masterekvation

Masterekvation, commonly called the master equation, is a mathematical framework for the time evolution of a probability distribution over the states of a stochastic system. It is used across physics, chemistry, biology and engineering to describe systems that hop randomly between discrete states.

In continuous-time, discrete-state systems, the probability P_i(t) of being in state i evolves as

dP_i/dt = sum_j [W_{ij} P_j - W_{ji} P_i],

where W_{ij} ≥ 0 is the transition rate from state j to state i. The matrix W = (W_{ij}) is the generator or rate matrix, with diagonal entries W_{ii} = -∑_{k≠i} W_{ki}, ensuring probability conservation.

The vector P(t) remains normalized, ∑_i P_i(t) = 1. If the system is irreducible, there exists a stationary distribution π with dπ/dt = 0, satisfying ∑_j (W_{ij} π_j - W_{ji} π_i) = 0; π is an eigenvector of W with eigenvalue 0.

Relation to other descriptions: For many states with small transitions, the master equation can be approximated by a Fokker-Planck equation for a continuous variable (diffusion approximation). It also underpins birth–death processes, chemical reaction networks and stochastic population dynamics.

Quantum analogue: In quantum systems the density operator ρ evolves according to the quantum master equation, often in Lindblad form: dρ/dt = -i[H, ρ] + ∑_k (L_k ρ L_k† - 1/2 {L_k† L_k, ρ}). This accounts for environmental interactions and non-unitary evolution.