12.3: Generalized Equations of Motion

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The most general way a system can be driven away from equilibrium by a forcing function $$F_e (t)$$ is according to the equations of motion:

\begin{align*} \dot {q}_i &= \dfrac{\partial H}{\partial p_i} + C_i({\rm x})F_e(t) \\[4pt] \dot {P}_i &=- \dfrac{\partial H}{\partial p_i} + D_i({\rm x})F_e(t) \end{align*}

where the $$3N$$ functions $$C_i$$ and $$D_i$$ are required to satisfy the incompressibility condition

$\sum_{i=1}^{3N}\left[ {\partial C_i \over \partial q_i} + {\partial D_i \over\partial p_i}\right] = 0 \nonumber$

in order to insure that the Liouville equation for $$f (x, t)$$ is still valid. These equations of motion will give rise to a distribution function $$f (x, t)$$ satisfying

${\partial f \over \partial t} + iLf = 0 \nonumber$

with $$\partial f/\partial t \neq 0$$. (We assume that $$f$$ is normalized so that $$\int d{\rm x}f({\rm x},t)=1$$.)

What does the Liouville equation say about the nature of $$f (x, t)$$ in the limit that $$C_i$$ and $$D_i$$ are small, so that the displacement away from equilibrium is, itself, small? To examine this question, we propose to solve the Liouville equation perturbatively. Thus, let us assume a solution of the form

$f({\rm x},t) = f_0(H({\rm x})) + \Delta f({\rm x},t) \nonumber$

Note, also, that the equations of motion $${\dot {x} }$$ take a perturbative form

$\dot{\rm x}(t) = \dot{\rm x}_0 + \Delta \dot{\rm x}(t) \nonumber$

and as a result, the Liouville operator contains two pieces:

$iL = \dot{\rm x}\cdot\nabla_{\rm x} = \dot{\rm x}_0\cdot \nabla _x +\Delta\dot{\rm x}\cdot\nabla_{\rm x} = iL_0 + i\Delta L \nonumber$

where $$iL_0 = \{...,H\}$$ and $$f_0 (H)$$ is assumed to satisfy

$iL_0 f_0(H({\rm x})) = 0 \nonumber$

$$\dot {x} _0$$ means the Hamiltonian part of the equations of motion

\begin{align*} \dot{q}_i &= \dfrac{\partial H}{\partial p_i} \\[4pt] \dot {P}_i &= - \dfrac{\partial H}{\partial q_i} \end{align*}

For an observable $$A (x)$$, the ensemble average of $$A$$ is a time-dependent quantity:

$\langle A(t)\rangle = \int d{\rm x}A({\rm x})f({\rm x},t) \nonumber$

which, when the assumed form for $$f (x , t)$$ is substituted in, gives

$\langle A(t)\rangle = \int d{\rm x}A({\rm x})f_0({\rm x}) + \int dx A (x) \Delta f (x, t ) = \langle A \rangle_0 + \int d{\rm x}A({\rm x})\Delta f({\rm x},t) \nonumber$

where $$\langle\cdot\rangle_0$$ means average with respect to $$f_0 (x)$$.

This page titled 12.3: Generalized Equations of Motion is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mark Tuckerman.