Appendix A: General transformation of the 1D Fokker-Planck equation

In this Appendix, I show how the drift and diffusion terms in a general Fokker-Planck equation [79] for evolution of a probability density function (PDF) transform when the variable (i.e. the spatial ordinate of the PDF) is transformed. It is an essential step to understanding the relation between ergodic energy spreading in the energy variable $E$ (in which the diffusion constant is most easily found) and in the enclosed-phase-space-volume variable $\Omega$ (the natural variable for discussing spreading because of its conservation under Liouville's theorem). I present known results; in particular, the results derived here are assumed by Jarzynski [106] and Cohen [46]. However it is complicated enough, especially in the case of a time-dependent density of states, to merit a clear explanation.

We start with a known evolution of the PDF $\eta(\Omega,t)$ in $\Omega$-space, and want to find the equivalent evolution of $\rho (E,t)$, the PDF in $E$-space which describes the same situation. We assume a one-to-one time-dependent mapping $\Omega = \Omega(E,t)$, which can be represented as a (fixed) surface in three-dimensional $(E,t,\Omega)$ space (Fig. A.1). Therefore $\eta(\Omega,t)$ and $\rho (E,t)$ are projections of the same function on the 3D surface, down the $E$ and $\Omega$ directions respectively. The PDFs are related as follows,

$$\eta \; = \; \frac{\rho}{g} , \hspace{0.5in} \mbox{Jacobea... ...\; \left. \frac{\partial \Omega}{\partial E}\right\vert _{t} .$$ (A.1)

From now on I will use prime to denote spatial derivative and dot time derivative, in the natural variables of each PDF. Namely,

$$\eta' \equiv \left. \frac{\partial \eta}{\partial \Omega}\r... ...quiv \left. \frac{\partial \rho}{\partial t}\right\vert _{E} .$$ (A.2)

Time-dependence will not be written explicitly. Partial derivatives will be taken with $t$ held constant unless otherwise specified.

Figure A.1: Projecting a probability density on the surface $\Omega (E,t)$ down to $\rho (E,t)$ in the $E-t$ plane, or across to $\eta(\Omega,t)$ in the $\Omega-t$ plane. The gradient $g$ is also shown.

The Fokker-Planck equations in the two variables arise as follows. The probability fluxes (i.e. probability per unit time flowing past a fixed point) are assumed linear in the local density and its local gradient:

$$j_\eta(\Omega,t) \; =\; u \eta - C \eta' ,$$ (A.3)

$$j_\rho(E,t) = v \rho - D \rho' .$$ (A.4)

For the PDF $\eta$, the coefficient $u(\Omega,t)$ is a `drift velocity' field and $C(\Omega,t)$ is a diffusion rate (both are generally space- and time-dependent). For the PDF $\rho$, similarly, $v(E,t)$ is the drift velocity and $D(E,t)$ the diffusion rate. We also have the probability conservation laws,

$$\dot{\eta} + \frac{\partial j_\eta}{\partial \Omega} = 0,$$ (A.5)

$$\dot{\rho} + \frac{\partial j_\rho}{\partial E} = 0 .$$ (A.6)

Combining (A.3) and (A.5) gives the Fokker-Planck equation obeyed by $\eta$,

$$\dot{\eta} \; = \; -(u \eta)' + (C \eta')' .$$ (A.7)

Similarly the Fokker-Planck equation obeyed by $\rho$ is,

$$\dot{\rho} \; = \; -(v \rho)' + (D \rho')' .$$ (A.8)

Given functions $u(\Omega,t)$, $C(\Omega,t)$ and (A.7), our task is to find $v(E,t)$ and $D(E,t)$ such that (A.8) and the Jacobean transformation (A.1) are obeyed for all time. The Jacobean results in the following rule for transforming derivatives:

$$\frac{\partial }{\partial \Omega} \; = \; \frac{1}{g} \frac{\partial }{\partial E} .$$ (A.9)

Use of this and of (A.1) allows (A.7) to be written as

$$\dot{\eta} \; = \; -\frac{1}{g}\frac{\partial }{\partial E}... ...rac{\partial }{\partial E}\left(\frac{\rho}{g}\right)\right) ,$$ (A.10)

where we have identified the transformed diffusion rate $D = C/g^2$. Note that the diffusion term has a modified form, which we examine later. The LHS now poses a problem. It is possible to transform the LHS, written as $(\partial/\partial t)_\Omega (\rho/g)$, using the Legendre rule for a change of variable:

$$\left. \frac{\partial }{\partial t}\right\vert _{\Omega} \ ... ...\right\vert _{\Omega} \! \cdot \frac{\partial }{\partial E} .$$ (A.11)

However the resulting sequence of algebra required to reach the desired form is very complicated^A.1and I will not give it here.

Instead I give a geometric approach which bypasses the algebra. We ask how the probability fluxes are related. If $\Omega(E)$ were independent of time, $j$ would transform without a Jacobean, that is $j_\rho = j_\eta$. Now with $\Omega(E)$ time-dependent, $\rho$ acquires an extra drift velocity equal to the gradient of the constant-$\Omega$ contour in the $E-t$ plane:

$$j_\rho \; = \; j_\eta + \left. \frac{\partial E}{\partial t}\right\vert _{\Omega} \rho.$$ (A.12)

This can easily be verified by considering Fig. A.1. The apparent lack of $E-\Omega$ symmetry in this relation is deceptive since the second (flux difference) term can also be written $-(\partial \Omega/\partial t)_E \, \eta$ using the relation

$$\left. \frac{\partial E}{\partial t}\right\vert _{\Omega} \... ...\frac{\partial \Omega}{\partial E}\right\vert _{t} \; = \; -1,$$ (A.13)

which is a geometric property of any surface in 3D.

Combining (A.12) with (A.5), (A.6) and (A.9) immediately gives

$$\dot{\eta} \; = \; \frac{1}{g} \dot{\rho} + \frac{1}{g} \fr... ...ac{\partial E}{\partial t}\right\vert _{\Omega} \rho \right) ,$$ (A.14)

which can now be substituted into the LHS of (A.10) to give the desired Fokker-Planck evolution (A.8). The resulting transformed coefficients are

$$D \; =\; \frac{C}{g^2} \hspace{1.45in} \mbox{transformed diffusion rate}$$ (A.15)

$$v \; =\; \frac{u}{g} + \left. \frac{\partial E}{\partial t}\right\vert _{\Omega} + \frac{g'}{g^3}C \hspace{0.5in} \mbox{transformed drift rate}.$$ (A.16)

Looking at the transformed drift, the first term is the (scaled) original drift rate, the second is `dragging' due to time-dependence of $\Omega(E)$, and the third, which can also be written as $(g'/g)D$, is a drift term entirely due to the modification of the diffusion term. The second term, being (minus) the parameter velocity $\dot{x}$ times the conservative force $F(x) \equiv -(\partial E/\partial x)_\Omega$, will only cause reversible energy changes. Note that the third term arises from the the transformation of the diffusion term

$$(C \eta')' \;\;\; \longrightarrow \;\;\; \frac{1}{g} \left( D g \left( \frac{\rho}{g} \right)' \right)' ,$$ (A.17)

In general, we see that a transformed diffusion term takes the form of the right-hand side of (A.17), which is biased. By `biased' I mean that it has nonzero local drift when there is zero local density gradient. The bias can be visualized if we imagine that diffusion in $\Omega$-space comes from a the continuum limit of an unbiased random walk on a uniformly-spaced lattice of sites. The same sites appear non-uniformly spaced in $E$-space, so the same random walk gives additional drift for the PDF in $E$-space. Only when a diffusion term is written in terms of its `natural' variable (here $\Omega$) does it take the unbiased form of the left-hand side of (A.17). This illustrates that great care should be taken in writing down an energy diffusion term in the theory of energy spreading.