Condensed Matter Physics, 2000, vol. 3, No. 3(23), p. 545-558, English
DOI:10.5488/CMP.3.3.545

Title: FRACTAL BEHAVIOUR OF QUANTUM PATHS IN STATISTICAL PHYSICS
Author(s): J.P.Badiali (Structure et Reactivite des Systemes Interfaciaux, Universite Pierre et Marie Curie, 4 Place Jussieu, 75230 Paris Cedex 05, France)

The path integral formalism is used to describe the statistical properties of an ideal gas of spinless particles. It is shown that the quantum paths exhibit the same properties in non-relativistic and relativistic domains provided the creation of new particles is avoided. Some quantities associated with the paths are introduced, they have a simple meaning if the quantity $\beta \hbar$, where $\beta$ is the reverse of the temperature, is considered as an ordinary time. The relation between the velocity on the path and the momentum is not the usual one, an extra term appears showing that the thermostat can not fix the average value of this velocity although all the thermodynamic quantities have their traditional values. The paths describe fluctuating trajectories on which the particles do not follow the equation of motion. For time intervals much shorter than $\beta \hbar$ we recover the properties of the Brownian motion. The trajectories are located in space in a volume restricted by the Compton wavelength for the short distances and the thermal de Broglie wavelength for the largest ones. It is shown that the time-energy uncertainty is verified on the quantum paths. This suggests that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route. Strong arguments are given showing that $\beta \hbar$ can be considered as an ordinary time and not as a formal quantity having the same dimension as time. This paper shows that for a time scale of 10 femtoseconds a totally new physics can be expected at room temperature. In addition it is suggested that the ratio $\hbar/k_{\rm B}$ may play a decisive role in the foundation of a covariant statistical physics.

Comments: Figs. 0, Refs. 19, Tabs. 0.

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