Ergodic hypothesis

The ergodic hypothesis ( often referred to as ergodic theorem ) states that, so that all energetically possible phase space regions are also reached thermodynamic systems generally behave randomly ( "molecular chaos "). The time that a trajectory in the phase space of microstates is located in a particular region is proportional to the volume of this region. The ergodic hypothesis is fundamental to statistical mechanics. It combines, among other things, the results of molecular dynamics simulations and Monte Carlo simulations.

In other words, the hypothesis states so that thermodynamic systems have the property of ergodicity.

Defined and limited

Precise is assumed that for almost all measures of time average is equal to the ensemble average:

Wherein the probability of state i, which is given by the probability distribution of the ensemble.

Is a prerequisite for validity, that the considered stochastic process is stationary and has a finite correlation time; then the ergodic hypothesis is valid in the limit of infinite time.

Further, a dynamic system only insofar ergodic (or more precisely quasi- ergodic ), as the trajectory ( ie the path of the system) to each point in phase space is arbitrarily close in finite time. In contrast, Ludwig Boltzmann formulated in his original paper in 1887 that the path meets each point.

Although the ergodic hypothesis clearly appears simple, its rigorous mathematical justification is extremely difficult.

Use in systems theory

We use the word in the system theory for the classification of systems and the signals generated by them. An ergodic signal is a stochastic ( randomly subjugated ) stationary signal that is both aperiodic and recurrent. This is for example the case when the signal has a distinctive waveform, without repeating these at fixed intervals. Ergodic systems tend to produce an output signal dependent only slightly from the initial excitation.

Injury

In the case of spontaneous symmetry breaking, the ergodic hypothesis is violated ( Ergodizitätsbrechung ). There are disjoint ergodic regions in phase space. This can be done during phase transitions in glass transitions, ie during solidification of a liquid, or in spin glasses.