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.
Contents
- Mixing in stochastic processes
- Types of mixing
- Mixing in dynamical systems
- L 2 displaystyle L2 formulation
- Products of dynamical systems
- Generalizations
- Examples
- Topological mixing
- References
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
Define a function
In this definition, P is the probability measure on the sigma algebra. The symbol
The process
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.
Types of mixing
Suppose {Xt} is a stationary Markov process, with stationary distribution Q. Denote
denote the conditional expectation operator on
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 < e−δt 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.
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 < γe−δt for some δ > 0, and it is “α-mixing with sub-exponential decay rate” if αt < ξ(t) for some non-increasing function ξ(t) satisfying t−1ln ξ(t) → 0 as t → ∞.
The α-mixing coefficients are always smaller than the ρ-mixing ones: αt ≤ ρt, 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 exponential decay rate” if βt < γe−δt for some δ > 0, and it is “β-mixing with sub-exponential decay rate” if βtξ(t) → 0 as t → ∞ for some non-increasing function ξ(t) satisfying t−1ln ξ(t) → 0 as t → ∞.
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 systems
A similar definition can be given using the vocabulary of measure-preserving dynamical systems. Let
For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with
To understand the above definition physically, consider a shaker
In such a situation, one would expect that after the liquid is sufficiently stirred (
where
A dynamical system is said to be weak mixing if one has
In other words,
in the Cesàro sense, and ergodic if
For a system that is weak mixing, the shift operator T will have no (non-constant) square-integrable eigenfunctions with associated eigenvalue of one. 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.
L 2 {displaystyle L^{2}} formulation
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
A dynamical system
A dynamical system
Since the system is assumed to be measure preserving, this last line is equivalent to saying that
Products of dynamical systems
Given two measured dynamical systems
Generalizations
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.
Examples
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 map considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature...
Topological mixing
A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map
where
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
A system is said to be topologically mixing if, given open sets
For a continuous-time system,
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.