Classical Fluctuation-Relaxation Theorem

A general expression is derived for the average infinitesimal-impulse-response matrix of a conservative classical system in a canonical ensemble. The equations of motion are taken as ${\dot{x}}_n=X_n$, where the $X\text{'}\mathrm{s}$, as well as the energy $E$, are functions of the $x\text{'}\mathrm{s}$ but not of their time derivatives. The Liouville equation $\Sigma \frac{\partial {\dot{x}}_n}{\partial x_n}=0\mathrm{}$ is assumed, but it is not required that the equations of motion be derivable from a Hamiltonian. If they are, the $x\text{'}\mathrm{s}$ are the canonical coordinates and momenta. The result found is $g_{\mathrm{mn}}\left(\tau \right)=\frac{{\Phi }_{\mathrm{mn}}\left(\tau \right)}{\mathrm{kT}}$, ($\tau \ge 0$), where $\epsilon g_{\mathrm{mn}}\left(\tau \right)$ is the average increment in $x_m$ at time $t$ resulting from an infinitesimal increment $\epsilon$ externally induced in $x_n$ at time $t-\tau$, ${\Phi }_{\mathrm{mn}}\left(\tau \right)$ is the covariance $〈x_m\mathrm{}\left(t\right)\mathrm{}{\left(\frac{\partial E}{\partial x_n}\right)}^{\prime }〉$, where the prime denotes argument $t-\tau$, $k$ and $T$ are Boltzmann's constant and absolute temperature. This relation is derived as a direct consequence of the fact that two initially isolated systems in equilibrium at identical temperatures remain in equilibrium when weakly coupled to each other.