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There are two approaches to understanding Boltzmann's ergodic hypothesis in statistical mechanics. The first one, purely mathematical, goes by way of theorems while the second one relies on physical measurements. By its own nature the former is universal whereas the latter is specific to a system. By all account they seem orthogonal to each other. But should not they meet at the end? If, for example, both conclude that the hypothesis is not valid in a given system, should not their conclusions be compatible? We illustrate in this work how the two cultures meet in the physics of ergodicity.

In recent years the term ergodicity has come into scientific vogue in various physical problems. In particular when a system exibits chaotic behavior, it is often said to be ergodic. Is it a correct usage of the term ergodicity? Does it not mean that the time and ensemble averages of a variable are equal? Are they really related one to one? We examine this issue via simple models of harmonic oscilators by means of the theorems of Birkhoff and Khinchin and also by our own physical theory of ergometry. This study also considers the chaotic behavior in the logistic map.

The physical theory for the ergodic hypothesis is premised on the idea that the hypothesis is measurable by scattering experiment. Therewith it proves the hypothesis by measurable properties of the response function. Birkhoff's theorem proves the ergodic hypothesis by abstract properties of phase space, the measure and transitivity. In this work we apply the two independent approaches to a particular many-body model (spin-1/2 XY model) to see whether they arrive at the same conclusions on the ergodicity question. The two approaches are compared to gain insight into their goals.

It is fair to say that our current mathematical understanding of the dynamics of gravitational collapse to a black hole is limited to the spherically symmetric situation and, in fact, even in this case much remains to be learned. The reason is that Einstein's equations become tractable only if they are reduced to a (1 + 1)-dimensional system of partial differential equations. Owing to this technical obstacle, very little is known about the collapse of pure gravitational waves because by Birkhoff's theorem there is no spherical collapse in vacuum. In this essay, we describe a new cohomogeneity-two symmetry reduction of the vacuum Einstein equations in five and higher odd dimensions which evades Birkhoff's theorem and admits time-dependent asymptotically flat solutions. We argue that this model provides an attractive (1 + 1)-dimensional geometric setting for investigating the dynamics of gravitational collapse in vacuum.

Oppenheimer and Snyder found in 1939 that gravitational collapse in vacuum produces a "frozen star", i.e. the collapsing matter only asymptotically approaches the gravitational radius (event horizon) of the mass, but never cross it within a finite time for an external observer. Based upon our recent publication on the problem of gravitational collapse in the physical universe for an external observer, the following results are reported here: (1) Matter can indeed fall across the event horizon within a finite time and thus black holes (BHs), rather than "frozen stars", are formed in gravitational collapse in the physical universe. (2) Matter fallen into an astrophysical BH can never arrive at the exact center; the exact interior distribution of matter depends upon the history of the collapse process. Therefore gravitational singularity does not exist in the physical universe. (3) The metric at any radius is determined by the global distribution of matter, i.e. not only by the matter inside the given radius, even in a spherically symmetric and pressureless gravitational system. This is qualitatively different from the Newtonian gravity and the common (mis)understanding of the Birkhoff's Theorem. This result does not contract the "Lemaitre–Tolman–Bondi" solution for an external observer.

There are two approaches to understanding Boltzmann's ergodic hypothesis in statistical mechanics. The first one, purely mathematical, goes by way of theorems while the second one relies on physical measurements. By its own nature the former is universal whereas the latter is specific to a system. By all account they seem orthogonal to each other. But should not they meet at the end? If, for example, both conclude that the hypothesis is not valid in a given system, should not their conclusions be compatible? We illustrate in this work how the two cultures meet in the physics of ergodicity.

In recent years the term ergodicity has come into scientific vogue in various physical problems. In particular when a system exibits chaotic behavior, it is often said to be ergodic. Is it a correct usage of the term ergodicity? Does it not mean that the time and ensemble averages of a variable are equal? Are they really related one to one? We examine this issue via simple models of harmonic oscilators by means of the theorems of Birkhoff and Khinchin and also by our own physical theory of ergometry. This study also considers the chaotic behavior in the logistic map.

The physical theory for the ergodic hypothesis is premised on the idea that the hypothesis is measurable by scattering experiment. Therewith it proves the hypothesis by measurable properties of the response function. Birkhoff's theorem proves the ergodic hypothesis by abstract properties of phase space, the measure and transitivity. In this work we apply the two independent approaches to a particular many-body model (spin-1/2 XY model) to see whether they arrive at the same conclusions on the ergodicity question. The two approaches are compared to gain insight into their goals.