# Mori-Zwanzig Theory: A more general derivation of the GLE

A derivation of the GLE valid for a general bath can be worked out. The details of the derivation are given in the book by Berne and Pecora called *Dynamic Light Scattering*. The system coordinate \(\underline {q} \) and its conjugate momentum \(\underline {p} \) are introduced as a column vector:

\[ {\textbf A} = \left(\matrix{q \cr p}\right) \]

and, in addition, one introduces statistical *projection operators* \(P\) and \(Q \) that project onto subspaces in phase space parallel and orthogonal to \(A\). These operators take the form

\[P = \underline {\langle ...{\textbf A}^{\rm T}\rangle \langle {\textbf A}{\textbf A}^{\rm T}\rangle ^{-1} }\]

\[Q = I-P \]

These operators are Hermitian and satisfy the property of idempotency:

\[\underline {P^2} = P\]

\[\underline {Q^2} = Q\]

Also, note that

\[P {\textbf A} ={\textbf A}\]

\[Q {\textbf A} = \underline {0} \]

The time evolution of \({\textbf A}\) is given by application of the classical propagator:

\[ {\textbf A}(t) = e^{iLt}{\textbf A}(0) \]

Note that the evolution of \({\textbf A} \) is unitary, i.e., it preserves the norm of \({\textbf A} \):

\[ \vert{\textbf A}(t)\vert^2 = \vert{\textbf A}(0)\vert^2 \]

Differentiating both sides of the time evolution equation for \({\textbf A} \) gives:

\[ {dA \over dt} = e^{iLt} iL{\bf A}(0) \]

Then, an identity operator is inserted in the above expression in the form \(I = P + Q \):

\[ {dA \over dt} = e^{iLt}(P+Q)iL{\textbf A}(0) = e^{iLt}PiL{\textbf A}(0) + e^{iLt}QiL{\textbf A}(0) \]

The first term in this expression defines a frequency matrix acting on \({\textbf A}\):

\(\underline {e^{iLt}PiL{\textbf A}(0)}\) |
\(\underline {e^{iLt}\langle iL{\textbf A}{\textbf A}^{\rm T}\rangle\langle {\textbf A}{\textbf A}^{\rm T}\rangle^{-1}{\textbf A} }\) |
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\(\underline {\langle iL{\textbf A}{\textbf A}^{\rm T}\rangle\langle {\textbf A}{\textbf A}^{\rm T}\rangle^{-1}e^{iLt}{\textbf A} } \) |
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\(\underline {\langle iL{\textbf A}{\textbf A}^{\rm T}\rangle\langle {\textbf A}{\textbf A}^{\rm T}\rangle^{-1}{\textbf A}(t) }\) |
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\(i{\bf \Omega}{\textbf A}(t) \) |

where

\[ {\bf\Omega} = \langle L{\bf A}{\bf A}^{\rm T}\rangle\langle {\bf A}{\bf A}^{\rm T}\rangle^{-1} \]

In order to evaluate the second term, another identity operator is inserted directly into the propagator:

\[ e^{iLt} = e^{i(P+Q)Lt} \]

Consider the difference between the two propagators:

\[ e^{iLt} - e^{iQLt} \]

If this difference is Laplace transformed, it becomes

\[ (s-iL)^{-1} - (s-iQL)^{-1} \]

which can be simplified via the general operator identity:

\[ {\rm A}^{-1} - {\rm B}^{-1} = {\rm A}^{-1}({\rm B}-{\rm A}){\rm B}^{-1} \]

Letting

\[A = (s - iL ) \]

\[B = (s - iQL ) \]

we have

\(\underline { (s-iL)^{-1}-(s-iQL)^{-1} } \) | \(\underline {(s-iL)^{-1}(s-iQL - s + iL)(s-iQL)^{-1} } \) | ||

\(\underline {(s-iL)^{-1}iPL(s-iQL)^{-1} } \) |

or

\[ (s-iL)^{-1} = (s-iQL)^{-1} + (s-iL)^{-1}(s-iQL - s + iL)(s-iQL)^{-1} \]

Now, inverse Laplace transforming both sides gives

\[ e^{iLt} = e^{iQLt} + \int_0^t\;d\tau\;e^{iL(t-\tau)}iPLe^{iQL\tau} \]

Thus, multiplying fromthe right by \(QiL{\textbf A} \) gives

\[ e^{iLt}QiL{\bf A}= e^{iQLt}QiL{\bf A}+\int_0^t\;d\tau\;e^{iL(t-\tau)}iPLe^{iQL\tau}QiL{\bf A} \]

Define a vector

\[ {\bf F}(t) = e^{iQLt}QiL{\bf A}(0) \]

so that

\[ e^{iLt}QiL{\bf A}= {\bf F}(t) +\int_0^t\;d\tau\;\langle iL {\bf F} (\tau) {\bf A }^T \rangle \langle {\bf A}{\bf A}^{\rm T}\rangle^{-1} {\bf A}(t-\tau) \]

Because \({\textbf F} (t) \) is completely orthogonal to \({\textbf A} (t) \), it is straightforward to show that

\[ Q{\bf F}(t) = {\bf F}(t) \]

Then,

\(\underline {\langle iL{\bf F}(\tau){\bf A}^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} } \) |
\(\underline {\langle iLQ{\bf F}(\tau){\bf A}^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }\) | ||

\(\underline {-\langle Q{\bf F}(\tau)(iL{\bf A})^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} } \) | |||

\(\underline {-\langle Q^2{\bf F}(\tau)(iL{\bf A})^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }\) | |||

\(\underline {-\langle Q{\bf F}(\tau)(QiL{\bf A})^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} } \) |
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\(\underline {-\langle {\bf F}(\tau){\bf F}^{\rm T}(0)\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} } \) |

Thus,

\[ e^{iLt}QiL{\bf A}= {\bf F}(t) - \int_0^t\;d\tau\;\langle {\bf F} (\tau){\bf F}^T (0) \rangle \langle {\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A}(t-\tau) \]

Finally, we define a memory kernel matrix:

\[ {\bf K}(t) = \langle {\bf F}(\tau){\bf F}^{\rm T}(0)\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1} \]

Then, combining all results, we find, for \({d {bf A} \over dt } \):

\[ {d{\bf A}\over dt} = i{\bf\Omega}(t){\bf A}-\int_0^t\;d\tau\;{\bf K}(\tau){\bf A}(t-\tau) + {\bf F}(t) \]

which equivalent to a generalized Langevin equation for a particle subject to a harmonic potential, but coupled to a general bath. For most systems, the quantities appearing in this form of the generalized Langevin equation are

\( i{\bf\Omega} \) | \({\bf K} (t) \) | ||

\({\bf F} (t) \) | \({\bf K} (t) \) | ||

\(\left(\matrix{0 \cr R(t)}\right) \) | \(\phi (q) = {m \omega ^2 q^2 \over 2} \) |

It is easy to derive these expressions for the case of the harmonic bath Hamiltonian when

\[ \langle R(0)R(t)\rangle = \langle R(0)e^{iLt}R(0)\rangle = kT\zeta(t) \]

For the case of a harmonic bath Hamiltonian, we had shown that the friction kernel was related to the random force by the fluctuation dissipation theorem:

\[ exp (iQLt ) \]

For a general bath, the relation is not as simple, owing to the fact that \({\textbf F} (t) \) is evolved using a modified propagator \(\langle R(0)e^{iQLt}R(0)\rangle = kT\zeta(t)\). Thus, the more general form of the fluctuation dissipation theorem is

\[ \langle R(0)e^{iQLt}R(0)\rangle \approx \langle R(0)e^{iL_{\rm cons}t}R(0)\rangle \]

so that the dynamics of \(R (t) \) is prescribed by the propagator \(\langle R(0)e^{iQLt}R(0)\rangle = kT\zeta(t)\). This more general relation illustrates the difficulty of defining a friction kernel for a general bath. However, for the special case of a stiff harmonic diatomic molecule interacting with a bath for which all the modes are soft compared to the frequency of the diatomic, a very useful approximation results. One can show that

\[ iL_{cons}\]

where \(C_{vv} (t) = {\langle \dot {q} (0) \dot {q} (t) \rangle \over \langle \dot {q} ^2 \rangle } \) is the Liouville operator for a system in which the diatomic is held rigidly fixed at some particular bond length (i.e., a constrained dynamics). Since the friction kernel is not sensitive to the details of the internal potential of the diatomic, this approximation can also be used for diatomics with stiff, *anharmonic* potentials. This approximation is referred to as the *rigid bond approximation* (see Berne, *et al*, *J. Chem. Phys.* **93**, 5084 (1990)).