Instanton paths Deep tunneling and the crossover temperature

Regime

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A ring of 32 beads represents the tunneling path, stretched between the two wells. Its action is minimized by plain gradient descent — no dedicated saddle search — which only finds the true instanton because the ring starts, and stays, exactly mirror-symmetric between the two wells.

Each bead is a copy of the same proton; together, the ring describes how the proton is spread out in space — its delocalization (see 1D quantum thermometer). The instanton is the single ring shape that dominates tunneling: the most likely way for the delocalized proton to reach across the barrier. It is not a route the proton travels along.

Preset

Controls

Readouts

Effective barrier: the classical barrier height that would give the same rate exponent as the tunneling path. Above T_c the ring sits on the saddle and this equals the true barrier V^‡; below T_c it drops under it — tunneling makes the barrier look lower. Energies in units of k_BT_c.

Temperature is given in units of T_c — the slider's value already is T/T_c, so T_c itself always reads 1.000.

The path is not the reaction coordinate. The beads are joined by springs whose stiffness grows with temperature: near T_c the springs are tight and hold the ring compact, so it cannot reach out to the classical minima; as T → 0 they slacken and the ring stretches until it does. A gap between the path and the minima markers is expected well below T_c, not a sign the path is cut off.

Standalone model, illustrative rather than fitted: a symmetric double well coupled to a transverse mode whose equilibrium position bulges near the barrier top. The bulge height (skew) bends the minimum-energy path without moving the saddle or either minimum — both always sit exactly on that curve, whatever the skew.