# Perturbative solution of the Liouville equation

As in the classical case, we assume a solution of the form

\[ \rho(t) = \rho_0(H_0) + \Delta \rho(t) \]

where

\[ [H_0,\rho_0]=0\;\;\;\;\;\Rightarrow\;\;\;\;\;{\partial \rho_0 \over \partial t}=0 \]

and we will assume

\[ \rho_0(H_0) = {e^{-\beta H_0} \over Q(N,V,T)} \]

Substituting into the Liouville equation and working to first order in small quantities, we find

\[ {\partial \Delta \rho \over \partial t} = {1 \over i\hbar}[H_0,\Delta \rho] -{1 \over i\hbar} [B,\rho_0]F_e(t) \]

which is a **first order inhomogeneous equation** that can be solved by using an integrating factor:

\[ \Delta \rho(t) = -{1 \over i\hbar}\int_{-\infty}^t\;ds\;e^{-iH_0(t-s)/\hbar}[B,\rho_0]e^{iH_0(t-s)/\hbar}F_e(s) \]

(Note that we have chosen the origin in time to be \( \underline {t = - \infty } \), which is an arbitrary choice.)

For an observable \(A\), the expectation value is

\[ \langle A(t)\rangle = {\rm Tr}(\rho A) = \langle A\rangle _0 + {\rm Tr}(\Delta \rho(t) A) \]

when the solution for \( \Delta \rho \) is substituted in, this becomes

\(\langle A(t) \rangle \) | \(\langle A \rangle _0 - {1 \over i\hbar}\int_{-\infty}^t\;ds\;{\rm Tr}\left[Ae^{-iH_0(t-s)/hbar}[B,\rho_0]e^{iH_0(t-s)/\hbar}\right]F_e(s)\) | ||

\(\langle A \rangle _0 - {1 \over i\hbar}\int_{-\infty}^t\;ds\;{\rm Tr}\left[e^{iH_0(t-s)/\hbar} Ae^{-iH_0(t-s)/hbar}[B,\rho_0]\right]F_e(s)\) | |||

\(\langle A\rangle _0 - {1 \over i\hbar} ds\;{\rm Tr}\left[A(t-s)[B,\rho_0]\right]F_e(s) \) |

where the cyclic property of the trace has been used and the Heisenberg evolution for \(A\) has been substituted in. Expanding the commutator gives

\(\langle A(t) \rangle \) | \(\langle A\rangle _0 - {1 \over i\hbar}\int_{-\infty}^t\;ds\;{\rm Tr}\left[A(t-s)B\rho_0 - A(t-s)\rho_0B\right]F_e(s) \) | ||

\(\langle A\rangle _0 - {1 \over i\hbar}\int_{-\infty}^t\;ds\;{\rm Tr}\left[\rho_0\left(A(t-s)B - BA(t-s)\right)\right]F_e(s)\) | |||

\(\langle A\rangle _0 - {1 \over i\hbar}\int_{-\infty}^t\;ds\;F_e(s)\langle [A(t-s),B(0)]_0\rangle \) |

where the cyclic property of the trace has been used again. Define a function

\[ \Phi_{AB}(t) = {i \over \hbar}\langle [A(t),B(0)]\rangle _0 \]

called the *after effect function*. It is essentially the antisymmetric quantum time correlation function, which involves the commutator between \(A (t) \) and \(B (0) \). Then the linear response result can be written as

\[ \langle A(t)\rangle = \langle A \rangle _0 + \int_{-\infty}^t\;ds F_e(s)\Phi_{AB}(t-s) \]

which is the starting point for the theory of quantum transport coefficients. If we choose to measure the operator \(B\), then we find

\[ \langle B(t)\rangle = \langle B\rangle _0 + \int_{-\infty}^t\;ds\;F_e(s)\Phi_{BB}(t-s) \]