already after one period the distribution of the logarithm of $E_n$ is close to the Gauss law.