Ergodicity

Ergodicity (Greek έργον: Work and όδος: path ) is a property of dynamical systems. The term goes back to the physicist Ludwig Boltzmann, who examined this property in conjunction with the statistical theory of heat. Ergodicity is studied in mathematics in ergodic theory.

General

The ergodicity refers to the average behavior of a system. Such a system is described by a model function which determines the temporal performance of the system depending on its current state. You can now submit in two ways:

Strictly ergodic a system is then called when the time average and ensemble average result with probability one to the same result. Clearly this means that during the development of the system, all possible states are reached, the state space is thus completely filled with time. This means in particular, that in such systems, the expected value is not dependent on the initial state.

A system is called weakly ergodic if, in both cases, only the expected value and the variance comply and higher order moments are neglected.

The exact mathematical proof of ergodicity, in particular the proof of strict ergodicity can be furnished only in special cases. In practice, the detection of weak ergodicity is carried out at one or a small number of pattern features.

Examples

A simple physical example of an ergodic system is a particle moving randomly in a closed container moves ( Brownian motion). The state of this particle can then be simplified by its position in three-dimensional space described, which is defined by the container. This space is then the state space, and the movement in the room, by a random function (more precisely, a Wiener process ) will be described. If now we follow the trajectory of the particle, this will happen every point of the container after a sufficiently long time. Therefore, it does not matter if you make an averaging over the time or the space - the system is ergodic. In statistical mechanics is the assumption that real particles actually behave ergodic central to the derivations of macroscopic thermodynamic quantities, see ergodic hypothesis.

Another example is the dice: The mean number of 1,000 eyes dice rolls can be both thus, find that one times one after the other rolls a die in 1000, as well as the fact that you roll dice at the same time in 1000. The reason is that the 1000 simultaneously rolled dice will all be in slightly different states ( position in space, orientation of edges, speed, etc. ) and thus represent an average over the state space. Hence the term ensemble average is: When ergodic system can follow the development of a whole " flock " of initial states simultaneously and thus gain the same statistical information, if considered as a correspondingly longer period. This is used for measurements in order to obtain reliable results in a short time for noisy data.

A simple example of a stationary process which is not ergodic, one obtains: a " fair coin " is littered. If "head" falls, take the constant sequence, otherwise the constant sequence. The ensemble average are here equal to the time average but 1 or 0 ( with probability ).

Formal definition

Let be a probability space and a maßerhaltende transformation, that is, is a measurable function and applies to all. Then is called ergodic on (or alternatively to ergodic ) if one of the following equivalent conditions is met:

• For each applies with either or.
• For anybody with either applies or (where the symmetric difference referred to ).
• Holds for every positive measure.
• For any two sets with positive measure and there is a so that.