This research activity was strictly related to his formulation of the ergodic hypothesis.

The ergodic hypothesis fails us here on any relevant timescale.

The ergodic hypothesis is often assumed in statistical analysis.

But the second law only makes sense for systems and timescales for which the ergodic hypothesis holds.

If the ergodic hypothesis holds, the system will explore each region of the phase space with a probability according to its volume.

In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.

However, using the ergodic hypothesis, the temperature can still be obtained to arbitrary precision by further averaging the momenta over a long enough time.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis.

Then Boltzmann assumed the ergodic hypothesis about wandering all over the configuration space, did the calculations, and, lo, statistical mechanics is upon us.

A statistical mechanical system in equilibrium can be modeled, via the ergodic hypothesis, as the stationary distribution of a stochastic process.

If entropy continues to increase in the contracting phase (see Ergodic hypothesis), the contraction would appear very different from the time reversal of the expansion.

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

The ergodic hypothesis does not seem to hold for the present universe and its rough timescale, at levels of complexity of molecular species and above.

The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking.

Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics.

(See also ergodic hypothesis.)

A microcanonical ensemble of classical systems provides a natural setting to consider the ergodic hypothesis, that is, the long time average coincides with the ensemble average.

It should perhaps be emphasized that while the microcanonical ensemble and Liouville's theorem are directly related, they should not be confused as being equivalent to the ergodic hypothesis.

Gibbs's analysis of irreversibility, and his formulation of Boltzmann's H-theorem and of the ergodic hypothesis, were major influences on the mathematical physics of the 20th century.

As we have noted, thanks to the ergodic hypothesis, the gas system ultimately visits each macrostate at a number of times proportional to the number of microstates in that macrostate.

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems.

The relationship between the microcanonical ensemble, Liouville's theorem, and ergodic hypothesis can be summarized as follows: The key assumption of a microcanonical ensemble is that all accessible microstates are equally probable.

The results of molecular dynamics simulations may be used to determine macroscopic thermodynamic properties of the system based on the ergodic hypothesis: the statistical ensemble averages are equal to time averages of the system.

The probabilistic model of LSA does not match observed data: LSA assumes that words and documents form a joint Gaussian model (ergodic hypothesis), while a Poisson distribution has been observed.

So long-term time averages and the ergodic hypothesis, despite the intense interest in them in the first part of the twentieth century, strictly speaking are not relevant to the probability assignment for the state one might find the system in.