Adiabatic invariance of phase-space volume

For a $d$-dimensional system with Hamiltonian ${\mathcal{H}}({\mathbf q},{\mathbf p};x)$, at a given fixed parameter value $x$ the surfaces of ${\mathcal{H}} = E$ form shells in phase-space $({\mathbf q},{\mathbf p})$. At an energy $E$, the phase-space volume enclosed by the shell is given by $\Omega (E;x)$, and the weight of the shell

$$g(E;x) \equiv \frac{d\Omega}{dE} (E;x) = \int d {\mathbf q... ... p} \; \delta ( {\mathcal{H}}({\mathbf q},{\mathbf p};x) - E)$$ (2.1)

can be thought of as the surface integral of the shell `thickness' $1/\vert\nabla {\mathcal{H}}\vert$. Note that to connect with quantum mechanics, phase space volume could be measured in Planck-cell units ( $d {\mathbf q} d {\mathbf p} \rightarrow d {\mathbf q} d {\mathbf p} / (2\pi \hbar)^d$), whereupon $g(E)$ is just the density of states. Following Jarzynski, I will not use these units here.

Ott[153] showed that in the adiabatic limit of slow time-dependent $x(t)$, an initial phase-space distribution on an energy shell which encloses phase-space volume $\Omega_0$ remains on an energy shell for all future times. What energy $E(x)$ the shell has at a future time (when the parameter has value $x$) is given by the condition that the enclosed phase-space volume remains constant:

$$\Omega(E(x);x) = \Omega_0 .$$ (2.2)

This is the reversible energy change. The gradient of this energy accounts for the elastic `force' $F(x)$ on the parameter $x$,

$$F(x)\; \equiv \; -\frac{dE}{dx} \; = \; \left\langle - \fr... ...bf q}(t),{\mathbf p}(t),x) \right\rangle_{{\mbox{\tiny E}}} .$$ (2.3)

The subscript ${\mbox{\tiny E}}$, wherever used, implies a microcanonical average at energy $E$. The second equality can be proved by differentiation of (2.2) with respect to $x$, with $\Omega$ written as a phase-space integral of the step function $\theta(E - {\mathcal{H}})$. As shown in Fig. 2.1b, the elastic force is the gradient of the tangent to the surface $\Omega (E;x)$ lying in a constant-$\Omega$ plane, that is, $F(x) = - -(\partial E/\partial x)_\Omega$. In the billiard case, $F(x)$ is zero when the deformation preserves the billiard volume.

Now this time-dependence of the shell's energy can be written in terms of the time-evolution of a probability density function. This is an effective equation for timescales longer than the ergodic (mixing) time, so that the distribution across the surface of an energy shell is assumed to have equilibrated (when viewed on a coarse-grained scale) between time-steps. This means that the distribution can be completely characterized by a function of energy alone: it is reduced to a one-dimensional time-dependent distribution $\rho (E,t)$. An equivalent representation is the distribution $\eta(\Omega,t)$ in the phase space volume variable $\Omega$. Keep in mind that there is a time-dependent (one-to-one) relationship between $\Omega$ and $E$: this can be visualized as constant-$x$ slices of a surface shown in Fig. 2.1b, at the value $x=x(t)$. Likewise the probability densities are related through the time-dependent Jacobean,

$$\eta(\Omega(E;x(t)),t) = \frac{1}{g(E;x(t))} \rho(E,t) ,$$ (2.4)

where the implied dependences on energy and time are shown in full.

There is no evolution in the adiabatic limit when written in terms of $\eta(\Omega,t)$:

$$\dot{\eta}(\Omega,t) = 0 .$$ (2.5)

The adiabatic invariance of $\Omega$ for each energy shell implies that the distribution is unchanging.