6 · Detailed balance and currents — the visual heart

Picture the space of joint choices — every combination of everyone's action — as a lattice, with the stationary probability of the logit-revision dynamics sitting on it like standing water.

In a potential game the water is still. Along every edge, forward flow equals backward flow — detailed balance — so the net probability current is zero everywhere, and the entropy production rate $$\sigma_{\mathrm{EP}} = \tfrac12\sum_{a,a'} \big[\pi(a)w(a{\to}a') - \pi(a')w(a'{\to}a)\big]\log\frac{\pi(a)w(a{\to}a')}{\pi(a')w(a'{\to}a)}$$ is zero. Strategic equilibrium is thermodynamic equilibrium.

In a harmonic game the water is still in total — the distribution is stationary — but it is turning: probability flows in closed loops around the lattice (rock beats scissors beats paper beats rock…), the current J* is nonzero, and the system dissipates continuously. This is a non-equilibrium steady state, the same object physics uses for driven systems, and it is where price cycles, allocation churn, and strategic circulation live. Measured: RPS at λ = 2 carries visible circulation and σ_EP ≈ 3.4; the same meter on any potential game reads below 10⁻¹².

Two things the meters taught us that we did not predict: dissipation and harmonic fraction co-move almost perfectly along the α axis, yet within the near-harmonic regime dissipation and response-asymmetry decouple entirely (they are different physical objects — a global flux versus a local derivative); and at high λ potential games escape criticality altogether, because play concentrates and the choice fluctuations that carry all these effects die out. Both are in the findings log with their regeneration seeds.