Ergodic
Systems / Theory |

**In General** > s.a. description
of chaos; irreversibility.

* __Ergodic system__: One
for which the trajectory of almost every point in phase space winds densely
around the energy shell, so trajectories
fill the available phase space (but not necessarily chaotically), and time
averages of observables equal averages over the energy shell.

* __Physical idea__: For
a system in (microcanonical) equilibrium, time and ensemble averages of physical
quantities can be treated as equivalent if we assume that the system explores
the accessible part of phase space, with the fraction of time spent near each
point being proportional to the value of the distribution function there
(ergodic hypothesis); Ergodic theory studies the validity of the hypothesis.

$ __Def__: Formally, ergodicity
is a property of the action of a group *G* of transformations on a
space *X* (with measure, topology, or smoothness preserved); An ergodic system is a
dynamical system (*X*, *μ*, *φ*)
such that *A* = *φ*(*A*)
implies *μ*(*A*) = 0 or 1.

* __Boltzmann-Sinai ergodic hypothesis__:
A gas of hard balls is ergodic; Proved in different stages (Sinai 1972, Bunimovich &
Sinai 1973, Sinai & Chernov 1987, ...), and now known to be true for any number of
balls in any number of dimensions *D* ≥ 2.

* __Local Ergodic Theorem__: (also known
as the Fundamental Theorem) A result giving sufficient conditions under which a phase point has
an open neighborhood that belongs (mod 0) to one ergodic component; It is a key ingredient
of many proofs of ergodicity for billiards and, more generally, for smooth
hyperbolic maps with singularities, but its proof relies
upon a delicate assumption (the Chernov-Sinai Ansatz), which is difficult
to check for some physically relevant models.

* __Use in statistical mechanics__:
It provides a link between thermodynamic observables and microcanonical probabilities;
The ergodic theorem demonstrates the equality of microcanonical
phase averages and infinite time averages for some systems, and one argues
that actual measurements of thermodynamic quantities yield time averaged quantities,
since measurements take a long time; The combination of these two points is held
to be an explanation of why calculating microcanonical phase averages is a
successful algorithm for predicting the values of thermodynamic observables;
It is also well known that this account is problematic.

@ __General references__: Billingsley 65; Arnold & Avez
67; Ornstein 74; Sinai 76; Cornfeld et al 82; Walters 82; Sinai 94; Kalikow
& McCutcheon 10; Einsiedler & Ward 11 [III]; Gaveau & Schulman a1401 [ergodicity is not reasonable and not needed]; Viana & Oliveira 16; Kerr & Li 16.

@ __Mathematical__: Pugh & Shub BAMS(04)
[stable ergodicity]; Bosco et al JSP(10)
[exponential rate of convergence in the ergodic theorem].

@ __Boltzmann-Sinai ergodic hypothesis__: Lee PhyA(06); Simányi a1007 [in full generality].

@ __And statistical mechanics__: Earman & Rédei BJPS(96);
van Lith SHPMP(01); > s.a. equilibrium.

@ __Conceptual__: Frigg & Werndl PhSc(11)#4-a1310 [*ε*-ergodicity and the approach to equilibrium].

> __Related topics__: see equilibrium statistical mechanics ["epsilon" ergodicity]; Mixing System; Recurrence.

**Specific Types of Systems** > s.a. semiclassical
quantum mechanics; stochastic processes.

* __Example__: *X* = S^{1},
*φ*(*x*) = exp(2πi *α*)*x*, with *α*
irrational, *μ* = Lebesgue measure.

@ __Physics examples__: Bolte et al AP(01)
[spinning particles]; Botelho a0912 [non-linear
wave propagation]; Chernov & Simányi JSP(10) [proof of the Local Ergodic Theorem
for two-dimensional billiards]; Birrell et al SP&A(12)-a1105 [transition from ergodic to explosive behavior]; > s.a. turbulence.

@ __Lack of ergodicity__:
Borgonovi et al JSP(04)
[classically chaotic spin chain]; Rebenshtok & Barkai JSP(08) [weakly non-ergodic]; Wang et al PRE(14)-a1309 [two elastic hard-point masses in 1D, with generic mass ratio].

@ __Ergodic hierarchy__: Berkovitz et al SHPMP(06)
[randomness, and chaos]; Castagnino & Gómez PhyA(13)-a1301 [quantum ergodic hierarchy, Kolmogorov and Bernoulli systems]; Gómez & Portesi a1607 [information geometric extension, entropic dynamics].

@ __Non-equilibrium systems__: Wang et al PRL(07) [shear flow with different time
and ensemble averages]; Magdziarz & Weron AP(11) [anomalous diffusion].

@ __Other applications__: Cowan AAP(78) [statistical geometry].

**In Quantum Theory** > s.a. localization [ergodicity breakdown in many-body systems]; states
in statistical mechanics.

@ __General references__: von Neumann ZP(29)-a1003 [proof
of ergodic theorem in quantum mechanics]; Farquhar & Landsberg PRS(57);
Matsuno JMP(75);
Kümmerer & Maassen JPA(04)
[pathwise ergodic theorem for quantum trajectories]; Narnhofer RPMP(05)
[von Neumann, type II and III algebras]; Zelditch mp/05-en
[and mixing]; Schubert AHP(06)mp/05 [semiclassical
behaviour of eigenfunctions]; Barnett CPAM(06)mp/05 [rate of quantum ergodicity]; Castagnino & Lombardi PhyA(09)
[general framework]; Zelditch a0911-ln
[survey]; Bauer & Mello JPA-a1312; Ozorio de Almeida JPA(14)-a1403 [negativity witness]; Zambrano et al PRE(15)-a1502 [local conjecture]; Lopes & Sebastiani a1507 [simplified proof of von Neumann's Quantum Ergodic Theorem]; Zhang et al PRE(16)-a1601 [and mixing]; Ho & Radičević a1701 [graph-based approach].

@ __Special systems__: Asadi et al JPA(15)-a1507; Gherardini a1604 [randomly perturbed quantum systems].

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