12.3: Generalized Equations of Motion
- Page ID
- 5286
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(f_0 (x) \).