OBSERVATION CHAMBER / ACTIVE
Kramers Escape Observatory
Watch thermal fluctuations carry independent paths to a barrier top. Compare survival, first-arrival times, and a continuum reference.
01 / OBSERVE FIRST ARRIVAL
One barrier, many waiting times
Focus the canvas: ← → barrier, ↑ ↓ noise; Space/P run or pause, S step, R reset. Every point is an independent path.
- Time
- 0.000
- Arrived at 0
- 0 / 256
- Survival
- 100%
- Arrival-only mean
- —
- Reference MFPT
- —
- Right-censored paths
- 256
03 / KEEP THE OBSERVATION
Record arrivals and unfinished paths
The arrival-only mean excludes unfinished paths. It is not the full mean first-passage time (MFPT). Reports include the observed counts, time limit, seed, and boundary definition.
The default timestep tends to overestimate arrival times. Agreement with the continuum MFPT or a fixed percentage accuracy is not guaranteed.
Download PNGObservation report
OBSERVATION MODEL
Reach the top for the first time
All paths start at x = −1. The left wall x = −1.6 reflects; the barrier top x = 0 absorbs. Once a path arrives, it stops. The right well is shown only to locate the barrier in the double-well potential.
This is thermally driven, overdamped first passage. No quantum tunneling, inertial motion, or full well-to-well transition is calculated.
FORMULA / BOUNDARIES
Drift competes with noise
dX = −U′(X)dt + √(2Ddt) N(0,1)
γ = 1 · x₀ = −1 · a = −1.6 · b = 0
Dτ″ − U′τ′ = −1
τ′(a) = 0 · τ(b) = 0
τ(x₀) = (1/D) ∫[x₀,b] e^(U(y)/D) ∫[a,y] e^(−U(z)/D) dz dy
The reference integrates this backward equation. The simulation uses Euler–Maruyama, mirror reflection at the left wall, and the first step endpoint at or above 0. A crossing and return inside one step can be missed; finite dt biases arrival times.
Gardiner / First-passage times and potential barriersHOW TO OBSERVE
Change one cause at a time
- Run the initial ensemble. Watch survival fall as paths reach 0 at different times.
- Increase B while keeping D fixed. The reference waiting time grows; many paths may remain unfinished at time 100.
- Increase D, then change the seed. Distinguish a systematic noise effect from the variation of one finite ensemble.
Kramers theory predicts exponential growth with B/D in the high-barrier regime. It is an approximation; no full-transition prefactor is applied to this barrier-top stopping rule.
NUMERICAL CHECKS
Check the model independently
The self-test checks the Brownian analytic limit, an independent nested quadrature, seeded replay, and input and boundary rules. It does not certify any one random ensemble as an exact MFPT measurement.
Not runOBSERVATION POLICY
This browser model follows independent overdamped stochastic paths. Model settings and generated reports stay local. It does not measure real particles, temperature, or visitors.
Route
This chamber belongs to Fields in Motion.
FIELD Back to Fields in Motion Route.