transitionrate

Transition rate is a measure of how quickly a system in a given state changes to another state. It is typically defined as the probability per unit time of making a transition, conditioned on being in the initial state. Transition rates can depend on external conditions such as temperature, fields, or concentrations.

In stochastic modeling, transition rates form a rate or generator matrix. For a system with states i and j, the rate of changing from i to j is q_ij. The diagonal elements q_ii are set so that each row sums to zero. The waiting time in state i follows an exponential distribution with parameter λ_i = sum_{j≠i} q_ij. The time evolution of state probabilities is governed by the master equation, which uses these rates to balance inflows and outflows for each state.

In chemistry, transition rates are encapsulated by rate constants for elementary steps. Under mass-action kinetics, the reaction rate r = k [A]^a [B]^b. First-order transitions have units of s^-1 and model single-molecule processes; second-order transitions have units of M^-1 s^-1 and involve two reactants. The coupled differential equations describing concentrations use these rate expressions to predict how the system evolves.

In quantum mechanics, transition rates between states due to a perturbation are described by transition amplitudes. Fermi’s golden rule gives the rate W_i→f = (2π/ħ) |⟨f|H′|i⟩|^2 ρ(E_f), where H′ is the perturbation and ρ is the density of final states. The total rate from an initial state is the sum over all accessible final states.

In nuclear and particle contexts, the transition rate is often called the decay rate, with lifetime τ = 1/λ and half-life t_1/2 = ln 2 / λ. The concept of transition rate thus links dynamical changes across physics, chemistry, and stochastic processes.