Hamiltonian analogs of combustion engines: A systematic exception to adiabatic decoupling

Workhorse theories throughout all of physics derive effective Hamiltonians to describe slow time evolution, even though low-frequency modes are actually coupled to high-frequency modes. Such effective Hamiltonians are accurate because of adiabatic decoupling: the high-frequency modes “dress” the low-frequency modes, and renormalize their Hamiltonian, but they do not steadily inject energy into the low-frequency sector. Here, however, we identify a broad class of dynamical systems in which adiabatic decoupling fails to hold, and steady energy transfer across a large gap in natural frequency (“steady downconversion”) instead becomes possible, through nonlinear resonances of a certain form. Instead of adiabatic decoupling, the special features of multiple time scale dynamics lead in these cases to efficiency constraints that somewhat resemble thermodynamics.

Figure 1

Evolution of a Hamiltonian daemon, under Hamiltonian $H_1$ of Eq. (1), with the parameter values stated in the text, and initial conditions $Q\left(0\right)=0, \dot{Q}\left(0\right)=11.1,q_{+}\left(0\right)=40/\sqrt{15}, q_{-}\left(0\right)=p_{+}\left(0\right)=p_{-}\left(0\right)=0$. (a) Position $Q\left(t\right)$ of a mass. (b) Velocity $\dot{Q}\left(t\right)$, with insets showing detail.

Figure 2

Effective potential $V(q,t)$ from $H_5$, which represents $H_1$ in new canonical variables, with linearized external potential and adiabatic elimination of nonresonant terms. The time $t=30$ is chosen for illustration; the potential looks similar for other $t$ during the driving phase of Fig. 1. Dashed lines are horizontal guides to the eye, to confirm that shallow wells exist. Unbound orbits correspond to the adiabatically decoupled phase of $H_1$; bound orbits correspond to the steady downconversion of the daemon's driving phase.

Figure 3

Schematic: Daemons as systems with three time scales, analogous to the dynamical time scales of a combustion engine. Energy is stored densely in degrees of freedom with a high dynamical frequency $\mathrm{\Omega }$. This energy is consumed at a slow rate $\gamma$ to do work at a steady speed $v_c$. The fact that $\gamma \ll \mathrm{\Omega }$ distinguishes engine-like dynamics (steady downconversion) from bomb-like (rapid energy release). The nontrivial mechanism which achieves steady downconversion involves cycles on the intermediate frequency scale $\omega$, with $\gamma \ll \omega \ll \mathrm{\Omega }$.

Figure 4

Area $S\left(t\right)$ enclosed in $(q,p)$ phase space by bound orbits under $H_5$, for the parameter values of Fig. 1. Thick curve: $S$ for the largest bound orbit that is possible at time $t$. This maximum area shrinks as the lattice tilt increases and the local minima become shallower. At time $t_F=52$ the harmonic force $-g\left(t\right)$ exceeds the maximum force the lattice can exert, and no minima exist. Solid curve: $S$ for the trajectory shown in Fig. 1. Since the explicit time dependence of $H_5$ is slow, $S$ is adiabatically invariant; when the potential minimum becomes too shallow to contain it, the dynamical transition to adiabatic decoupling must occur. Dashed curve: $S$ for a less efficient trajectory, under $H_1$ but with different initial conditions. Here the adiabatic invariance of $S$ forces the daemon to stop even though further driving is not forbidden by force limits or energy conservation. Inset: The adiabatic invariant $S$ is not strictly constant, but exhibits small, rapid oscillations around an average value that holds steady over long times.

Figure 5

Adiabatic evolution $(q_5\left(t\right),p_5\left(t\right))$ (solid curves) versus exact evolution $(q_1\left(t\right),p_1\left(t\right))$ (dashed curves), over the interval $16\le t\le 16.4$. Initial conditions for the exact evolution are those stated in the caption of Fig. 1 in the main text, and apply at $t=0$. Initial conditions for the adiabatic evolution are $q_5\left(16\right)=-0.01,p_5\left(16\right)=-100$. Fitting was done simply by eye. The suppression of high-frequency components from the adiabatic evolution is not error: it is what the adiabatic evolution is supposed to do.

Figure 6

Same as Fig. 5 above, but for time intervals $[25,25.4], [35,35.4]$, and $[45,45.4]$, respectively, from top to bottom. Solid curves are the same adiabatic solution plotted in Fig. 5, but with fitted time shifts $\mathrm{\Delta }t=-0.001, +0.025$, and $-0.001$, from top to bottom.

Figure 7

(a) Same as Fig. 1 of main text, but with the dashed trajectory from main text Fig. 4 for comparison. When both trajectories are in the driving phase, they are impossible to distinguish on this scale; this is a consequence of how the second trajectory was selected, as explained in the Appendix text. The dashed trajectory ceases steady downconversion at $t\doteq 43.7$, in agreement with Fig. 4. (b) Work $W\left(t\right)=E_S\left(t\right)-E_S\left(0\right)$ done by the daemon on the slow mass, where $E_S=P^2/\left(2M\right)+(M{\nu }^2/2)Q^2$ is the energy of the slow oscillator. Final values $W\left(65\right)$ are $3.77\times {10}^7$ (solid) and $2.31\times {10}^7$ (dashed).