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Fractal behaviour of quantum paths in statistical physics

J. P. Badiali

Published Jan 1, 2000• 0 citations
Publisher

Abstract

The path integral formalism is used to describe the statistical properties of \nan ideal gas of spinless particles. It is shown that the quantum paths exhibit \nthe same properties in non-relativistic and relativistic domains provided the \ncreation of new particles is avoided. Some quantities associated with the \npaths are introduced, they have a simple meaning if the quantity βh , where \nβ is the reverse of the temperature, is considered as an ordinary time. The \nrelation between the velocity on the path and the momentum is not the \nusual one, an extra term appears showing that the thermostat can not fix \nthe average value of this velocity although all the thermodynamic quantities \nhave their traditional values. The paths describe fluctuating trajectories on \nwhich the particles do not follow the equation of motion. For time intervals \nmuch shorter than βh we recover the properties of the Brownian motion. \nThe trajectories are located in space in a volume restricted by the Compton \nwavelength for the short distances and the thermal de Broglie wavelength \nfor the largest ones. It is shown that the time-energy uncertainty is verified \non the quantum paths. This suggests that the density matrix obtained by \nquantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual \nroute. Strong arguments are given showing that βh can be considered as \nan ordinary time and not as a formal quantity having the same dimension \nas 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 \nsuggested that the ratio h/kB may play a decisive role in the foundation of \na covariant statistical physics.

DOI
10.5488/cmp.3.3.545

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The path integral formalism is used to describe the statistical properties of \nan ideal gas of spinless particles. It is shown that the quantum paths exhibit \nthe same properties in non-relativistic and relativistic domains provided the \ncreation of new particles is avoided. Some quantities associated with the \npaths are introduced, they have a simple meaning if the quantity βh , where \nβ is the reverse of the temperature, is considered as an ordinary time. The \nrelation between the velocity on the path and the momentum is not the \nusual one, an extra term appears showing that the thermostat can not fix \nthe average value of this velocity although all the thermodynamic quantities \nhave their traditional values. The paths describe fluctuating trajectories on \nwhich the particles do not follow the equation of motion. For time intervals \nmuch shorter than βh we recover the properties of the Brownian motion. \nThe trajectories are located in space in a volume restricted by the Compton \nwavelength for the short distances and the thermal de Broglie wavelength \nfor the largest ones. It is shown that the time-energy uncertainty is verified \non the quantum paths. This suggests that the density matrix obtained by \nquantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual \nroute. Strong arguments are given showing that βh can be considered as \nan ordinary time and not as a formal quantity having the same dimension \nas 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 \nsuggested that the ratio h/kB may play a decisive role in the foundation of \na covariant statistical physics.
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 β , where β 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 β 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 β 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 /k B may play a decisive role in the foundation of a covariant statistical physics.
The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest 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.
The analogy between quantum 'paths' and random walks of a Brownian particle is extended to the relativistic regime. This yields a coherent picture of the fine structure of quantum paths: these can be viewed as analogies of the trajectory of a random walker. The fractal dimension of quantum paths is df=2 at scales large compared with the Compton wavelength Lambda but small compared with the de Broglie wavelength lambda and becomes df=1 at scales smaller than Lambda . The transition between the two regimes is exhibited explicitly and is analogous to the cross-over from ballistic to diffusive motion of Brownian particles. The analogue of the collisions in Brownian motion is electron-positron pair creations and annihilations, and the unit thermodynamic free energy kT corresponds to the rest mass mc2 of the particle.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.
A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.
In Feynman's space-time approach to nonrelativistic quantum mechanics, the wave function may be considered to be the sum of path integrals over Brownian-motion trajectories. This formalism is entirely equivalent to the Schr\"odinger-equation treatment with its intrinsic indeterminacy. Pursuing the analogy with Brownian motion further, one may introduce a hidden variable corresponding to particle velocity in order to represent the wave function as a sum of path integrals taken over phase-space trajectories. By means of a hidden parameter corresponding to the time scale of fictitious interactions of the particle with the vacuum, one can define generalized propagators which over very short times produce wave functions which are localizable in phase space and hence are deterministic. For long times the conventional Feynman formalism is recovered. This short-time causality does not violate the von Neumann injunction against hidden variables since the new velocity variable turns out to be non-Hermitian over the long time scales normally considered. In contrast to the models proposed by Bohm and others, the present Brownian model is linear and preserves the usual statistical interpretation of the wave function for sufficiently long times.