softmatter:Autocorelation function

An autocorrelation function indicates the extent to which a system retains a “memory” of itself at a previous time (or, conversely, how long it takes the system to “lose” its memory and “decorrelate” from itself.) The autocorrelation function for a particular property tells you how long you must simulate to generate a series of configurations uncorrelated from the previous series with respect to that property.

• In many instances, autocorrelation functions decay exponentially: C(t) = exp(t/τC)
• Then the relaxation time is typically the time it takes for the function to fall to 1/e of its initial value.
• It is possible to construct an autocorrelation function for any particle quantity (e.g. vi, ui,...) or any system quantity (e.g. U, E ...).

The general formulation is: $C_{AA}(t) = \frac{1}{N}\sum_{i=1}^{N}\frac{\left \langle \mathbf{A}_i(t)\cdot \mathbf{A}_i(0) \right \rangle}{\left \langle \mathbf{A}_i(0) \cdot \mathbf{A}_i(0) \right \rangle}$

where A is the quantity of interest.