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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 : : where ''Pk'' is the probability for the system to be in the state ''k'', while the Matrix is filled with a grid of transition-rate Constant s. In probability theory, this identifies the evolution as a Continuous-time Markov Process , with the integrated master equation obeying a Chapman-Kolmogorov Equation . Note that : (i.e. probability is conserved), so the equation may also be written: : In this form, it closely resembles Liouville's Equation in Classical Mechanics , and Lindblad's Equation in Quantum Mechanics . The master equation exhibits Detailed Balance if each of the terms of the summation disappears separately at equilibrium — ie if, for all states ''k'' and ''ℓ'' having equilibrium probabilities and , . If the matrix is symmetric, ie all the microscopic transition dynamics are state- Reversible so : 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 ). One generalization of the master equation is the Fokker-Planck Equation which describes the time evolution of a continuous probability distribution. EXTERNAL LINKS
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