Fokker-Planck equation for energy evolution

Stochastic evolution is generally governed by a Fokker-Planck equation, describing drift and diffusion of a density function (see Appendix A and [79]). The derivation of the correct Fokker-Planck equation for the time-evolution of the energy distribution $\rho (E,t)$ is not trivial. Now that we have the energy diffusion rate $D_{{\mbox{\tiny E}}}$ at a given energy $E$, it is tempting to use it directly as a standard (energy-dependent) diffusion term $\frac{\partial }{\partial E} (D_{{\mbox{\tiny E}}} \frac{\partial \rho}{\partial E} )$. This simple answer is wrong, and would give the wrong heating rate. As discussed in Appendix A, care is needed in writing the diffusion term. The equation was first written down correctly (without derivation) by Wilkinson [201]. Jarzynski [106] derives it in a clear fashion (which I will follow), and later he [108] and Berry and Robbins [27] rederive it using expansions of the full phase-space density in powers of a `slowness parameter' ($\propto V$).

It must be realised that, although the spreading rate was derived in energy, it is $\Omega$ that is the `natural' variable in which to discuss spreading. Not only is the evolution of $\eta(\Omega,t)$ simple, but the consideration of Liouville's theorem becomes obvious. In the adiabatic limit $V\rightarrow0$ we have seen that $\eta(\Omega,t)$ is constant (2.5). At finite $V$ the evolution generally becomes

$$\dot{\eta} = -\frac{\partial }{\partial \Omega} ( u \eta) +... ... \left(D_\Omega \frac{\partial \eta}{\partial \Omega} \right).$$ (2.19)

Liouville's equation of motion [126] for $f({\mathbf q},{\mathbf p},t)$, the full phase-space probability density, is $\dot{f} = -\{ f, {\mathcal{H}} \}$. Regardless of the Hamiltonian and its time-dependence, this means that a constant density must remain so for all times. Recalling that $\eta$ is just the average of $f$ over an energy-shell, this means $\eta(\Omega,t) = \mbox{const}$ must be a solution of (2.19). It must also be a solution with zero flux $j_\eta$ everywhere in $\Omega$ (if this we not so, a discontinuity would arise at $\Omega=0$; see also Appendix B of [106] ). This implies that in (2.19) the drift term $u$ is zero, and the diffusion is unbiased in $\Omega$-space. By `unbiased' I mean that at every value of $\Omega$, zero density gradient results in zero flux. Note that the $D_\Omega$ may still depend on $\Omega$.

Now this ``Liouville trick'' has been performed, we can transform the evolution back to energy-space; this is done using Appendix A, and gives a Fokker-Planck equation with non-zero drift,

$$v \; = \; \dot{x}\left. \frac{\partial E}{\partial x}\right... ...\partial g}{\partial E}\right\vert _{t} D_{{\mbox{\tiny E}}} .$$ (2.20)

The first term gives the usual adiabatic energy change, while the second is a drift term entirely due to diffusion. This gives the final energy evolution, written with the $D_{{\mbox{\tiny E}}}$ terms grouped together to give a modified diffusion term,

$$\dot{\rho} \; = \; -\dot{x} \frac{\partial }{\partial E} \... ...{\partial }{\partial E} \left( \frac{\rho}{g} \right) \right].$$ (2.21)

Remember that (2.21) is an effective equation for timescales longer than the ergodic time. In the context of billiards, energy evolution can be thought of as the biased `random walk' in energy space due to the sum of `kicks' due to the particle hitting the moving walls [107].