Mix

Mix, mixes, mixture, or mixing may refer to:

In mathematics, science, and technology

In electronics and telecommunications

Other uses in mathematics, science, and technology

Mixing (mathematics)

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

Mixing (physics)

In physics, a dynamical system is said to be mixing if the phase space of the system becomes strongly intertwined, according to at least one of several mathematical definitions. For example, a measure-preserving transformation T is said to be strong mixing if

whenever A and B are any measurable sets and μ is the associated measure. Other definitions are possible, including weak mixing and topological mixing.

The mathematical definitions of mixing are meant to capture the notion of physical mixing. A canonical example is the Cuba libre: suppose that a glass initially contains 20% rum (the set A) and 80% cola (the set B) in separate regions. After stirring the glass, any region of the glass contains approximately 20% rum. Furthermore, the stirred mixture is in a certain sense inseparable: no matter where one looks, or how small a region one looks at, one will find 80% cola and 20% rum.

Every mixing transformation is ergodic, but there are ergodic transformations which are not mixing.