The time average of a function along a trajectory starting from almost everywhere equals the ensemble average.
$X(t)$ is ergodic if $\overline{X}=⟨X⟩$. In other words, when the average along a single trajectory and the average over many different trajectories at a fixed time are equivalent.

Hypothesis: $w[t]$ eventually visits all of $\mathrm{\Omega }$ regardless of $w[0]$. If true, then Birkhoff's equality holds: $$\lim_{T\to\infty} \frac{1}{T}\int_0^T f(w[t])dt = \int_\Omega f(w)P(w)dw$$

• The ergodic theorem states that for typical Hamiltonian systems with finite dynamics, the time average is equal to the ensemble average.
• Mixing (self-correlation of a well-behaved function) means that correlation decays in time.
  • Weak mixing is when the time average of the squared correlation decays
• Chaos means the phase space orbits exhibit local instability, namely that phase space trajectories exiting from nearby points diverge exponentially.
  • Trajectories in an ergodic flow diverge algebraically, in contrast to the exponential divergence of chaos. Thus, 'chaos' is stronger than 'ergodicity'. A chaotic flow is definitely mixing.

Importance of all three

• Ergodicity links time and ensemble averages.
• Mixing and chaos justify certain key assumptions in the derivation of the Boltzmann equation and the proof of the H-Theorem.

Microcanonical ensemble is the largest region of the phase space. Given that entropy only increases ($d{\mathrm{\Omega }}_{eq}/dt\ge 0$), ${\mathrm{\Omega }}_{eq}$ grows until it reaches ${\mathrm{\Omega }}_{eq}(E,V,N)$. The only constraints to how much it can grow are the conserved quantities like energy, volume, and number of particles.
Symmetry creates conservation laws, and conservation laws prevent ergodicity