Transfer operator
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Anticipation of a New Lover's Arrival, The
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grammar
: ''The transfer operator is different from the [[Transfer (group `theory)|transfer homomorphism]].''
In [[mathematics]], the '''transfer operator''' encodes information about an iterated map and is frequently used to study the behavior of [[dynamical systems]], [[statistical mechanics]], [[quantum chaos]] and [[fractals]]. The transfer operator is sometimes called the '''Ruelle operator''', after [[David Ruelle]], or the '''Ruelle-Perron-Frobenius operator''' in reference to the applicability of the [[Frobenius-Perron theorem]] to the determination of the eigenvalues of the operator.
The iterated function to be studied is a map
<math>f:X\rightarrow X</math> for an arbitrary set <math>X</math>. The transfer operator is defined as an operator <math>\mathcal{L}</math> acting on the space of functions
<math>\Phi:X\rightarrow \mathbb{C}</math> as
:<math>(\mathcal{L}\Phi)(x) = \sum_{y\in f^{-1}(x)} g(y) \Phi(y)</math>
where <math>g:X\rightarrow\mathbb{C}</math> is an auxiliary valuation function. When <math>f</math> has a [[Jacobian]] determinant, then <math>g</math> is usually taken to be <math>g=1/|J|</math>.
Some questions about the form and nature of a transfer operator are addressed in the theory of [[composition operator]]s.
The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic [[pushforward]] of ''g'': in essence, the transfer operator is the [[direct image functor]] in the category of measureable spaces.
==Applications==
Whereas the iteration of a function <math>f</math> naturally leads to a study of the orbits of points of X under iteration (the study of [[Chaos theory|point dynamics]]), the transfer operator defines how (smooth) maps evolve under iteration. Thus, transfer operators typically appear in [[physics]] problems, such as [[quantum chaos]] and [[statistical mechanics]], where attention is focused on the time evolution of smooth functions.
It is often the case that the transfer operator is positive, has discrete positive real-valued [[eigenvalue]]s, with the largest eigenvalue being equal
to one. For this reason, the transfer operator is sometimes called the
Frobenius-Perron operator.
The [[eigenfunction]]s of the transfer operator are usually fractals.
When the logarithm of the transfer operator corresponds to a quantum [[Hamiltonian (quantum theory)|Hamiltonian]], the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selected [[quantum ensemble|ensemble]] of quantum states will encompass a large number of very different fractal eigenstates with non-zero [[support (mathematics)|support]] over the entire volume. This can be used to explain many results from classical statistical mechanics, including the irreversibility of time and the increase of [[entropy]].
The transfer operator of the Bernoulli map <math>b(x)=2x-\lfloor 2x\rfloor</math> is exactly solvable and is a classic example of [[chaos theory|deterministic chaos]]; the discrete eigenvalues correspond
to the [[Bernoulli polynomials]]. This operator also has a continuous
spectrum consisting of the [[Hurwitz zeta function]].
The transfer operator of the Gauss map <math>h(x)=1/x-\lfloor 1/x \rfloor</math> is called the [[Gauss-Kuzmin-Wirsing operator|Gauss-Kuzmin-Wirsing (GKW) operator]] and due to its extraordinary difficulty, has not been fully solved. The theory of the GKW dates back to a hypothesis by Gauss on [[continued fraction]]s and is closely related to the [[Riemann zeta function]].
==See also==
* [[Bernoulli scheme]]
* [[Shift of finite type]]
==References==
* {{cite book | author=David Ruelle | title=Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics | publisher=Addison-Wesley, Reading | year=1978 | id=ISBN 0-201-13504-3}}
* {{cite book | author=Dieter H. Mayer | title=The Ruelle-Araki transfer operator in classical statistical mechanics | publisher=Springer-Verlag | year=1978 | id=ISBN 0-387-09990-5}}
* David Ruelle, ''[http://www.maths.ex.ac.uk/~mwatkins/zeta/ruelle.pdf Dynamical Zeta Functions and Transfer Operators]'', (2002) Institut des Hautes Etudes Scientifiques preprint IHES/M/02/66. ''(Provides an introductory survey).''
[[Category:Chaos theory]]
[[Category:Dynamical systems]]
[[Category:Operator theory]]
[[Category:Spectral theory]]