# 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:

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

\(\underline {m_{\alpha}\ddot{x}_{\alpha} } \) | \(\underline {-m_{\alpha}\omega_{\alpha}^2x_{\alpha}- g_{\alpha}q } \) |

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

\(\tilde{q}(s)\) | \(\int_0^{\infty}\;dt\;e^{-st}q(t) \) | ||

\(\underline {\tilde{x}_{\alpha} } \) | \(\int_0^{\infty}\;dt\;e^{-st}x_{\alpha}(t) \) |

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 \(q (S) \) 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.