Ergodic hypothesis
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In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a particle in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equally probable over a long period of time.
The ergodic hypothesis is often assumed in statistical analysis. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. Right or not, the analyst assumes that it is as good to observe a process for a long time as sampling many independent realisations of the same process. The assumption seems inevitable when only one stochastic process can be observed, such as variations of a price on the market.
Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the total or convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. Liouville's theorem ensures that the notion of time average makes sense, but ergodicity does not follow from Liouville's theorem.
In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase space can be sufficiently large that the thermodynamic equilibrium state exhibits some form of ergodicity breaking. A common example is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero. The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking. However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments.
[edit] Mathematics
In mathematics, ergodic theory is a branch which deals with dynamical systems which satisfy a version of this hypothesis, phrased in the language of measure theory.
