Mixing (mathematics)

Repeated application of the baker's map to points colored red and blue, initially separated. The baker's map is mixing, which can be seen qualitatively as the red and blue points seem to be completely mixed after several iterations.

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 processesEdit

Let   be a stochastic process on a probability space  . The sequence space into which the process maps can be endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology.

Define a function  , called the strong mixing coefficient, as

 

for all  . The symbol  , with   denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times a and b, i.e. the σ-algebra generated by  .

The process   is said to be strongly mixing if   as  . That is to say, a strongly mixing process is such that, in a way that is uniform over all times   and all events, the events before time   and the events after time   tend towards being independent as  ; more colloquially, the process, in a strong sense, forgets its history.

Types of mixingEdit

Suppose   were a stationary Markov process with stationary distribution   and let   denote the space of Borel-measurable functions that are square-integrable with respect to the measure  . Also let

 

denote the conditional expectation operator on   Finally, let

 

denote the space of square-integrable functions with mean zero.

The ρ-mixing coefficients of the process {xt} are

 

The process is called ρ-mixing if these coefficients converge to zero as t → ∞, and “ρ-mixing with exponential decay rate” if ρt < eδt for some δ > 0. For a stationary Markov process, the coefficients ρt may either decay at an exponential rate, or be always equal to one.[1]

The α-mixing coefficients of the process {xt} are

 

The process is called α-mixing if these coefficients converge to zero as t → ∞, it is “α-mixing with exponential decay rate” if αt < γeδt for some δ > 0, and it is α-mixing with a sub-exponential decay rate if αt < ξ(t) for some non-increasing function   satisfying

 

as  .[1]

The α-mixing coefficients are always smaller than the ρ-mixing ones: αtρt, therefore if the process is ρ-mixing, it will necessarily be α-mixing too. However, when ρt = 1, the process may still be α-mixing, with sub-exponential decay rate.

The β-mixing coefficients are given by

 

The process is called β-mixing if these coefficients converge to zero as t → ∞, it is β-mixing with an exponential decay rate if βt < γeδt for some δ > 0, and it is β-mixing with a sub-exponential decay rate if βtξ(t) → 0 as t → ∞ for some non-increasing function   satisfying

 

as  .[1]

A strictly stationary Markov process is β-mixing if and only if it is an aperiodic recurrent Harris chain. The β-mixing coefficients are always bigger than the α-mixing ones, so if a process is β-mixing it will also be α-mixing. There is no direct relationship between β-mixing and ρ-mixing: neither of them implies the other.

Mixing in dynamical systemsEdit

A similar definition can be given using the vocabulary of measure-preserving dynamical systems. Let   be a dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing if, for any  , one has

 

For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with   replaced by   with g being the continuous-time parameter.

To understand the above definition physically, consider a shaker   full of an incompressible liquid, which consists of 20% wine and 80% water. If   is the region originally occupied by the wine, then, for any region   within the shaker, the percentage of wine in   after   repetitions of the act of stirring is

 

In such a situation, one would expect that after the liquid is sufficiently stirred ( ), every region   of the shaker will contain approximately 20% wine. This leads to

 

where  , because measure-preserving dynamical systems are defined on probability spaces, and hence the final expression implies the above definition of strong mixing.

A dynamical system is said to be weak mixing if one has

 

In other words,   is strong mixing if   in the usual sense, weak mixing if

 

in the Cesàro sense, and ergodic if   in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing.

For a system that is weak mixing, the shift operator T will have no (non-constant) square-integrable eigenfunctions with associated eigenvalue of one.[citation needed] In general, a shift operator will have a continuous spectrum, and thus will always have eigenfunctions that are generalized functions. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.

  formulationEdit

The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system   is equivalent to the property that, for any function  , the sequence   converges strongly and in the sense of Cesàro to  , i.e.,

 

A dynamical system   is weakly mixing if, for any functions   and  

 

A dynamical system   is strongly mixing if, for any function   the sequence   converges weakly to   i.e., for any function  

 

Since the system is assumed to be measure preserving, this last line is equivalent to saying that   so that the random variables   and   become orthogonal as   grows. Actually, since this works for any function   one can informally see mixing as the property that the random variables   and   become independent as   grows.

Products of dynamical systemsEdit

Given two measured dynamical systems   and   one can construct a dynamical system   on the Cartesian product by defining   We then have the following characterizations of weak mixing:

Proposition. A dynamical system   is weakly mixing if and only if, for any ergodic dynamical system  , the system   is also ergodic.
Proposition. A dynamical system   is weakly mixing if and only if   is also ergodic. If this is the case, then   is also weakly mixing.

GeneralizationsEdit

The definition given above is sometimes called strong 2-mixing, to distinguish it from higher orders of mixing. A strong 3-mixing system may be defined as a system for which

 

holds for all measurable sets A, B, C. We can define strong k-mixing similarly. A system which is strong k-mixing for all k = 2,3,4,... is called mixing of all orders.

It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong m-mixing implies ergodicity.

ExamplesEdit

Irrational rotations of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.

Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, and the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature.

Topological mixingEdit

A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map   is said to be topologically transitive if, for every pair of non-empty open sets  , there exists an integer n such that

 

where   is the nth iterate of f. In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator. A related idea is expressed by the wandering set.

Lemma: If X is a complete metric space with no isolated point, then f is topologically transitive if and only if there exists a hypercyclic point  , that is, a point x such that its orbit   is dense in X.

A system is said to be topologically mixing if, given open sets   and  , there exists an integer N, such that, for all  , one has

 

For a continuous-time system,   is replaced by the flow  , with g being the continuous parameter, with the requirement that a non-empty intersection hold for all  .

A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.

Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.

ReferencesEdit

  • Chen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). "Nonlinearity and temporal dependence". Journal of Econometrics. 155 (2): 155–169. CiteSeerX 10.1.1.597.8777. doi:10.1016/j.jeconom.2009.10.001.CS1 maint: ref=harv (link)
  • Achim Klenke, Probability Theory, (2006) Springer ISBN 978-1-84800-047-6
  • V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (1968) W. A. Benjamin, Inc.