12.3: Generalized Equations of Motion

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Page ID: 5286

Mark Tuckerman
New York University

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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 \(f_0 (x) \).