ergodicity

He made important contributions in settling Quantum unique ergodicity conjecture.

The question of ergodicity is whether they coincide.

The ergodicity requirement is that the ensemble average coincide with the time average.

A sufficient condition for ergodicity is that the time evolution of the system is a mixing.

One can discuss the ergodicity of various properties of a stochastic process.

The equivalence of the two methods is due to the ergodicity of the chaotic system.

Ergodicity is the assumption that a system, at a given temperature, explores the full phase space, just each state takes different probabilities.

Ergodicity of a continuous dynamical system means that its trajectories "spread around" the phase space.

Verbally, ergodicity means that time and space averages are equal, formally:

It is known that strong m-mixing implies ergodicity.

This transformation has even stronger properties of unique ergodicity, minimality, and equidistribution.

Ergodicity is where ensemble average equals time average.

Unique ergodicity of the flow was established by Hillel Furstenberg in 1972.

Mutation alone can provide ergodicity of the overall genetic algorithm process (seen as a Markov chain).

He has also used systems of a few coupled spins to illustrate the general requirements for equilibrium and ergodicity in isolated systems.

These images provide a graphic, intuitive demonstration of the strong ergodicity of the system.)

Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner.

One of Hedlund's early results was an important theorem about the ergodicity of geodesic flows.

In short, thermalization could not occur because of a certain "soliton symmetry" in the system which broke ergodicity.

He proved unique ergodicity of horocycle flows on a compact hyperbolic Riemann surfaces in the early 1970s.

This shows that some sort of mixing of energies, formally called ergodicity, is important for the law of equipartition to hold.

The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory.

A property of continuous dynamical systems that is the opposite of ergodicity is complete integrability.

Hence, strong mixing implies weak mixing, which implies ergodicity.

Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind.