Why Detailed Balance?
	Statistical Mechanics: Entropy, Order Parameters, and Complexity
		James P. Sethna, Cornell University, Spring 2020
				Lecture 23


Detailed balance, the in-class topic for today, is a crucial feature for the sophisticated algorithms we use to numerically explore equilibrium statistical mechanics. But why is it true? What is the physics behind it?  We can get a bit of an clue by studying sensible physical systems that do not satisfy detailed balance. 

Detailed balance has to do with time-reversal symmetry. Detailed balance says that the net flux P rho* from alpha to beta is the same as that from beta to alpha. In a general Markov chain, this net flux would change sign under time-reversal. But if the Markov states alpha and beta are invariant under time-reversal, and if the equilibrium state $\rho*$ is time invariant, this difference formula is unchanged under time-reversal. So it must be zero for each alpha and beta, so the system must obey detailed balance.

Magnetic fields break time-reversal invariance (we'll see in the Chiral Wave Equation exercise), so detailed balance need not hold. Also, the states in phase space are not invariant (since momenta change sign when time flips), and clearly the flows in phase space do not obey detailed balance.

See you Wednesday!