deletions | additions       diff --git a/time_series_convergence.tex b/time_series_convergence.tex  index 5a11137..d0968c5 100644  --- a/time_series_convergence.tex  +++ b/time_series_convergence.tex 
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\documentclass[11pt]{article}  \usepackage{amsmath,amssymb, amsfonts}  \usepackage{graphicx}  \begin{document}  \section{Time series convergence}  Early visual inspection of the square displacement, $(x-x_0)^2$, histograms showed a sudden, innatural, truncation of the histogram at some large displacements. The point at which the truncation occurrs grows as the number of iterations increases. This observation suggests that the system has not reached equilibration and, as a matter of fact, it is still diffusing within the basin. This is a serious concern because when measuring the mean square displacement we want to make sure that the number of iterations is sufficiently large to reach the boundaries of the basin of attraction when the walker is not coupled to the origin ($k=0$).   Asenjo et al. \cite{Asenjo_2014} found that the weakly coupled replicas have the tendency to get trapped into far and tiny regions of the basin. Swapping these replicas with the more strongly coupled replicas (with larger $k$), that oscillate around the origin, then improves equilibration. Yet, the timescales over which we were performing the calculations (1e5-1e6) did not seem to be sufficient to reach the boundaries of the basin. See Fig.~(\ref{fig:timeseries}) for an example.  \begin{figure}[t]  \centering  \includegraphics[width=\linewidth]{./figures/time_series.png}  \caption{Time series for a $n=32$ particle system at packing fraction $\phi_{HS}=0.7$ and $\phi_{SS}=0.88$ ran for about $2 \times 10^6$ MCMC steps, one $1/100$ of all points are shown.}  \label{fig:timeseries}  \end{figure}  Since we do not know how long it will take to reach the boundary of the basin and we cannot afford to run the calculations for an exceedingly large number of steps, we would like to develop an automated way of adapting the calculation length to reach some target relative statistical error. The statistical error $\Delta_A$, the root-mean square deviation of the sample mean $\overline{A}$ from the true expectation value $\langle A \rangle$, is given by:  \begin{equation}  \label{eq:delta1} 
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We would now expect to be dealing with a stationary time series while, as a matter of fact, the system is probably still slowly drifting. In any case, we want to estimate the integrated autocorrelation time which depends strongly on the length of the time series. When truncating the time series we might have been left with only a short array of point. For this reason we return as the predicted number of PT iterations \texttt{eq\_max\_ptiter}. When the time series is at least $10^5$ in length we then start updating the number of predicted PT iterations from Eq.~(\ref{eq:ptiter_prediction}). Since all replicas run for the same amount of time we take the largest number of steps from one of the replicas, typically the one corresponding to $k=0$.  \end{document}