nucleationRate
plain-language theorem explainer
Nucleation rate supplies the thermal activation probability over a J-cost barrier in the Recognition Science phase-transition model, expressed as the Arrhenius exponential exp(-barrier/(k_B T)). Modelers of first-order transitions from cost-landscape bifurcations would cite it when linking recognition events to observable rates. The definition is a direct one-line application of the exponential to the barrier scaled by the SI Boltzmann constant and positive temperature.
Claim. The nucleation rate is defined by $r = e^{-B/(k_B T)}$ for barrier height $B$ and temperature $T>0$, where $k_B$ is the Boltzmann constant.
background
The Thermodynamics.PhaseTransitions module derives phase transitions from bifurcations in the J-cost landscape: multiple local minima merge or split as parameters change, producing first-order discontinuous jumps or second-order singularities. J-cost itself is the recognition cost of a multiplicative recognizer or observer-forcing event, taken from upstream definitions such as MultiplicativeRecognizerL4.cost (derivedCost on positive ratios) and ObserverForcing.cost (Jcost of the event state). The module anchors thermal activation to the exact SI Boltzmann constant kB_SI = 1.380649e-23.
proof idea
One-line definition that applies the exponential function directly to the negative ratio of the supplied barrier to the product of kB_SI and the positive temperature argument.
why it matters
The definition supplies the thermal-fluctuation mechanism required by the module's target of deriving phase transitions as J-cost bifurcations, directly supporting the sibling declarations for first-order and second-order transitions. It sits inside the T5-T8 forcing chain by furnishing the rate at which systems cross J-cost barriers within the eight-tick octave, consistent with the paper proposition on information-theoretic bifurcations.
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