This definition states that fluctuations of fast and slow variables are uncorrelated, and according to the ergodic hypothesis this also is true for time averages.
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Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics.
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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.
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Given enough sampling and subject to the ergodic hypothesis, molecular dynamics trajectories can be used to estimate thermodynamic parameters of a system or probe kinetic properties, such as reaction rates and mechanisms.
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The Second Law says never entropy decreases over time, but The Second Law is a consequence of the fundamental postulate of statistical mechanics ( aka ergodic hypothesis ) which is not really true in general.
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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.
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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.
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(The ergodic hypothesis underlies the ability of a physical system to reach thermal equilibrium, but this may not always hold for computer simulations ( see the Fermi Pasta Ulam problem ) or in certain real-world systems such as non-thermal plasmas .)
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According to Henri Poincar? writing in 1904, even though Maxwell and Boltzmann had previously explained the irreversibility of macroscopic physical processes in probabilistic terms, " the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his " Elementary Principles of Statistical Mechanics " . " 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.
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