Section 30.1

Invariants and Symmetry Arguments

Quantities that never change (red-black pairs), ants that pass through each other, the wine-and-water symmetry.

Toolkit

Many “will this process reach a certain state” or “what quantity results” questions collapse once you find a conserved or exactly-tracked quantity, or re-describe the dynamics in an equivalent but simpler form.

Key techniques:

Invariant counting: find a quantity (parity, difference, sum mod $k$ ) unchanged by every allowed move; if the target state has a different value, it is unreachable.
Conservation of swapped amount: if two containers of equal total volume exchange fluid and end at their original volumes, the amount of A in B’s container equals the amount of B in A’s container.
Parity obstruction: if a condition forces a sum of several quantities to be even, but the multiset of values has an odd count of odd entries, no configuration satisfies it.
Collision = pass-through relabelling: for identical particles exchanging velocities/directions on collision, the multiset of trajectories is the same as if they passed through each other; so any symmetric functional (e.g. last exit time, set of exit times) can be computed from the non-interacting system, $\max_i X_i$ for independent exit times $X_i$ , with $\mathbb{E}[\max]$ computable via order statistics.

Problems