states to k, and is given by: where is the probability for the system to be in the state , while the matrix is filled with a grid of transition-rate constants. Similarly, contributes to the occupation of all other states In probability theory, this identifies the evolution as a continuous-time Markov process, with the integrated master equation obeying a Chapman–Kolmogorov equation. The master equation can be simplified so that the terms with ℓ = k do not appear in the summation. This allows calculations even if the main diagonal of the is not defined or has been assigned an arbitrary value. The final equality arises from the fact that because the summation over the probabilities yields one, a constant function. Since this has to hold for any probability (and in particular for any probability of the form for some k) we get Using this we can write the diagonal elements as . The master equation exhibits detailed balance if each of the terms of the summation disappears separately at equilibrium—i.e. if, for all states k and ℓ having equilibrium probabilities and , These symmetry relations were proved on the basis of the time reversibility of microscopic dynamics (microscopic reversibility) as Onsager reciprocal relations. Examples of master equations 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). The Lindblad equation in quantum mechanics is a generalization of the master equation describing the time evolution of a density matrix. Though the Lindblad equation is often referred to as a master equation, it is not one in the usual sense, as it governs not only the time evolution of probabilities (diagonal elements of the density matrix), but also of variables containing information about quantum coherence between the states of the system (non-diagonal elements of the density matrix). Another special case of the master equation is the Fokker–Planck equation which describes the time evolution of a continuous probability distribution. Complicated master equations which resist analytic treatment can be cast into this form (under various approximations), by using approximation techniques such as the system size expansion. Stochastic chemical kinetics are yet another example of the Master equation. A chemical Master equation is used to model a set of chemical reactions when the number of molecules of one or more species is