In physics, a master equation is a phenomenological first-order differential equation describing the time-evolution of the probability of a system to occupy each one of a discrete set of states: A Superconductor demonstrating the Meissner Effect. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... The word probability derives from the Latin probare (to prove, or to test). ... The word discrete comes from the Latin word discretus which means separate. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In physics, the term state is used in several related senses, each of which expresses something about the way a physical system is. ...

where P_k is the probability for the system to be in the state k, while the matrix is filled with a grid of transition-rate constants. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

In probability theory, this identifies the evolution as a continuous-time Markov process, with the integrated master equation obeying a Chapman-Kolmogorov equation. In probability theory, a continuous-time Markov process is a stochastic process { X(t) : t ≥ 0 } that enjoys the Markov property and takes values from amongst the elements of a discrete set called the state space. ... In mathematics, specifically in probability theory, and yet more specifically in the theory of stochastic processes, the Chapman-Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. ...

Note that

(i.e. probability is conserved), so the equation may also be written:

If the matrix is symmetric, ie all the microscopic transition dynamics are state-reversible so A reversible process (or reversible cycle if the process is cyclic) , in thermodynamics, is a process that can be reversed by means of infinitesimal changes in some property of the system (Sears and Salinger, 1986). ...

this gives:

Many physical problems in classical, quantum mechanics and problems in other sciences, can be reduced to the form of a master equation, thereby performing a great simplification of the problem (see mathematical model). It has been suggested that this article or section be merged with Newtonian mechanics. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ...

One generalization of the master equation is the Fokker-Planck equation which describes the time evolution of a continuous probability distribution. The Fokker-Planck equation (named after Adriaan Fokker and Max Planck; also known as the Kolmogorov Forward equation) describes the time evolution of the probability density function of position and velocity of a particle. ...

See also

• Markov process
• Fermi's golden rule
• Lindblad equation
• Detailed balance
• Boltzmann's H-theorem