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Derivation of the GLE

The GLE can be derived from the harmonic bath Hamiltonian by simply solving Hamilton's equations of motion, which take the form

\[ \underline {\dot{q} } = {P \over m} \]

\[ \underline {\dot{p}} = \underline {-{\partial \phi \over \partial q} - \sum_{\alpha}g_{\alpha}x_{\alpha}-\sum_{\alpha}{g_{\alpha}^2 \over m_{\alpha}\omega_{\alpha}^2}q }\]

\[ \underline {\dot{x}_{\alpha} } = {P_{\alpha} \over m_{\alpha }} \]

\[ \underline {\dot{p}_{\alpha} } = \underline {-m_{\alpha}\omega_{\alpha}^2x_{\alpha}- g_{\alpha}q } \]


This set of equations can also be written as second order differential equation:

 

\(\underline {m\ddot{q} } \) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\underline {-{\partial \phi \over \partial q} - \sum_{\alpha}g_{\alpha}x_{\alpha}-\sum_{\alpha}{g_{\alpha}^2 \over m_{\alpha}\omega_{\alpha}^2}q}\)

 
\(\underline {m_{\alpha}\ddot{x}_{\alpha} } \) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\underline {-m_{\alpha}\omega_{\alpha}^2x_{\alpha}- g_{\alpha}q } \)

 


In order to derive an equation for \(\underline {q} \), we solve explicitly for the dynamics of the bath variables and then substitute into the equation for \(\underline {q} \). The equation for  \(\underline {x_{\alpha}} \) is a second order inhomogeneous differential equation, which can be solved by Laplace transforms. We simply take the Laplace transform of both sides. Denote the Laplace transforms of \(\underline {q} \) and \(\underline {x_{\alpha}} \) as

 

\(\tilde{q}(s)\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(\int_0^{\infty}\;dt\;e^{-st}q(t) \)  
\(\underline {\tilde{x}_{\alpha} } \) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(\int_0^{\infty}\;dt\;e^{-st}x_{\alpha}(t) \)  


and recognizing that

\[ \int_0^{\infty}\;dt\;e^{-st}\ddot{x}_{\alpha}(t) = s^2\tilde{x}_{\alpha}(s) - sx_{\alpha}(0) -\dot{x}_{\alpha}(0) \]


we obtain the following equation for \( \tilde{x}_{\alpha}(s) \):

\[ (s^2 + \omega_{\alpha}^2)\tilde{x}_{\alpha}(s) = sx_{\alpha} (0) + \dot{x}_{\alpha}(0) - {g_{\alpha}\over m_{\alpha}}\tilde{q}(s) \]


or

\[ \tilde{x}_{\alpha}(s) = {s \over s^2 + \omega_{\alpha}^2}x_{\alpha}(0) + {1 \over s^2 + \omega _{\alpha}^2} \dot {x} _{\alpha} (0) - {g_{\alpha} \over m_{\alpha}}{\tilde{q}(s) \over s^2 + \omega_{\alpha}^2} \]


\(x_{\alpha} (t) \)can be obtained by inverse Laplace transformation, which is equivalent to a contour integral in the complex \(s\)-plane around a contour that encloses all the poles of the integrand. This contour is known as the Bromwich contour. To see how this works, consider the first term in the above expression. The inverse Laplace transform is

\[ {1 \over 2\pi i}\oint\;ds\;{se^{st} \over s^2 + \omega_{\alpha}^2} = {1 \over 2 \pi i} \oint \;ds\;{se^{st} \over (s+i\omega_{\alpha})(s-i\omega_{\alpha})}\]


The integrand has two poles on the imaginary \(s\)-axis at \(\pm i \omega _{\alpha}\). Integration over the contour that encloses these poles picks up both residues from these poles. Since the poles are simple poles, then, from the residue theorem:

\[ {1 \over 2\pi i}\oint\;ds\;{se^{st} \over (s+i\omega_{\alpha} )(s - i\omega _{\alpha})} = {1 \over 2 \pi i} \left [ 2 \pi i \left ({i\omega_{\alpha}e^{i\omega _{\alpha}t } \over 2i\omega _{\alpha} } + {-i\omega_{\alpha}e^{i\omega _{\alpha}t } \over -2i\omega_{\alpha}}\right)\right] =\cos\omega_{\alpha}t \]


By the same method, the second term will give \( (\sin\omega_{\alpha}t)/\omega_{\alpha} \). The last term is the inverse Laplace transform of a product of  \(q (S) \)$\tilde{q}(s)$ and \(1/(s^2+\omega_{\alpha}^2) \). From the convolution theorem of Laplace transforms, the Laplace transform of a convolution gives the product of Laplace transforms:

\[ \int_0^{\infty}\;dt\;e^{-st}\int_0^t\;d\tau\;f(\tau)g(t-\tau) =\tilde{f}(s)\tilde{g}(s) \]


Thus, the last term will be the convolution of \(q (t) \) with \((\sin\omega_{\alpha}t)/\omega_{\alpha} \). Putting these results together, gives, as the solution for \(x_{\alpha} (t) \):

\[ x_{\alpha}(t) = x_{\alpha}(0)\cos\omega_{\alpha}t + {\dot{x} _{\alpha} (0) \over \omega _{\alpha} } \sin \omega _{\alpha} t - {g_{\alpha} \over m_{\alpha}\omega_{\alpha}}\int_0^t d\tau q(\tau)\sin\omega_{\alpha}(t-\tau) \]


The convolution term can be expressed in terms of \(\underline {\dot{q}} \) rather than \(\underline {q} \) by integrating it by parts:

\[ {g_{\alpha}\over m_{\alpha}\omega_{\alpha}}\int_0^t\;d\tau\;q (\tau) \sin \omega _{\alpha} (t - \tau ) = {g_{\alpha} \over m_{\alpha} \omega _{\alpha}^2 } \left [ q (t) - q (0) \cos \omega _{\alpha} t \right ] - {g_{\alpha} \over m_{\alpha} \omega _{\alpha}^2}\int_0^t\;d\tau\;\dot{q}(\tau)\cos\omega_{\alpha}(t-\tau) \]


The reasons for preferring this form will be made clear shortly. The bath variables can now be seen to evolve according to

\[ x_{\alpha}(t) = x_{\alpha}(0)\cos\omega_{\alpha}t + {\dot{x} (0) \over \omega _{\alpha}} \sin \omega_{\alpha} t + {g_{\alpha} \over m_{\alpha} \omega _{\alpha}^2}\int_0^t\;d\tau\;\dot{q}(\tau)\cos\omega_{\alpha}(t-\tau) - {g_{\alpha} \over m_{alpha}\omega_{\alpha}^2}\left[q(t)-q(0)\cos\omega_{\alpha}t\right] \]


Substituting this into the equation of motion for \(\underline {q} \), we find

\[ m\ddot{q} = -{\partial \phi \over \partial q} -\sum_{\alpha} g_{\alpha} \left [ x_{\alpha} (0) \cos \omega _{\alpha} t + {P_{\alpha} (0) \over m_{\alpha} \omega _{\alpha} } \sin \omega _{\alpha} t + {g _{\alpha} \over m_{\alpha} \omega _{\alpha}^2 } q(0) \cos \omega _{\alpha} t \right ] - \sum _{\alpha} {g^2_{\alpha} \over m_{\alpha} \omega ^2_{\alpha} } \int _0^t d \tau \dot {q} (\tau) \cos \omega_{\alpha} (t - \tau )\]

                   \(+ \sum _{\alpha} {g_{\alpha}^2 \over m_{\alpha}\omega_{\alpha}^2}q(t) - \sum _{\alpha} {g_{\alpha}^2 \over m_{\alpha}\omega_{\alpha}^2}q(t) \)


We now introduce the following notation for the sums over bath modes appearing in this equation:

1.
Define a dynamic friction kernel
\[\zeta(t) = \sum_{\alpha}{g_{\alpha}\over m_{\alpha}\omega_{\alpha}^2}\cos\omega_{\alpha}t\]
2.
Define a random force
\[R(t) = -\sum_{\alpha}g_{\alpha}\left[\left(x_{\alpha}(0) + {g_{\alpha} \over m_{\alpha} \omega_{\alpha}^2 } q (0) \right ) \cos \omega_{\alpha} t + {P_{\alpha} (0) \over m_{\alpha}\omega_{\alpha}}\sin\omega_{\alpha}t\right] \]

Using these definitions, the equation of motion for \(\underline {q} \) reads

\[m\ddot{q} = -{\partial \phi \over \partial q} - \int_0^t\;d\tau\;\dot{q}(\tau)\zeta(t-\tau)+ R(t) \] (1)

Eq. (1) is known as the generalized Langevin equation. Note that it takes the form of a one-dimensional particle subject to a potential \(\phi (q) \), driven by a forcing function \(R (t) \) and with a nonlocal (in time) damping term \(\underline {-\int_0^t\;d\tau\;\dot{q}(\tau)\zeta(t-\tau) } \), which depends, in general, on the entire history of the evolution of \(\underline {q} \). The GLE is often taken as a phenomenological equation of motion for a coordinate \(\underline {q} \) coupled to a general bath. In this spirit, it is worth taking a moment to discuss the physical meaning of the terms appearing in the equation.