C Forms of Convergence

In the online simulations, there are four histograms plotted: two in velocity and two in position. The left column shows a histogram calculated at time \(t\), whereas the right column shows a histogram calculated on \([0,t]\). Here we briefly explain the difference, and why both are of interest.

Convergence in Law

The histograms at time \(t\) are so that we can assess whether the system is converging in law. Recall that a stochastic process \(X_t\in \mathbb{R}^n\) converges in law to \(X\in\mathbb{R}^n\) if \[ \lim_{t\to\infty} \mathbb{P}(X_t \in A) = \mathbb{P}(X\in A), \] for a measurable subset \(A\in \mathbb{R}^n\). If the particle system is converging in law, after a long simulation time we should see that the histogram produced looks very similar to our predicted invariant distribution: \(\mu_{\pm}\).

Convergence of Ergodic Averages

If instead we take a histogram on the interval \([0,t]\), we are assessing converge of ergodic average. This can be described intuitively as the average over time converging to the average in space, or more formally, for a process \(X_t\in\mathbb{R}^n\), \[ \frac{1}{t}\int_0^t X(s)\mathrm{d}s \to \mathbb{E}[X_t] \qquad \text{ as } t\to\infty. \] By taking the histogram over the interval, we are calculating a discrete version of left hand side of this limit.

The difference is important in the present of periodic behaviours. Consider the deterministic particle system, which after a long simulation, consists of one large cluster of particles moving around the torus. This system has not converged in law to the uniform distribution in position. If we observe the particles at any given time, the distribution will be far from uniform. If instead however we compute the space average by calculating the histogram on the interval, we will see and almost uniform distribution (if the cluster is moving at constant speed). Hence there is an important distinction to be made between different forms of convergence for the particle system.