Nota de FísicaPublicado
Consideraciones criticas de los fundamentos de la mecanica estadistica: nueva solucion al problema ergodico
Felix Cernuschi
1987
Datos de la publicación
- Autores
- Felix Cernuschi
- Tipo
- Nota de Física
- Estado
- Publicado
- Fecha de publicación
- 1987
- Descripción
- A representative ensemble of gaseous systems of equal macros copic values does not represent enough knowledge to define precisely the state of some system of interest as a function of time. Each member of the ensemble may be id a very great number of different states. On account of the incompleteness of the specifications defining the states of each system of the ensemble, it is not possible to construct the foundations of statistical mechanics on only the exact and deterministic principles of classical mechanics. The molecules of a gas move in all directions with great velocity, about 10 cm /sec and each one produces about <math>10^{11}\,\!</math> collisions per second with other particles and strikes the walls of the vessel; therefore it is impossible to predict their trajectories, which follow irregular zig-zag courses. Consequently the representative point of a close and isolated system in <math>\Gamma</math> space cannot have a continuous and derivable path, as it is admitted in the quasi-ergodic theorem. For this reason we infer that the hypothesis by which the generalized operator of hydrodynamics is applied to the density of a "fine dust" of representative points of systems of an ensemble in any element of phase space, is not correct. Those points are not material ones, but mathematical points; and it is not impossible, though it is extremely improbable, that two or more of such points coincide in an instant of time, if the number of systems in the ensemble is great enough, and a short lapse later they are far apart from each other. We think that from the physical point of view, those points in phase space would have a rather turbulent behaviour. Consequently we shall avoid to use Liouville's theorem in our endeavour to find the foundation of classical statistic mechanics. The quasi-ergodic theorem of George D. Birkhoff is undoubtly a masterpiece in mathematics; but from the physical point of view we think that it is objectable. This theorem is established on the existance of measure-preserving transformations of a set of points into itself. This implies in its physical application the acceptance of Liouville's Theorem. We indicate why this theorem is not on a solid physical basis. Birkhoff's theorem, if we consider it valid in Physics, would demonstrate, in general, the equality between the time average and the space average in the accesible <math>\Gamma</math> - space, but this result in secured for an extremely long time, while it is only necessary a very short period of time to obtain the corresponding results experimentally. The fundamental theorems of statistical mechanics are deduced using the deterministic and reversible laws of classical mechanics, ignoring the role played by the hazardous, intensive and completely unforeseen collision amongst the particles of the fluid system considered. We show that Boltzmann's famous principle - which expresses entropy as proportional to the logaritm of the member of "complexions" that the state of an isolated system has at a given time - is equivalent to the definition of entropy given by Shannon"s theory of information. R. C. Tolman in his book "The Principles of Statistical Mechanics" points out that on the only basis of the rigorous, deterministic and reversible laws of classical mechanics it is not, possible to set up on a permanent basis the guiding principles of statistical mechanics. He considers necessary to introduce some postulate as to a priori probabilities. For this purpose he introduces "the hypothesis of equal a priori probabilities for different regions of the phase space that correspond to extensions of the same magnitude". Tolman generalized Boltzmann H-theorem valid in <math>\Gamma</math> phase space, employing Liouville's theorem. We obtain a similar generalization without using Liouville’s theorem. With the mentioned generalized 11-theorem and information theory we prove the above mentioned Tolman's hypothesis as to a priori-probabilities. If we divide the accesible <math>\Gamma</math> space, of the representative point P(t) of a close and isolated system, in <math>\mathcal{N}</math> cells of equal measure, at equilibrium, we prove that each cell has the probability <math>1/\mathcal{N}</math> of being occupied by the representative point P(t) of the-system; and that <math>p_{ij}=p_{ji}\,\!</math> i.e. the probability of passing from cell i to cell j , at equilibrium, equals the reverse transit. Our conclusion ensures that the time average of a function of the coordinates q and conjugate momenta p of the particles of a system, at equilibrium with constant energy E, is equivalent to the average over the accessible. phase, space.
- Año
- 1987
- Número
- CBPF-NF-053/87
- Vínculo con tesis
- No
- Vínculo con beca
- No