# 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 $${\textbf A}$$:

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

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,

 $$\underline {\langle iL{\bf F}(\tau){\bf A}^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }$$ $$\underline {\langle iLQ{\bf F}(\tau){\bf A}^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }$$ $$\underline {-\langle Q{\bf F}(\tau)(iL{\bf A})^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }$$ $$\underline {-\langle Q^2{\bf F}(\tau)(iL{\bf A})^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }$$ $$\underline {-\langle Q{\bf F}(\tau)(QiL{\bf A})^{\rm T}\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }$$ $$\underline {-\langle {\bf F}(\tau){\bf F}^{\rm T}(0)\rangle\langle{\bf A}{\bf A}^{\rm T}\rangle^{-1}{\bf A} }$$

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)).