Ergodicity

Measure-preserving dynamical systems

The mathematical definition of ergodicity aims to capture ordinary every-day ideas about randomness and ground them in measure theory and dynamical systems. ^[7] This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as diffusion and Brownian motion, as well as common-sense notions of mixing.^[8]

As an example consider the mixing of the dust in Saturn's rings: the equation of motions are deterministic, the system is in general not integrable [9], there is an effective dynamics which is essentially the Navier Stokes equations [10], it is about statistical distributions of particles in stable orbits but the dynamics is hyperbolic and unstable[11], therefore the Kolmogorov-Sinai entropy is not zero[12]

The second important motivation behind ergodicity is the analogy with mixing, ergodic theory can be motivated by phase transitions, and phase transitions are deeply interlinked with the concept of mixing of different fluids and different phases.^[13] There is a hierarchy of types of mixing that lead from weak mixing, to strong mixing and to chaos^[14], for example a maxwell distribution at equilibrium, implies a fully mixed gas and microstates, the existence of local and global averages both in time and space, where instead a fully developed turbulence can be interpreted as fluids that do not mix at any energy scale.

The extremal equilibrium states are τ-ergodic measures. They are interpreted as pure thermodynamic phases. Since the equilibrium states correspond to tangents to the graph of P, the discontinuities of the derivative of P correspond to phase transitions. One would thus like to know if P is piecewise analytic (in a suitable sense). An extremal equilibrium state σ may have a non-trivial decomposition into extremal Gibbs states. This is an example of symmetry breaking (the broken symmetry is the invariance under τ).[15]

Ergodic processes

The measure theoretic discussion assumes a concept of volume and integral, typically from a Lebesgue measure. Intutively when mixing different liquids the volume of each liquid is preserved. More generally in a statistical system the measure is a probability measure and is normalized to one. In this sense measure theory is dual to Kolmogorov axioms of probabilty^[16] and is a global approach on the full space of avaialble states. Locally instead one can see time evolution as a stochastic process, with a transfer operator from one state to the following state, such as in a Markov chain or as subsequent coin tosses.

In statistical mechanics

This section reviews ergodicity in statistical mechanics. The above abstract definition of a volume is required as the appropriate setting for definitions of ergodicity in physics. Consider a container of liquid, or gas, or plasma, or other collection of atoms or particles. Each and every particle x i {\displaystyle x_{i}} has a 3D position, and a 3D velocity, and is thus described by six numbers: a point in six-dimensional space R 6 . {\displaystyle \mathbb {R} ^{6}.} If there are N {\displaystyle N} of these particles in the system, a complete description requires 6 N {\displaystyle 6N} numbers. Any one system is just a single point in R 6 N . {\displaystyle \mathbb {R} ^{6N}.} The physical system is not all of R 6 N {\displaystyle \mathbb {R} ^{6N}} , of course; if it's a box of width, height and length W × H × L {\displaystyle W\times H\times L} then a point is in ( W × H × L × R 3 ) N . {\displaystyle \left(W\times H\times L\times \mathbb {R} ^{3}\right)^{N}.} Nor can velocities be infinite: they are scaled by some probability measure, for example the Boltzmann–Gibbs measure for a gas. Nonetheless, for N {\displaystyle N} close to the Avogadro number, this is obviously a very large space. This space is called the canonical ensemble.

A physical system is said to be ergodic if any representative point of the system eventually comes to visit the entire volume of the system. For the above example, this implies that any given atom not only visits every part of the box W × H × L {\displaystyle W\times H\times L} with uniform probability, but it does so with every possible velocity, with probability given by the Boltzmann distribution for that velocity (so, uniform with respect to that measure). The ergodic hypothesis states that physical systems actually are ergodic. Multiple time scales are at work: gases and liquids appear to be ergodic over short time scales. Ergodicity in a solid can be viewed in terms of the vibrational modes or phonons, as obviously the atoms in a solid do not exchange locations. Glasses present a challenge to the ergodic hypothesis; time scales are assumed to be in the millions of years, but results are contentious. Spin glasses present particular difficulties. Approach to ergodicity in this cases are shown to be anomalous ^[27].

Formal mathematical proofs of ergodicity in statistical physics are hard to come by; most high-dimensional many-body systems are assumed to be ergodic, without mathematical proof. Exceptions include the dynamical billiards, which model billiard ball-type collisions of atoms in an ideal gas or plasma. The first hard-sphere ergodicity theorem was for Sinai's billiards, which considers two balls, one of them taken as being stationary, at the origin. As the second ball collides, it moves away; applying periodic boundary conditions, it then returns to collide again. By appeal to homogeneity, this return of the "second" ball can instead be taken to be "just some other atom" that has come into range, and is moving to collide with the atom at the origin (which can be taken to be just "any other atom".) This is one of the few formal proofs that exist; there are no equivalent statements e.g. for atoms in a liquid, interacting via van der Waals forces, even if it would be common sense to believe that such systems are ergodic (and mixing). More precise physical arguments can be made, though.

Simple dynamical systems

The formal study of ergodicity can be approached by examining fairly simple dynamical systems. Some of the primary ones are listed here.

The irrational rotation of a circle is ergodic: the orbit of a point is such that eventually, every other point in the circle is visited. Such rotations are a special case of the interval exchange map. The beta expansions of a number are ergodic: beta expansions of a real number are done not in base-N, but in base- β {\displaystyle \beta } for some β . {\displaystyle \beta .} The reflected version of the beta expansion is tent map; there are a variety of other ergodic maps of the unit interval. Moving to two dimensions, the arithmetic billiards with irrational angles are ergodic. One can also take a rectangle, squash it, cut it and reassemble it; this is the previously mentioned baker's map. Its points can be described by the set of bi-infinite strings in two letters, that is, extending to both the left and right; as such, it looks like two copies of the Bernoulli process. If one deforms sideways during the squashing, one obtains Arnold's cat map. In most ways, the cat map is prototypical of any other similar transformation.

In classical mechanics and geometry

Ergodicity is a widespread phenomenon in the study of symplectic manifolds and Riemannian manifolds. Symplectic manifolds provide the generalized setting for classical mechanics, where the motion of a mechanical system is described by a geodesic. Riemannian manifolds are a special case: the cotangent bundle of a Riemannian manifold is always a symplectic manifold. In particular, the geodesics on a Riemannian manifold are given by the solution of the Hamilton–Jacobi equations.

The geodesic flow of a flat torus following any irrational direction is ergodic (however, this requires the state space of the flow to be taken as the torus itself and not as its unit tangent bundle, which is the usual state space for the geodesic flow). Informally this means that when drawing a straight line in a square starting at any point, and with an irrational angle with respect to the sides, if every time one meets a side one starts over on the opposite side with the same angle, the line will eventually meet every subset of positive measure (however, the direction of the tangent vectors of this trajectory remains constant, which is why if this flow is taken on the unit tangent bundle, it is non-ergodic and actually an integrable system). More generally, on any flat surface there are many ergodic directions for the geodesic flow.

For non-flat surfaces, one has that the geodesic flow of any negatively curved compact Riemann surface is ergodic. A surface is "compact" in the sense that it has finite surface area. The geodesic flow is a generalization of the idea of moving in a "straight line" on a curved surface: such straight lines are geodesics. One of the earliest cases studied is Hadamard's billiards, which describes geodesics on the Bolza surface, topologically equivalent to a donut with two holes. Ergodicity can be demonstrated informally, if one has a sharpie and some reasonable example of a two-holed donut: starting anywhere, in any direction, one attempts to draw a straight line; rulers are useful for this. It doesn't take all that long to discover that one is not coming back to the starting point. (Of course, crooked drawing can also account for this; that's why we have proofs.)

These results extend to higher dimensions. The geodesic flow for negatively curved compact Riemannian manifolds is ergodic. A classic example for this is the Anosov flow, which is the horocycle flow on a hyperbolic manifold. This can be seen to be a kind of Hopf fibration. Such flows commonly occur in classical mechanics, which is the study in physics of finite-dimensional moving machinery, e.g. the double pendulum and so-forth. Classical mechanics is constructed on symplectic manifolds. The flows on such systems can be deconstructed into stable and unstable manifolds; as a general rule, when this is possible, chaotic motion results. That this is generic can be seen by noting that the cotangent bundle of a Riemannian manifold is (always) a symplectic manifold; the geodesic flow is given by a solution to the Hamilton–Jacobi equations for this manifold. In terms of the canonical coordinates ( q , p ) {\displaystyle (q,p)} on the cotangent manifold, the Hamiltonian or energy is given by

H = 1 2 ∑ i j g i j ( q ) p i p j {\displaystyle H={\tfrac {1}{2}}\sum _{ij}g^{ij}(q)p_{i}p_{j}}

with g i j {\displaystyle g^{ij}} the (inverse of the) metric tensor and p i {\displaystyle p_{i}} the momentum. The resemblance to the kinetic energy E = 1 2 m v 2 {\displaystyle E={\tfrac {1}{2}}mv^{2}} of a point particle is hardly accidental; this is the whole point of calling such things "energy". In this sense, chaotic behavior with ergodic orbits is a more-or-less generic phenomenon in large tracts of geometry.

Ergodicity results have been provided in translation surfaces, hyperbolic groups and systolic geometry. Techniques include the study of ergodic flows, the Hopf decomposition, and the Ambrose–Kakutani–Krengel–Kubo theorem. An important class of systems are the Axiom A systems.

A number of both classification and "anti-classification" results have been obtained. The Ornstein isomorphism theorem applies here as well; again, it states that most of these systems are isomorphic to some Bernoulli scheme. This rather neatly ties these systems back into the definition of ergodicity given for a stochastic process, in the previous section. The anti-classification results state that there are more than a countably infinite number of inequivalent ergodic measure-preserving dynamical systems. This is perhaps not entirely a surprise, as one can use points in the Cantor set to construct similar-but-different systems. See measure-preserving dynamical system for a brief survey of some of the anti-classification results.

In wave mechanics

All of the previous sections considered ergodicty either from the point of view of a measurable dynamical system, or from the dual notion of tracking the motion of individual particle trajectories. A closely related concept occurs in (non-linear) wave mechanics. There, the resonant interaction allows for the mixing of normal modes, often (but not always) leading to the eventual thermalization of the system. One of the earliest systems to be rigorously studied in this context is the Fermi–Pasta–Ulam–Tsingou problem, a string of weakly coupled oscillators.

A resonant interaction is possible whenever the dispersion relations for the wave media allow three or more normal modes to sum in such a way as to conserve both the total momentum and the total energy. This allows energy concentrated in one mode to bleed into other modes, eventually distributing that energy uniformly across all interacting modes.

Resonant interactions between waves helps provide insight into the distinction between high-dimensional chaos (that is, turbulence) and thermalization. When normal modes can be combined so that energy and momentum are exactly conserved, then the theory of resonant interactions applies, and energy spreads into all of the interacting modes. When the dispersion relations only allow an approximate balance, turbulence or chaotic motion results. The turbulent modes can then transfer energy into modes that do mix, eventually leading to thermalization, but not before a preceding interval of chaotic motion.

In quantum mechanics

As to quantum mechanics, there is no universal quantum definition of ergodicity or even chaos (see quantum chaos).^[28] However, there is a quantum ergodicity theorem stating that the expectation value of an operator converges to the corresponding microcanonical classical average in the semiclassical limit ℏ → 0 {\displaystyle \hbar \rightarrow 0} . Nevertheless, the theorem does not imply that all eigenstates of the Hamiltonian whose classical counterpart is chaotic are features and random. For example, the quantum ergodicity theorem does not exclude the existence of non-ergodic states such as quantum scars. In addition to the conventional scarring,^{[29][30][31][32]} there are two other types of quantum scarring, which further illustrate the weak-ergodicity breaking in quantum chaotic systems: perturbation-induced^{[33][34][35][36][37]} and many-body quantum scars.^[38]

Invariant measure

Let ( X , B ) {\displaystyle (X,{\mathcal {B}})} be a measurable space. If T {\displaystyle T} is a measurable function from X {\displaystyle X} to itself and μ {\displaystyle \mu } a probability measure on ( X , B ) {\displaystyle (X,{\mathcal {B}})} , then a measure-preserving dynamical system is defined as a dynamical system for which μ ( T − 1 ( A ) ) = μ ( A ) {\displaystyle \mu {\mathord {\left(T^{-1}(A)\right)}}=\mu (A)} for all A ∈ B {\displaystyle A\in {\mathcal {B}}} . Such a T {\displaystyle T} is said to preserve μ ; {\displaystyle \mu ;} equivalently, that μ {\displaystyle \mu } is T {\displaystyle T} -invariant.

Ergodic measure

A measurable function T {\displaystyle T} is said to be μ {\displaystyle \mu } -ergodic or that μ {\displaystyle \mu } is an ergodic measure for T {\displaystyle T} if T {\displaystyle T} preserves μ {\displaystyle \mu } and the following condition holds:

For any A ∈ B {\displaystyle A\in {\mathcal {B}}} such that T − 1 ( A ) = A {\displaystyle T^{-1}(A)=A} either μ ( A ) = 0 {\displaystyle \mu (A)=0} or μ ( A ) = 1 {\displaystyle \mu (A)=1} .

In other words, there are no T {\displaystyle T} -invariant subsets up to measure 0 (with respect to μ {\displaystyle \mu } ).

Some authors^[39] relax the requirement that T {\displaystyle T} preserves μ {\displaystyle \mu } to the requirement that T {\displaystyle T} is a non-singular transformation with respect to μ {\displaystyle \mu } , meaning that if N {\displaystyle N} is a subset so that T − 1 ( N ) {\displaystyle T^{-1}(N)} has zero measure, then so does T ( N ) {\displaystyle T(N)} .

Examples

The simplest example is when X {\displaystyle X} is a finite set and μ {\displaystyle \mu } the counting measure. Then a self-map of X {\displaystyle X} preserves μ {\displaystyle \mu } if and only if it is a bijection, and it is ergodic if and only if T {\displaystyle T} has only one orbit (that is, for every x , y ∈ X {\displaystyle x,y\in X} there exists k ∈ N {\displaystyle k\in \mathbb {N} } such that y = T k ( x ) {\displaystyle y=T^{k}(x)} ). For example, if X = { 1 , 2 , … , n } {\displaystyle X=\{1,2,\ldots ,n\}} then the cycle ( 1 2 ⋯ n ) {\displaystyle (1\,2\,\cdots \,n)} is ergodic, but the permutation ( 1 2 ) ( 3 4 ⋯ n ) {\displaystyle (1\,2)(3\,4\,\cdots \,n)} is not (it has the two invariant subsets { 1 , 2 } {\displaystyle \{1,2\}} and { 3 , 4 , … , n } {\displaystyle \{3,4,\ldots ,n\}} ).

Equivalent formulations

The definition given above admits the following immediate reformulations:

• for every A ∈ B {\displaystyle A\in {\mathcal {B}}} with μ ( T − 1 ( A ) △ A ) = 0 {\displaystyle \mu {\mathord {\left(T^{-1}(A)\bigtriangleup A\right)}}=0} we have μ ( A ) = 0 {\displaystyle \mu (A)=0} or μ ( A ) = 1 {\displaystyle \mu (A)=1\,} (where △ {\displaystyle \bigtriangleup } denotes the symmetric difference);
• for every A ∈ B {\displaystyle A\in {\mathcal {B}}} with positive measure we have μ ( ⋃ n = 1 ∞ T − n ( A ) ) = 1 {\textstyle \mu {\mathord {\left(\bigcup _{n=1}^{\infty }T^{-n}(A)\right)}}=1} ;
• for every two sets A , B ∈ B {\displaystyle A,B\in {\mathcal {B}}} of positive measure, there exists n > 0 {\displaystyle n>0} such that μ ( ( T − n ( A ) ) ∩ B ) > 0 {\displaystyle \mu {\mathord {\left(\left(T^{-n}(A)\right)\cap B\right)}}>0} ;
• Every measurable function f : X → R {\displaystyle f:X\to \mathbb {R} } with f ∘ T = f {\displaystyle f\circ T=f} is constant on a subset of full measure.

Importantly for applications, the condition in the last characterisation can be restricted to square-integrable functions only:

• If f ∈ L 2 ( X , μ ) {\displaystyle f\in L^{2}(X,\mu )} and f ∘ T = f {\displaystyle f\circ T=f} then f {\displaystyle f} is constant almost everywhere.

Further examples

Ergodic theorems

If μ {\displaystyle \mu } is a probability measure on a space X {\displaystyle X} which is ergodic for a transformation T {\displaystyle T} the pointwise ergodic theorem of G. D. Birkhoff states that for every measurable function f : X → R {\displaystyle f:X\to \mathbb {R} } and for μ {\displaystyle \mu } -almost every point x ∈ X {\displaystyle x\in X} the time average on the orbit of x {\displaystyle x} converges to the space average of f {\displaystyle f} . Formally this means that lim k → + ∞ ( 1 k + 1 ∑ i = 0 k f ( T i ( x ) ) ) = ∫ X f d μ . {\displaystyle \lim _{k\to +\infty }\left({\frac {1}{k+1}}\sum _{i=0}^{k}f\left(T^{i}(x)\right)\right)=\int _{X}fd\mu .}

The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions.

Related properties

Examples

As in the discrete case the simplest example is that of a transitive action, for instance the action on the circle given by T t ( z ) = e 2 i π t z {\displaystyle T_{t}(z)=e^{2i\pi t}z} is ergodic for Lebesgue measure.

An example with infinitely many orbits is given by the flow along an irrational slope on the torus: let X = S 1 × S 1 {\displaystyle X=\mathbb {S} ^{1}\times \mathbb {S} ^{1}} and α ∈ R {\displaystyle \alpha \in \mathbb {R} } . Let T t ( z 1 , z 2 ) = ( e 2 i π t z 1 , e 2 α i π t z 2 ) {\displaystyle T_{t}(z_{1},z_{2})=\left(e^{2i\pi t}z_{1},e^{2\alpha i\pi t}z_{2}\right)} ; then if α ∉ Q {\displaystyle \alpha \not \in \mathbb {Q} } this is ergodic for the Lebesgue measure.

Ergodic flows

Further examples of ergodic flows are:

• Billiards in convex Euclidean domains;
• the geodesic flow of a negatively curved Riemannian manifold of finite volume is ergodic (for the normalised volume measure);
• the horocycle flow on a hyperbolic manifold of finite volume is ergodic (for the normalised volume measure)

Functional analysis interpretation

A very powerful alternate definition of ergodic measures can be given using the theory of Banach spaces. Radon measures on X {\displaystyle X} form a Banach space of which the set P ( X ) {\displaystyle {\mathcal {P}}(X)} of probability measures on X {\displaystyle X} is a convex subset. Given a continuous transformation T {\displaystyle T} of X {\displaystyle X} the subset P ( X ) T {\displaystyle {\mathcal {P}}(X)^{T}} of T {\displaystyle T} -invariant measures is a closed convex subset, and a measure is ergodic for T {\displaystyle T} if and only if it is an extreme point of this convex.^[43]

Unique ergodicity

The transformation T {\displaystyle T} is said to be uniquely ergodic if there is a unique Borel probability measure μ {\displaystyle \mu } on X {\displaystyle X} which is ergodic for T {\displaystyle T} .

In the examples considered above, irrational rotations of the circle are uniquely ergodic;^[47] shift maps are not.

The dynamical system associated with a Markov chain

Let S {\displaystyle S} be a finite set. A Markov chain on S {\displaystyle S} is defined by a matrix P ∈ [ 0 , 1 ] S × S {\displaystyle P\in [0,1]^{S\times S}} , where P ( s 1 , s 2 ) {\displaystyle P(s_{1},s_{2})} is the transition probability from s 1 {\displaystyle s_{1}} to s 2 {\displaystyle s_{2}} , so for every s ∈ S {\displaystyle s\in S} we have ∑ s ′ ∈ S P ( s , s ′ ) = 1 {\textstyle \sum _{s'\in S}P(s,s')=1} . A stationary measure for P {\displaystyle P} is a probability measure ν {\displaystyle \nu } on S {\displaystyle S} such that ν P = ν {\displaystyle \nu P=\nu }  ; that is ∑ s ′ ∈ S ν ( s ′ ) P ( s ′ , s ) = ν ( s ) {\textstyle \sum _{s'\in S}\nu (s')P(s',s)=\nu (s)} for all s ∈ S {\displaystyle s\in S} .

Using this data we can define a probability measure μ ν {\displaystyle \mu _{\nu }} on the set X = S Z {\displaystyle X=S^{\mathbb {Z} }} with its product σ-algebra by giving the measures of the cylinders as follows: μ ν ( ⋯ × S × { ( s n , … , s m ) } × S × ⋯ ) = ν ( s n ) P ( s n , s n + 1 ) ⋯ P ( s m − 1 , s m ) . {\displaystyle \mu _{\nu }(\cdots \times S\times \{(s_{n},\ldots ,s_{m})\}\times S\times \cdots )=\nu (s_{n})P(s_{n},s_{n+1})\cdots P(s_{m-1},s_{m}).}

Stationarity of ν {\displaystyle \nu } then means that the measure μ ν {\displaystyle \mu _{\nu }} is invariant under the shift map T ( ( s k ) k ∈ Z ) ) = ( s k + 1 ) k ∈ Z {\displaystyle T\left(\left(s_{k}\right)_{k\in \mathbb {Z} })\right)=\left(s_{k+1}\right)_{k\in \mathbb {Z} }} .

Criterion for ergodicity

The measure μ ν {\displaystyle \mu _{\nu }} is always ergodic for the shift map if the associated Markov chain is irreducible (any state can be reached with positive probability from any other state in a finite number of steps).^[48]

The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix P {\displaystyle P} a sufficient condition for this is that 1 be a simple eigenvalue of the matrix P {\displaystyle P} and all other eigenvalues of P {\displaystyle P} (in C {\displaystyle \mathbb {C} } ) are of modulus <1.

Note that in probability theory the Markov chain is called ergodic if in addition each state is aperiodic (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).^[49]

Moreover, the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures.

Examples