OBSERVATION CHAMBER / ACTIVE
Nonequilibrium Markov Cycle Observatory
Change the transition rates of a three-state cycle. Observe probability flow, stationary circulation, cycle affinity, and entropy production.
ROUTE 100 / STRUCTURE / MARKOV CYCLE
A steady distribution can still carry a current
Watch probabilities relax toward π. At nonzero affinity, the stationary distribution keeps a circulating current. Arrows show signed net probability currents, not particle trajectories.
PAUSEDCanvas 2D is unavailable. Controls, tables, reports and checks still work. PNG is unavailable.
Click a state to restart with 98% probability there and 1% in each other state. The buttons below do the same. Focus the canvas: Space/P = play or pause, S = step, R = reset.
- TIME
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- σ NOW
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- σ AT π
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- LOG RATE RATIO
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- MASS DRIFT
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- MAX |πQ|
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Probability ledger
| State | p(t) | π | E |
|---|
Rates and signed edge currents
| Positive edge | kᵢⱼ / s⁻¹ | kⱼᵢ / s⁻¹ | Jᵢⱼ / s⁻¹ | Jᵢⱼ(π) / s⁻¹ |
|---|
Positive orientation is 0 → 1 → 2 → 0. Negative J means reverse flow. In stationarity all three signed currents agree, although they need not be zero.
Download PNGCurrent observation report
DETERMINISTIC CHECKS
Check conservation and limiting cases —
Checks cover stationarity, positive probabilities, mass conservation, composition, equilibrium, flat energies, current reversal and invalid input.
| Check | Error | Limit | Result |
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OBSERVATION GUIDE / STRUCTURE
Compare equilibrium and stationarity
- Choose EQUILIBRIUM and STATIONARY. Every edge current vanishes: detailed balance holds.
- Choose CLOCKWISE, then STATIONARY. π stays fixed while a current circulates. COUNTERCLOCKWISE reverses its sign.
- Choose FLAT. π is exactly one third in each state; J = (2/3)sinh(a/6) and stationary σ = aJ.
REPORT and PNG pause playback. Edits and reset clear stale outputs. Hidden tabs stop without catching up or restarting. Reduced motion disables playback, while STEP remains available. RESET restores a = 0, δ = 1, biased state 0, probability lens and empty checks and outputs.
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MODEL AND SCOPE
Three positive states, continuous time
E = [0, δ, 2δ], k₀ = 1 s⁻¹
kᵢⱼ = k₀ exp((Eᵢ − Eⱼ)/2 + a/6)
kⱼᵢ = k₀ exp((Eⱼ − Eᵢ)/2 − a/6)
dpᵢ/dt = Σⱼ (pⱼkⱼᵢ − pᵢkᵢⱼ)
Jᵢⱼ = pᵢkᵢⱼ − pⱼkⱼᵢ
a = ln(k₀₁k₁₂k₂₀ / (k₁₀k₂₁k₀₂))
σ = Σ{01,12,20} Jᵢⱼ ln(pᵢkᵢⱼ / pⱼkⱼᵢ) ≥ 0
The rates above use the positive edges 01, 12 and 20. Each unordered pair enters σ once. E and a are dimensionless; entropy uses kB = 1 and σ is shown in kB/s. At a = 0, π is proportional to exp(−E).
Stationary probabilities use directed spanning trees. Uniformization evolves probabilities with P = I + Q/q and bounded Poisson tails. It preserves positivity without Euler clipping. This model does not simulate particle walks, eigenmodes or personal estimates. It does not measure heat, work, organisms or real networks.
OBSERVATION POLICY
This deterministic three-state continuous-time Markov model runs in your browser. Rates, calculations, reports, and images stay local. It does not sample particle paths or measure visitors or real systems.
Route
This chamber belongs to Structures & Symmetries.
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