Energy spreading--constant velocity case

We now derive the next-order correction to the adiabatic limit, namely spreading in energy (or equivalently in $\Omega$), which causes the irreversible heating. This was first found by Ott [153] using `multiple-time-scale analysis'. To derive the spreading rate, we now assume constant parameter velocity $x = Vt$ over a time interval long enough to establish diffusive behaviour (following [47,46]). The longer-time evolution for a general $x(t)$ can then be built from these short constant-velocity segments.

For any given particle trajectory $({\mathbf q}(t),{\mathbf p}(t))$ (ensemble member launched at $({\mathbf q_0},{\mathbf p_0})$ at $t=0$) there is an associated stochastic time-dependent `force' $-\frac{\partial \mathcal{H}}{\partial x}({\mathbf q}(t),{\mathbf p}(t),x(t))$ on the parameter $x$. The time-dependence arises from that of the trajectory. In the billiard case, this is simply the impulses the particle exerts on the deforming part of the wall. We extract the fluctuation of this quantity about its average (which from (2.3) is just the conservative force $F(x)$), giving the definition

$${\mathcal{F}}(t) \; \equiv \; -\frac{\partial \mathcal{H}... ...tial x}({\mathbf q}(t),{\mathbf p}(t),x(t)) \; - \; F(x(t)) .$$ (2.6)

The external work done in changing the parameter from 0 to $x$ can be found by integrating force over distance to give $Q = - \int_0^x dx' [ {\mathcal{F}}(t(x')) + F(x') ]$, which must be converted to an increase in the particle energy. Writing this as a time integral, the particle's energy difference at time $t$ from the adiabatic value $E(x(t))$ is

$${\mathcal{H}}(t) - E(x(t)) = - V \; \int_0^t dt_1 {\mathcal{F}}(t_1).$$ (2.7)

Now ${\mathcal{F}}(t)$ has a finite correlation time $\tau_{{\mbox{\tiny cl}}}$ which is similar to or less than the ergodic time. For $t \gg \tau_{{\mbox{\tiny cl}}}$ then the above expression is simply the final energy change resulting from a `random walk' of step size $\sim \tau_{{\mbox{\tiny cl}}}$. Squaring this and taking a microcanonical average (at energy $E$) gives the energy variance

$$\langle [ {\mathcal{H}}(t) - E(x(t)) ]^2 \rangle_{{\mbox{\tiny E}}} \; =\; V^2 \, \int_0^t \int_0^t dt_1 dt_2 \; \langle {\mathcal{F}}(t_1) {\mathcal{F}}(t_2) \rangle_{{\mbox{\tiny E}}}$$ (2.8)

$$\approx 2 D_{{\mbox{\tiny E}}} \cdot t \hspace{0.5in} \mbox{for} \; t \gg \tau_{{\mbox{\tiny cl}}}.$$ (2.9)

This last approximation is good for $t \gg \tau_{{\mbox{\tiny cl}}}$, where the double integral grows linearly in $t$ (with fractional error from linearity dying like $\tau_{{\mbox{\tiny cl}}}/ t$). This is simply the variance of a random walk growing linearly in time. It can also be shown using a transform of variables to $t' = (t_2+t_1)/2$ and $\tau = t_2-t_1$, with the range of $t'$ being $[0,t]$, and the range of $\tau$ can be taken to $[-\infty,\infty]$, as illustrated in Fig. 2.2a. We will write $D_{{\mbox{\tiny E}}}$ in terms of $C_{{\mbox{\tiny E}}}(\tau)$, the autocorrelation function of the fluctuating force. $C_{{\mbox{\tiny E}}}(\tau)$ depends on both the classical motion at energy $E$ and on the particular deformation chosen, and is defined by

$$C_{{\mbox{\tiny E}}}(\tau) \; \equiv \; \langle {\mathcal{F}}(0) {\mathcal{F}}(\tau) \rangle_{{\mbox{\tiny E}}}$$ (2.10)

$$= \langle {\mathcal{F}}(t') {\mathcal{F}}(t' + \tau) \rangle_{{\mbox{\tiny E}}} , \hspace{0.5in} 0 < t' < t$$ (2.11)

where the second equality states the assumption that the autocorrelation does not change over the timescale $t$. This latter condition restricts $Vt$ to be much smaller than $\delta x_c^{cl}$, the parametric change which changes the classical Hamiltonian (and hence the statistical properties of the system) significantly. In combination with the limit $t \gg \tau_{{\mbox{\tiny cl}}}$, this gives

$$V \; \ll \; \delta x_c^{cl} / \tau_{{\mbox{\tiny cl}}}$$ (2.12)

(this is the `trivial slowness condition' of Cohen [46]).

We now have established diffusive energy spreading with a rate (given by substitution of (2.10) into (2.8)) of

$$D_{{\mbox{\tiny E}}} \; = \; \mbox{\small$\frac{1}{2}$}\nu_{{\mbox{\tiny E}}}V^2$$ (2.13)

where $\nu_{{\mbox{\tiny E}}}$ is the (zero-frequency) ``noise intensity'' of the fluctuating force, equal to the first moment of the autocorrelation function:

$$\nu_{{\mbox{\tiny E}}}\; \equiv \; \int_{-\infty}^\infty d\... ...tau) \; \equiv \; \tilde{C}_{{\mbox{\tiny E}}}(\omega = 0) .$$ (2.14)

The correlation power spectrum or spectral density of ${\mathcal{F}}(t)$ is called $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$, and is the fourier transform of the autocorrelation function. I use the convention

$$\tilde{C}_{{\mbox{\tiny E}}}(\omega)\; \equiv \; \int_{-\... ...y}^\infty C_{{\mbox{\tiny E}}}(\tau) e^{i \omega \tau} d\tau .$$ (2.15)

Now that diffusion is established for short time-steps $t$, the evolution over longer times with a general $x(t)$ can be composed of independent diffusive steps each operating on the probability distribution $\rho (E,t)$ given by the previous step. This `memory-less' stochastic approximation is called Markovian [79]. Note that, in this picture, if $x(t)$ ever becomes comparable to $\delta x_c^{cl}$ then the diffusion rate should be treated as parameter-dependent $D_{{\mbox{\tiny E}}}(x(t))$.

Figure: a) Plot of the microcanonical average of ${\mathcal{F}}(t_1){\mathcal{F}}(t_2)$, showing how its double integral (Eq. 2.8) from time 0 to $t$ can be rewritten as $t$ times the first moment of the autocorrelation function $C_{{\mbox{\tiny E}}}(\tau)$. $C(\tau)$ itself is shown shaded, and it is significant only for $\vert\tau\vert<\tau_{{\mbox{\tiny cl}}}$. b) Evolution of $\rho (E,t)$ (shaded) showing drift and diffusion. Energy-dependence of both the diffusion constant and the density of states causes an asymmetric distribution, and additional drift. This causes the centroid $\langle E \rangle$ (shown by black dot) to be become higher than the adiabatic value $E(x(t))$ (shown by dashed line). This drift never becomes larger than the width due to spreading.