12.4: General Properties of Time Correlation Functions
- Page ID
- 5288
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Define a time correlation function between two quantities \(A (x) \) and \(B (x) \) by
\[\begin{align*} C_{AB} (t) &= \langle A(0)B(t)\rangle \\[4pt] &= \int d{\rm x}f({\rm x})A({\rm x})e^{iLt}B({\rm x}) \end{align*} \]
The following properties follow immediately from the above definition:
Property 1
\[\langle A(0)B(t)\rangle = \langle A(-t)B(0)\rangle \nonumber \]
Property 2
\[C_{AB}(0) = \langle A({\rm x})B({\rm x})\rangle \nonumber \] Thus, if \(A = B \), then
\[C_{AA}(t) = \langle A(0)A(t)\rangle \nonumber \]
known as the autocorrelation function of \(A\), and
\[C_{AA}(0) = \langle A^2\rangle \nonumber \]
If we define \(\delta A = A - \langle A \rangle \), then
\[C_{\delta A\delta A}(0) = \langle (\delta A)^2\rangle =\langle ( A - \langle A \rangle )^2 \rangle = \langle A^2\rangle - \langle A\rangle^2 \nonumber \]
which just measures the fluctuations in the quantity \(A\).
Property 3
A time correlation function may be evaluated as a time average, assuming the system is ergodic. In this case, the phase space average may be equated to a time average, and we have
\[C_{AB}(t) = \lim_{T\rightarrow\infty}{1 \over T-t}\int_0^{T-t}ds A({\rm x}(s))B({\rm x}(t+s)) \nonumber \]
which is valid for \( t<<T \). In molecular dynamics simulations, where the phase space trajectory is determined at discrete time steps, the integral is expressed as a sum
\[ C_{AB}(k\Delta t) = {1 \over N-k}\sum_{j=1}^{N-k}A({\rm x}_k)B({\rm x}_{k+j})\;\;\;\;\;\;\;\;\;\;k=0,1,2,...,N_c \nonumber \]
where \(N\) is the total number of time steps, \(\Delta t \) is the time step and \(N_c << N \).
Property 4: Onsager regression hypothesis
In the long time limit, \(A \) and \(B\) eventually become uncorrelated from each other so that the time correlation function becomes
\[ C_{AB}(t) = \langle A(0)B(t)\rangle \rightarrow \langle A\rangle\langle B\rangle \nonumber \]
For the autocorrelation function of \(A\), this becomes
\[ C_{AA}(t)\rightarrow \langle A\rangle^2 \nonumber \]
Thus, \(C_{AA} (t) \) decays from \(\langle A^2 \rangle \) at \( t = 0\) to \( \langle A^2 \rangle \) as \(t \rightarrow \infty \).
An example of a signal and its time correlation function appears in Figure \(\PageIndex{1}\). In this case, the signal is the magnitude of the velocity along the bond of a diatomic molecule interacting with a Lennard-Jones bath. Its time correlation function is shown beneath the signal:
Over time, it can be seen that the property being autocorrelated eventually becomes uncorrelated with itself.