The world according to Renyi: thermodynamics of fractal systems

We discuss a basic thermodynamic properties of systems with multifractal structure. This is possible by extending the notion of Gibbs-Shannon's entropy into more general framework - Renyi's information entropy. We show a connection of Renyi's parameter q with the multifractal singularity spectrum f(\alpha) and clarify a relationship with the Tsallis-Havrda-Charvat entropy. Finally, we generalize Hagedorn's statistical theory and apply it to high-energy particle collisions.Comment: 8 LaTeX pages, Proc. of the International Symposium on Non-equilibrium and Nonlinear Dynamics in Nuclear and Other Finite System

Bateman's dual system revisited: I. Quantization, geometric phase and relation with the ground-state energy of the linear harmonic oscillator

By using the Feynman-Hibbs prescription for the evolution amplitude, we quantize the system of a damped harmonic oscillator coupled to its time-reversed image, known as Bateman's dual system. The time-dependent quantum states of such a system are constructed and discussed entirely in the framework of the classical theory. The corresponding geometric (Pancharatnam) phase is calculated and found to be directly related to the ground-state energy of the 1D linear harmonic oscillator to which the 2D system reduces under appropriate constraint.Comment: 24 pages, RevTeX, 4 figures. Fully revised versio

Special Relativity induced by Granular Space

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.Comment: 6 pages, RevTeX4. Some revisions in DSR section and in Conclusions. References adde