Ergodic
Theory| Ergodic
Theory |
In General > s.a. description
of chaos; irreversibility.
* Ergodic system: The
trajectory of almost every point in phase space winds densely around the
energy shell, so the time average equals the average over the energy shell;
Trajectories
fill the available phase space, but not necessarily chaotically.
* 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, 
, 
)
such that A = (A)
implies (A)
= 0 or 1.
* 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.
@ Mathematical: Pugh & Shub BAMS(04)
[stable ergodicity].
@ And statistical mechanics: Earman & Rédei BJPS(96);
van Lith SHPMP(01);
Lee PhyA(06)
[Boltzmann's ergodic hypothesis].
@ Quantum: Farquhar & Landsberg PRS(57);
Zelditch mp/05-in
[and
mixing]; Narnhofer RPMP(05)
[von Neumann, type II and III algebras]; > s.a. states
in statistical mechanics.
@ Conceptual: Berkovitz et al SHPMP(06)
[ergodic hierarchy, randomness, and chaos].
Specific Types of Systems > s.a. semiclassical
quantum mechanics; stochastic processes.
* Example: X = S1,
(x)
= exp(2
i
)x, with
irrational, =
Lebesgue measure.
@ Examples: Bolte et al AP(01)
[spinning particles]; > s.a. turbulence.
@ Lack of ergodicity:
Borgonovi et al JSP(04) [classically chaotic spin chain].
@ Non-equilibrium systems: Wang et al PRL(07) [shear flow with different time
and ensemble averages].
@ Quantum: Matsuno JMP(75);
Kümmerer & Maassen JPA(04)
[pathwise ergodic theorem for quantum trajectories]; Schubert mp/05 [semiclassical
behaviour of eigenfunctions]; Barnett CPAM(06)mp/05
[rate of quantum ergodicity].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
21 jun 2008