 ### Corr description

parent a6cc9e55
 ... ... @@ -47,7 +47,7 @@ from a separate `.py`-file, put that file into `code`. It will then be visible d Changes you make to the documentation are directly deployed to the webserver after pushing. If you work in a separate branch called `branch_name`, the changes are deployed to `https://compphys.quantumtinkerer.tudelft.nl/test_builds/branch_name`. In this way you can `https://computationalphysics.quantumtinkerer.tudelft.nl/test_builds/branch_name`. In this way you can check intermediate stages without exposing this to the students directly. ## Testing the build locally ... ...
 ... ... @@ -7,7 +7,7 @@ observable \$A\$ as \$\$\langle A\rangle = \frac{1}{N} \sum_{n>0}^{N}{A_n}\$\$ where \$N\$ the total simulation time (we only consider the data *after* where \$N\$ is the total simulation time (we only consider the data *after* the equilibration phase). This average would converge to the actual thermodynamic average if we could go to \$N=\infty\$. Since in our simulation \$N\$ is always finite, the time average is only an *estimate* of the ... ... @@ -19,7 +19,7 @@ If we had statistically independent random data for an physical observable \$A\$, we could compute the standard deviation of an average \$\langle A\rangle\$ with the well-known formula \$\$\sigma_A = \sqrt{\frac{1}{N-1} \left(\langle A^2 \rangle - \langle A \rangle^2\right)}\$\$. \$\$\sigma_A = \sqrt{\frac{1}{N-1}\sum_n\left(A_n - \langle A \rangle \right)^2}\$\$. However, this does **not** work with molecular dynamics, as we do **not** have statistically independent random data. We are doing a time-evolution, meaning that every new ... ... @@ -29,14 +29,14 @@ configuration is a small variation of the previous configuration - the data thus ### The autocorrelation function Correlated data means that the data sequence has a memory of the previous configurations. Let's now assume that we have a sequence of correlated data \$A_n\$. We can quantify the amount of correlation can by the autocorrelation function: previous configurations. Let's now assume that we have a sequence of correlated data \$A_n\$. We can quantify the amount of correlation can by the normalized autocorrelation function (also seen as Pearson correlation coefficient in the literature): \$\$ \chi_A(t) = \sum_n \left(A_n - \langle A \rangle \right) \$\$ \chi_A(t) = \frac{1}{\sigma_A^2} \sum_n \left(A_n - \langle A \rangle \right) \left(A_{n + t} - \langle A \rangle\right) \$\$ that compares the fluctuations at a certain time distance. Typically that compares the fluctuations at a certain time distance, assuming a stationary process. Typically the autocorrelation function has an exponential decay \$e^{-t/\tau}\$, where \$\tau\$ is the correlation time of the simulation (Note that in my definition here the correlation "time" refers to the index \$n\$ and ... ... @@ -52,12 +52,10 @@ \$\$ The formula above is valid for computing the autocorrelation function for an infinitely long time-series. To compute it from your finite length simulation data, you can use the formula \$\$ \chi_A(t) = \frac{1}{N - t} \sum_{n=1}^{N-t} A_n A_{n+t} - \frac{1}{N-t} \sum_{n=1}^{N-t} A_n \times \frac{1}{N-t} \sum_{n=1}^{N-t} A_{n+t}\,. \$\$ To get \$\tau\$ you would then have to fit \$\chi_A(t)\$ to \$e^{-t/\tau}\$. \chi_A(t) = \frac{\left(N-t\right)\sum_{n} A_n A_{n+t} - \sum_{n} A_n \times \sum_{n} A_{n+t}}{\sqrt{\left(N-t\right)\sum_{n} A_n^2 - \left(\sum_{n} A_n\right)^2}\sqrt{\left(N-t\right)\sum_{n} A_{n+t}^2 - \left(\sum_{n} A_{n+t}\right)^2}} \$\$ for \$1 \leq n\leq N-t\$. To get \$\tau\$ you would then have to fit \$\chi_A(t)\$ to \$e^{-t/\tau}\$. #### Example ... ...
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