Statistical Mechanics and Quantum Fields on Fractals (original) (raw)
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Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals
Journal of Physics A: Mathematical and Theoretical, 2012
We investigate the spectral zeta function of a self-similar Sturm-Liouville operator associated with a fractal self-similar measure on the half-line and C. Sabot's work connecting the spectrum of this operator with the iteration of a rational map of several complex variables. We obtain a factorization of the spectral zeta function expressed in terms of the zeta function associated with the dynamics of the corresponding renormalization map, viewed as a rational function on the complex projective plane, P 2 (C). The result generalizes to several complex variables and to the case of fractal Sturm-Liouville operators a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended to the Sierpinski gasket and some other decimable fractals by A. Teplyaev. As a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in P 2 (C), thereby extending to several variables an analogous result by A. Teplyaev. The above fractal Hamiltonians and their spectra are relevant to the study of diffusions on fractals and to aspects of condensed matters physics, including to the key notion of density of states.
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Fractal Behaviour of Quantum Paths in Statistical Physics
Condensed Matter Physics, 2000
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.
Physica D: Nonlinear Phenomena, 1999
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Weyl's problem for the spectral distribution of Laplacians on P.C.F. self-similar fractals
Communications in Mathematical Physics, 1993
We establish an analogue of WeyΓs classical theorem for the asymptotics of eigenvalues of Laplacians on a finitely ramified (i.e., p.c.f.) self-similar fractal K, such as, for example, the Sierpinski gasket. We consider both Dirichlet and Neumann boundary conditions, as well as Laplacians associated with Bernoulli-type ("multifractal") measures on K. From a physical point of view, we study the density of states for diffusions or for wave propagation in fractal media. More precisely, let Q(X) be the number of eigenvalues less than x. Then we show that ρ(x) is of the order of x ds / 2 as x-» +00, where the "spectral exponent" d s is computed in terms of the geometric as well as analytic structures of K. Further, we give an effective condition that guarantees the existence of the limit of x~d s / 2 ρ(x) as x->-hoc; this condition is, in some sense, "generic". In addition, we define in terms of the above "spectral exponents" and calculate explicitly the "spectral dimension" of K.
Quantum field-theory models on fractal spacetime
Communications in Mathematical Physics, 1989
The present work explores the possibility of giving a nonperturbative definition of the quantum field-theory models in non-integer dimensions, which have been previously studied by Wilson and others using analytic continuation of dimension in perturbation integrals. The method employed here is to base the models on fractal point-sets of non-integer Hausdorff-Besicovitch dimension. Two types of scalar-field models are considered: the one has its propagator (= covariance operator kernel) given by a proper-time or heat-kernel representation and the other has a hierarchical propagator. The fractal lattice version of the proper-time propagator is shown to be reflection-positive. The hierarchical models are introduced and their properties discussed on an informal basis.