the verification of the Boltzmann-sinaiErgodic Hypothesis.
An overview of the history of Ludwig Boltzmann's more than one hundred year old ergodic hypothesis is given.
We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i.
For systems of elastic hard balls on a torus Yakov Sinai, in 1963, [Sin(1963)] gave a mathematically rigorous version of Boltzmann's celebrated ergodic hypothesis: the system of an arbitrar
The path toward greater theoretical rigor, explains University of Warwick mathematics professor Tom Ward, reached a key milestone in the early 1930s when American mathematician George D. Birkhoff and Austrian-Hungarian (and later, American) mathematician and physicist John von Neumann separately reconsidered and reformulated Boltzmann’s ergodic hypothesis, leading to the pointwise and mean ergodic theories, respectively (see ref. ).
Conditional Proof of the Boltzmann-Sinai Ergodic Hypothesis
This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.
The justification for this hypothesis is a problem that the originators of statistical mechanics, J. C. Maxwell () and L. Boltzmann (), wrestled with beginning in the 1870s as did other early workers, but without mathematical success. J. W. Gibbs in his 1902 work () argued for his version of the hypothesis based on the fact that using it gives results consistent with experiments. The 1931–1932 ergodic theorem applied to the phase space of a mechanical system that arises in statistical mechanics and to the one-parameter group of homeomorphisms representing the time evolution of the system asserts that for almost all orbits, the time average of an integrable function on phase space is equal to its phase average, provided that the one-parameter group is metrically transitive. Hence, the ergodic theorem transforms the question of equality of time and phase averages into the question of whether the one-parameter group representing the time evolution of the system is metrically transitive.
Boltzmann's ergodic hypothesis, ..
In their influential 1911 article, Ehrenfest and Ehrenfest () summarized and discussed problems with the ergodic hypothesis and then proposed instead the quasi-ergodic hypothesis as a replacement. This hypothesis states that some orbit of the flow will pass arbitrarily close to every point of phase space, or in other words this orbit is topologically dense in the phase space. This hypothesis is a far more plausible one than the old ergodic hypothesis, and it does imply that any continuous function invariant under the flow is constant. Some authors [von Plato ()] have argued that, despite what Boltzmann had written down most of the time in his articles about the ergodic hypothesis, that he probably really meant something like what was later termed the quasi-ergodic hypothesis. However, the quasi-ergodic hypothesis does not imply metric transitivity. For instance, it is not even true that a minimal flow (every orbit is dense) with an invariant measure is metrically transitive [see Furstenberg ()]. for examples. Therefore, although the original ergodic hypothesis was too strong to be plausible, the quasi-ergodic hypothesis was too weak to establish equality of time and phase averages. Further mathematical, progress had to await the concept of metric transitivity and the ergodic theorems of 1931 and 1932. For more details, see the survey article of Mackey ().
It is interesting to look back at the early history of statistical mechanics to see how the founders of the subject handled the topic of time averages and space averages. Boltzmann () coined the terms ergoden or ergodische (which we translate as ergodic) from the Greek (energy) and (path) or energy path. He put forth what he called the ergodic hypothesis, which postulated that the mechanical system, say for gas dynamics, starting from any point, under time evolution , would eventually pass through every state on the energy surface. Maxwell and his followers in England called this concept the continuity of path (). It is clear that under this assumption, time averages are equal to phase averages, but it is also equally clear to us today that a system could be ergodic in this sense only if phase space were one dimensional. Plancherel () and Rosenthal () published proofs of this, and earlier, Poincare () had expressed doubts about Boltzmann’s ergodic hypothesis. Certainly part of the problem Maxwell and Boltzmann faced was that the mathematics necessary for a proper discussion of the foundations of statistical mechanics, such as the measure theory of Borel and Lebesgue, and elements of modern topology had not been discovered until the first decade of the 20th century and were hence unavailable to them.
Proof of the Boltzamnn-Sinai ergodic hypothesis for ..
Both the verification of the Boltzmann-Sinai ergodic hy ..
Kubo proposed a physical approach to proving the validity of Boltzmann's ergodic hypothesis
Proof of the Boltzmann-Sinai ergodic hypothesis for …
in the last 10-15 years his school has obtained the strongest results on the Boltzmann-Sinai ergodic hypothesis.
in particular with Boltzmann's ergodic hypothesis .
07/04/2006 · Kubo proposed a physical approach to proving the validity of Boltzmann's ergodic hypothesis
of the Boltzmann-Sinai ergodic hypothesis ..
The proper mathematical formulation of Ludwig Boltzmann’s ergodic hypothesis, which has incited so much interest and discussion in the last hundred years, is still not clear.
This “strong” form of Boltzmann’s ergodic hypothesis ..
A brief discussion of some key developments in the foundations of statistical mechanics will serve to provide a deeper appreciation for the notion of an abstract dynamical system and its role in ergodic theory. The theory emerged as a new abstract field of mathematical physics beginning with the ergodic theorems of von Neumann and Birkhoff in the early 1930s; see Moore (2015) for a historical discussion of von Neumann’s and Birkhoff’s theorems. The theorems have their roots in Ludwig Boltzmann’s ergodic hypothesis, which was first formulated in the late 1860s (Boltzmann 1868, 1871). Boltzmann introduced the hypothesis in developing classical statistical mechanics; it was used to provide a suitable basis for identifying macroscopic quantities with statistical averages of microscopic quantities, such as the identification of gas temperature with the mean kinetic energy of the gas molecules. Although ergodic theory was inspired by developments in classical mechanics, classical statistical mechanics, and even to some extent quantum mechanics (as will be shown shortly), it became of substantial interest in its own right and developed for the most part in an autonomous manner.
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