Measure-preserving dynamical system explained

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium.

Definition

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system

(X, l{B}, \mu, T)

with the following structure:

X is a set,l B is a σ-algebra over X ,\mu: l{B} → [0,1] is a probability measure, so that \mu (X) = 1 , and \mu(\varnothing) = 0 ,T:X → X is a measurable transformation which preserves the measure \mu , i.e., \forall A\in l{B} \mu(T ^{-1} (A)) = \mu(A) .

Discussion

One may ask why the measure preserving transformation is defined in terms of the inverse

\mu(T ^{-1} (A)) = \mu(A) instead of the forward transformation \mu(T(A)) = \mu(A) . This can be understood intuitively.

Consider the typical measure on the unit interval

[ 0 , 1 ] , and a map Tx = 2 x\mod 1 = \begin{cases} 2 x if x < 1 / 2 \\ 2 x -1 if x > 1 / 2 \\ \end{cases} . This is the Bernoulli map. Now, distribute an even layer of paint on the unit interval [ 0 , 1 ] , and then map the paint forward. The paint on the [ 0 , 1 / 2 ] half is spread thinly over all of [ 0 , 1 ] , and the paint on the [ 1 / 2 , 1 ] half as well. The two layers of thin paint, layered together, recreates the exact same paint thickness.

More generally, the paint that would arrive at subset

A \subset [ 0 , 1 ] comes from the subset T ^{-1} (A) . For the paint thickness to remain unchanged (measure-preserving), the mass of incoming paint should be the same: \mu(A) = \mu(T ^{-1} (A)) .

Consider a mapping

l{T} of power sets:l{T}:P(X)\to P(X) Consider now the special case of maps l{T} which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends X to X (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective map T:X\to X by writing l{T}(A)=T ^{-1} (A) . Of course, one could also define l{T}(A)=T(A) , but this is not enough to specify all such possible maps l{T} . That is, conservative, Borel-preserving maps l{T} cannot, in general, be written in the form l{T}(A)=T(A); .\mu(T ^{-1} (A)) has the form of a pushforward, whereas \mu(T(A)) is generically called a pullback. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the transfer operator is defined in terms of the pushforward of the transformation map T ; the measure \mu can now be understood as an invariant measure; it is just the Frobenius–Perron eigenvector of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.)

There are two classification problems of interest. One, discussed below, fixes

(X, l{B}, \mu) and asks about the isomorphism classes of a transformation map T . The other, discussed in transfer operator, fixes (X, l{B}) and T , and asks about maps \mu that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into dissipative systems and the route to equilibrium.

In terms of physics, the measure-preserving dynamical system

(X, l{B}, \mu, T) often describes a physical system that is in equilibrium, for example, thermodynamic equilibrium. One might ask: how did it get that way? Often, the answer is by stirring, mixing, turbulence, thermalization or other such processes. If a transformation map T describes this stirring, mixing, etc. then the system (X, l{B}, \mu, T) is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure \mu is the one mode that does not decay away. The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.

Informal example

The microcanonical ensemble from physics provides an informal example. Consider, for example, a fluid, gas or plasma in a box of width, length and height

w x l x h, consisting of N atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in w x l x h x R ^{3.} A given collection of N atoms would then be a single point somewhere in the space (w x l x h) ^{N} x R ^{3N} . The "ensemble" is the collection of all such points, that is, the collection of all such possible boxes (of which there are an uncountably-infinite number). This ensemble of all-possible-boxes is the space X above.

In the case of an ideal gas, the measure

\mu is given by the Maxwell–Boltzmann distribution. It is a product measure, in that if p _{i(x,y,z,v} _{x,v} _{y,v} ^{3 xd} ^{z)d} ^{3p} is the probability of atom i having position and velocity x,y,z,v _{x,v} _{y,v} _{z} , then, for N atoms, the probability is the product of N of these. This measure is understood to apply to the ensemble. So, for example, one of the possible boxes in the ensemble has all of the atoms on one side of the box. One can compute the likelihood of this, in the Maxwell–Boltzmann measure. It will be enormously tiny, of order l{O}\left(2 ^{-3N} \right). Of all possible boxes in the ensemble, this is a ridiculously small fraction.

The only reason that this is an "informal example" is because writing down the transition function

T is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if there are interactions between the particles themselves, like a van der Waals interaction or some other interaction suitable for a liquid or a plasma; in such cases, the invariant measure is no longer the Maxwell–Boltzmann distribution. The art of physics is finding reasonable approximations.

This system does exhibit one key idea from the classification of measure-preserving dynamical systems: two ensembles, having different temperatures, are inequivalent. The entropy for a given canonical ensemble depends on its temperature; as physical systems, it is "obvious" that when the temperatures differ, so do the systems. This holds in general: systems with different entropy are not isomorphic.