As shown in the lecture, the TASEP with parallel update corresponds to rule \(184\). Here we consider the TASEP with \(N=50\) sites and periodic boundary conditions.

In the Metropolis algorithm (as applied to the Ising model), random changes in the spin configuration are either accepted or rejected. It might appear strange that, if a change is rejected, the same spin configuration appears twice (or even three times etc.) in a row of the Markov chain. It is essential, however, that the unchanged spin configurations are included in the Markov chain to get the proper probability distribution. This can be analysed in a much simpler context: the application of the Metropolis algorithm to a one-dimensional integral.

Consider the one-dimensional integral \[I=\int\limits_{-\infty}^{\infty}\mathrm{d}x\,w(x)\,g(x),\hspace{10px}\text{with}\,\,w(x)=\frac{1}{\sqrt{\pi}}e^{-x^2}.\] The Metropolis algorithm generates a Markov chain of random numbers \(\{x_i\}\), with a distribution given by \(w(x)\), from which the integral can be approximated as \[I\approx\frac{1}{N}\sum_{i=1}^Ng(x_i).\] The \(\bar{x}_{i+1}\) in the Metropolis algorithm can be calculated as \[\bar{x}_{i+1}=x_i+h\delta,\] wiht \(\delta\) a random number between \(-1\) and \(1\) (equally distributed). The maximum step site \(h\) can be set to \(h=1\).