Relation to susceptibility

Here, for the sake of intuition, I briefly make contact with the traditional LRT notation arising in condensed matter physics [127,122,160,99,65]. I consider first the `response' (time-dependent expectation value) of a general measurable whose operator is $\mathcal{G}$. Looking at Eq.(2.40) we can write

$$\langle {\mathcal{G}}(t) \rangle \; = \; {\mathcal{G}}_{mm}... ...} \left[ a_n^* (t) {\mathcal{G}}_{nm}(t) + \mbox{c.c.} \right]$$ (2.54)

where the first term is the constant equilibrium value $\langle {\mathcal{G}}(0) \rangle$ and the other terms have been kept only up to first order in the driving $x$. The response is linear but is not generally local in time, i.e. it has memory. However, because of translational invariance in time, the response must be local in frequency. So we can relate the Fourier transform of response to that of driving by $\langle \tilde{\mathcal{G}}(\omega) \rangle = \chi_{{\mbox{\tiny GF}}}(\omega) \tilde{x}(\omega)$, where $\chi_{{\mbox{\tiny GF}}}(\omega)$ is a generalized susceptibility. Numerous symmetry relations of $\chi_{{\mbox{\tiny GF}}}(\omega)$ follow from causality[127]; in particular $\chi_{{\mbox{\tiny GF}}}(-\omega) = \chi^*_{{\mbox{\tiny GF}}}(\omega)$. Substitution of (2.40) gives

$$\chi_{{\mbox{\tiny GF}}}(\omega) = \frac{i}{\hbar} \int_0^{... ...langle [ {\mathcal{G}}(t), {\mathcal{F}}(0) ] \right\rangle ,$$ (2.55)

where the latter form is averaged over $m$ with the appropriate initial distribution $p_m$, and is a standard result of LRT [65].

We now choose ${\mathcal{G}} = {\mathcal{F}}$ whereupon the subscript ${\mbox{\tiny GF}}$ is no longer needed. The susceptibility is split into real and imaginary parts, $\chi = \chi' + i\chi''$. The dissipation (energy absorption) is determined by $\chi''$ [127], since, using the driving (2.41) we have

$$\dot{Q}_{{\mbox{\tiny i}}rrev} = \langle \dot{\mathcal{H}}(t) \rangle \; = \; \dot{x}\langle{\math... ...ga t}) \left[\chi(\omega)e^{ i \omega t} - \chi(-\omega)e^{-i \omega t} \right]$$

$$= \mbox{\small$\frac{1}{2}$}(w A)^2 \cdot \frac{\chi''(\omega)}{\omega},$$ (2.56)

where at the last stage the oscillatory terms average to zero. This heating rate expression results entirely from the definition of $\chi$. However now our friction coefficient from the previous section can be seen to be simply $\mu(\omega) = \chi''(\omega)/\omega$.

We should also note that in the condensed matter physics context $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ is proportional to the dynamic form factor $S_{\mathbf q}(\omega)$ if we associate $\partial {\mathcal{H}}/\partial x$ with the density operator of a mode labelled ${\mathbf q}$ [160]. In the case where $x$ is flux enclosed by a mesoscopic sample (quantum dot), $\partial {\mathcal{H}}/\partial x$ is a current operator, thus $\mu$ determines the conductivity of the system. As one would expect, the conductivity is then given by the time-integral of the current-current correlation $e^2\langle v(0) v(t) \rangle$ [122,99,8]. We have performed such a calculation for a chaotic mesoscopic dot [15].