Physical Consequences of Complex Dimensions of Fractals

Abstract

It has recently been realized that fractals may be characterized by complex dimensions, arising from complex poles of the corresponding zeta function, and we show here that these lead to oscillatory behavior in various physical quantities. We identify the physical origin of these complex poles as the exponentially large degeneracy of the iterated eigenvalues of the Laplacian, and discuss applications in quantum mesoscopic systems such as oscillations in the fluctuation ${\Sigma }^2(E)$ of the number of levels, as a correction to results obtained in Random Matrix Theory. We present explicit expressions for these oscillations for families of diamond fractals, also studied as hierarchical lattices.

4 pages, 3 figures; v2: references added, as published in Europhysics Letters

References in corpus (5)

• Spectral Dimension of the Universe
• Spectral zeta functions of fractals and the complex dynamics of polynomials
• Semiclassical theory of the Ehrenfest-time dependence of quantum transport
• The Zeta Function of the Laplacian on Certain Fractals
• Electronic shot noise in fractal conductors

Cited by in corpus (33)

• Quantum Einstein Gravity
• Probing a dissipative phase transition via dynamical optical hysteresis
• Geometry and field theory in multi-fractional spacetime
• Fractal space-times under the microscope: A Renormalization Group view on Monte Carlo data
• Thermodynamics of photons on fractals
• Multifractional theories: an unconventional review
• Multi-scale gravity and cosmology
• Asymptotic Safety, Fractals, and Cosmology
• Laplace Operators on Fractals and Related Functional Equations
• Derivations and Dirichlet forms on fractals
• Dirac and magnetic Schrödinger operators on fractals
• Spatial Log Periodic Oscillations of First-Passage Observables in Fractals
• Explicit formulas for heat kernels on diamond fractals
• Perfect quantum state transfer on diamond fractal graphs
• Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
• Spectra of Perfect State Transfer Hamiltonians on Fractal-Like Graphs
• Symmetries and propagator in multifractional scalar field theory
• Energy and Laplacian on Hanoi-type fractal quantum graphs
• Log-periodic critical amplitudes: a perturbative approach
• Power dissipation in fractal AC circuits
• Fractal snowflake domain diffusion with boundary and interior drifts
• Spontaneous Emission from a Fractal Vacuum
• Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials
• Dynamics of Disordered Mechanical Systems with Large Connectivity, Free Probability Theory, and Quasi-Hermitian Random Matrices
• Observation of Log-Periodic Oscillations in the Quantum Dynamics of Electrons on the One-Dimensional Fibonacci Quasicrystal
• Entanglement entropy on finitely ramified graphs
• Finite entanglement entropy and spectral dimension in quantum gravity
• Regularized Laplacian determinants of self-similar fractals
• Quantum Vacuum Energy of Self-Similar Configurations
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs
• Evaluation of Spectral Zeta-Functions with the Renormalization Group
• First-passage times to a fractal boundary: local persistence exponent and its log-periodic oscillations
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets