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In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, ''etc''. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity). ==Mixing in stochastic processes== Let be a sequence of random variables. Such a sequence is naturally endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a sigma algebra, the Borel sigma algebra; it is the smallest (coarsest) sigma algebra that contains the topology. Define a function , called the strong mixing coefficient, as : In this definition, ''P'' is the probability measure on the sigma algebra. The symbol , with denotes a subalgebra of the sigma algebra; it is the set of cylinder sets that are specified between times ''a'' and ''b''. Given specific, fixed values , , ''etc.'', of the random variable, at times , , ''etc.'', then it may be thought of as the sigma-algebra generated by : The process is strong mixing if as . One way to describe this is that strong mixing implies that for any two possible states of the system (realizations of the random variable), when given a sufficient amount of time between the two states, the occurrence of the states is independent. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mixing (mathematics)」の詳細全文を読む スポンサード リンク
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