How do we use ergodic theory in physics?

Question

I am Mathematics student taking a graduate Ergodic Theory class. We are going over a lot of mathematical theory, but I would like to understand (at least at a superficial level) the connection with physics.

Here's what I know about ET: we usually study a measure space $(X, B, \mu)$ with a measure preserving transformation $f : X \to X$. We may also have function $\alpha:X \to \mathbb{R}$ or $\mathbb{C}$. Under various hypotheses we have lots of theorems about the behavior of points of $x$. For example, almost all points will have a dense orbit; almost every point is recurrent; the Ergodic Theorem; etc.

What I'm not sure about is exactly how this connects to the behavior of gasses trapped in a box (say, the unit cube $[0, 1]^3$). Each molecule has $3$ components of velocity. My confusion is:

• What is the space $X$? Is it $[0, 1]^3 \times \mathbb{R}^3?$ Or is it a finite subset of $[0, 1]^3 \times \mathbb{R}^3?$ consisting of those points which describe the finite number of molecules?
• What is the function $f$? If molecule $A$ is at point $p$ in $[0, 1]^3 \times \mathbb{R}^3$, is $f(p)$ the state of the molecule after (say) 1 second? What about $f(p)$ for points $p$ that do not correspond to any molecules?
• What about the observable $\alpha$? Similar questions about the domain of $\alpha$ as about the domain of $f$.
• What is the measure here?

Thank you very much.

Answer 1

I don't know anything about ergodic theory per se, but looking it up from Wikipedia, I got the following:

$X$ is the phase space of the system. In your example, if there are $N$ particles in the box, $X = [0,1]^{3N} \times \mathbb R^{3N}$ where $q \in [0,1]^3$ stands for the position in three dimensions and $p \in \mathbb R^3$ for the momentum (or velocity) of any particle (along the three directions, again). This means that every point $x \in X$ correspond to a valid configuration of the system in hand, where each particle $i \in \{0,..,N-1\}$ has position $q_i = [x_{3i},x_{3i+1},x_{3i+2}]^T$ and momentum $p_i = [x_{3i+3N},x_{3i+3N+1},x_{3i+3N+2}]^T$, both 3D vectors.

$f$ is... well, essentially the time evolution function. It would be weird if I tried to define it for a continuous-time system, but its role is the same in the discrete time systems, where, if one starts from a point $x_0 \in X$, you end up at $x_n = f^n(x_0)$, where $f^{n}(x) := f^{n-1}(f(x))$ and $f^1(x) = f(x)$ as usual.

In a dynamical system driven by a differential equation I guess it would be given via a infinitesimal time increment $dt$ at time $t$, e.g. in Hamiltonian mechanics you can get $p(t+dt) = p(t) + \dot p(t)\,dt$ and $q(t+dt) = q(t) + \dot q(t)\,dt$, where $\dot p(t)$ and $\dot q(t)$ are found by Hamilton's equations. Notice that in a continuous time system you'd need to integrate the changes over time (in better terms, solve the governing system of differential equations) rather than applying $f$ operation $n$ times. Also, since $\forall x \in X$ corresponds to a valid system state, this means that we can deduce the state of the system deterministically from any given initial configuration (not sure if this was necessary to be stated).

The observable $\alpha$ can be a multitude of things, an especially easy to see example of which being the position or momentum of the $i^\text{th}$ particle. I already defined them before, so it is obvious that $\forall x \in X$ any such observables are defined.

And for the measure, I found that, surprisingly, $6N$ dimensional Lebesgue measure works fine, refer to this section in Liouville's theorem wiki page