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- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06%3A_Classical_Interacting_Systems/6.04%3A_Liquid_State_PhysicsThe virial expansion is typically applied to low-density systems. When the density is high, where a is a typical molecular or atomic length scale, the virial expansion is impractical. There are to m...The virial expansion is typically applied to low-density systems. When the density is high, where a is a typical molecular or atomic length scale, the virial expansion is impractical. There are to many terms to compute, and to make progress one must use sophisticated resummation techniques to investigate the high density regime. To elucidate the physics of liquids, it is useful to consider the properties of various correlation functions.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Essential_Graduate_Physics_-_Statistical_Mechanics_(Likharev)/05%3A_Fluctuations/5.08%3A_Back_to_the_correlation_functionOn the other hand, since the average \(\langle f(t)f(t +\tau )\rangle\) of a stationary process should not depend on \(t\), instead of \(W_m(0)\) we may take the stationary probability distribution \(...On the other hand, since the average \(\langle f(t)f(t +\tau )\rangle\) of a stationary process should not depend on \(t\), instead of \(W_m(0)\) we may take the stationary probability distribution \(W_m(\infty )\), independent of the initial conditions, which may be found as the same special solution, but at time \(\tau \rightarrow \infty \).
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Essential_Graduate_Physics_-_Statistical_Mechanics_(Likharev)/05%3A_Fluctuations/5.04%3A_Fluctuations_as_functions_of_time\[\left\langle f_{\omega} f_{\omega^{\prime}}^{*}\right\rangle=\frac{1}{(2 \pi)^{2}} \int_{-\infty}^{+\infty} d t \int_{-\infty}^{+\infty} d \tau K_{f}(\tau) e^{i\left(\omega-\omega^{\prime}\right) t}...\[\left\langle f_{\omega} f_{\omega^{\prime}}^{*}\right\rangle=\frac{1}{(2 \pi)^{2}} \int_{-\infty}^{+\infty} d t \int_{-\infty}^{+\infty} d \tau K_{f}(\tau) e^{i\left(\omega-\omega^{\prime}\right) t} e^{i \omega^{\prime} \tau} \equiv \frac{1}{(2 \pi)^{2}} \int_{-\infty}^{+\infty} K_{f}(\tau) e^{i \omega^{\prime} \tau} d \tau \int_{-\infty}^{+\infty} e^{i\left(\omega-\omega^{\prime}\right) t} d t . \label{56}\]
