# 10: Time-Correlation Functions

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Time-correlation functions are an effective and intuitive way of representing the dynamics of a system, and are one of the most common tools of time-dependent quantum mechanics. They provide a statistical description of the time evolution of an internal variable or expectation value for an ensemble at thermal equilibrium. They are generally applicable to any time-dependent process, but are commonly used to describe random (or stochastic) and irreversible processes in condensed phases. We will use them extensively in the description of spectroscopy and relaxation phenomena. Although they can be used to describe the oscillatory behavior of ensembles of pure quantum states, our work is motivated by finding a general tool that will help us deal with the inherent randomness of molecular systems at thermal equilibrium. They will be effective at characterizing irreversible relaxation processes and the loss of memory of an initial state in a fluctuating environment.

• 10.1: Definitions, Properties, and Examples of Correlation Functions
Returning to the microscopic fluctuations of a molecular variable A , there seems to be little information in observing the trajectory for a variable characterizing the time-dependent behavior of an individual molecule. However, this dynamics is not entirely random, since they are a consequence of time-dependent interactions with the environment. We can provide a statistical description of the characteristic time scales and amplitudes to these changes by comparing the value of A at time t w
• 10.2: Correlation Function from a Discrete Trajectory
In practice classical correlation functions in molecular dynamics simulations or single molecule experiments are determined from a time-average over a long trajectory at discretely sampled data points.
• 10.3: Quantum Time-Correlation Functions
Quantum correlation functions involve the equilibrium (thermal) average over a product of Hermitian operators evaluated two times
• 10.4: Transition Rates from Correlation Functions
Here we will show that the rate of leaving an initially prepared state, typically expressed by Fermi’s Golden Rule through a resonance condition in the frequency domain, can be expressed in the time-domain picture in terms of a time-correlation function for the interaction of the initial state with others.

This page titled 10: Time-Correlation Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Andrei Tokmakoff via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.