Ivan Bonamassa

This is a list of informal seminars held within some courses attended at the University of Lecce during the B.Sc. and M.Sc. studies. They are the result of personal interests; some of them are associated with short term papers.

 

• The Physics of Phyllotaxis: How plants play with Math (2014)

We give an overview of the mathematical and physical aspects behind the

study of phyllotaxis (i.e. the study of pattern arising from the distribution

of plant organs). In particular, we review the fundamental theorem of phyllotaxis and describe some physical models which aim to explain the abundance of Fibonacci patterns in plants; the latter approach will lead us to introduce

some fascinating physical mechanism, like symmetry breaking mechanism

and quasi-bifurcation diagrams.

 

• Semiclassical potentials for prime numbers (2013)

In the last twenty years there have been many applications of Physical tools coming from Quantum Mechanics or Statistical Mechanics to Number Theory, mainly inspired by the Hilbert–Pólya conjecture. The problem has been mostly discussed in the context of the semi–classical approximation of Quantum Mechanics or, viceversa, in semiclassical quantization of Classical Mechanics. Here we review the findings made by G. Mussardo in 1998 with following the WKB formalism and apply the same procedure to the critical zeros of the Riemann zeta.

 

• Quantum signature of chaos in many body systems (2013)

This talk was presented for the Nuclear Physics (M.Sc.) class exam. It represents a part of a personal review given to the subject (still unfinished) entitled: "Quantum chaos, a travel through billiards, primes and nuclei, riding periodic orbits". In the latter, the main features of Quantum Chaos are lenghtly explained, starting from some relevant techniques of Random Matrix Theory, then moving towards the theory of periodic orbits where the WKB approximation and the semi-classical quantisation procedures of integrable, separately integrable and chaotic classical systems are described. These results allows one to arrive eventually at the Gutzwiller trace formula, which relates the spectral density of the quantized system to the primitive classical periodic orbits of its classical analogue. This remarkable formula is then applied to obtain some semi-classical expressions in RMT that have been recently applied to study the distribution of  the binding energies fluctuations in nuclei.

 

• Black Hole Thermodynamics (2012)

Thanks to the seminal papers of J. Bekenstein (1973) and S. Hawking (1974), black-holes seem to behave like black–bodies characterized by a temperature

and an entropy S=k A/4L^2 , where A represents the area of the event horizon,

k is the Boltzmann constant and L^2 is the Planck area. These results pose the problem to identify the microscopic degrees of freedom responsible for such an entropy, which is attainable only within a proper theory of quantum gravity. With this talk we review some classical and semi–classical aspects of the problem, from the general relativistic “principles” of black–hole thermodynamics, to the “semi–classical” Hawking calculations for the black–hole evaporation, ending with a brief description of some recent counting techniques of space–time microscopic degrees of freedom introduced in Loop Quantum Gravity.

 

• An introduction to Ergodic Number Theory (2011)

Recently, a lot of research has involved three different branches of Mathematics: Number Theory, Ergodic Theory and the Theory of Dynamical Systems. The existence of profound connections between the last two was already known in late XIX century, while the possibility for a connection between Ergodic Theory and Number Theory begun only in the second half of the XX century. Today, different ergodic concepts of dynamical systems fit naturally in number theoretical problems and are giving rise to a new area of Mathematics, baptized Ergodic Number Theory. The aim of the present talk is to give a brief introduction to the main results obtained so far in this direction, speculating about some possible connection with Physics.

 

• The Fermi-Pasta-Ulam problem (2011) 

In 1953 E. Fermi, J. Pasta and S. Ulam conducted some numerical calculations, aiming to study the long–time behavior of the state for a linear chain of N oscillators coupled with non–linear (cubic or quartic) nearest–neighbor interactions. Remarkably, after having excited the first normal mode with a certain amount of energy, they found that the system never thermalizes, but oscillates periodically among phase–space states arbitrarily near to the initial one, eventually returning exactly to the starting configuration. Later on, M. Kruskal and N. Zabusky explained this apparent paradox as the appearance of stable solitonic–excitations in the continuum limit of the FPU model. In this talk we introduce the main aspects of the problem, further showing the details of the continuum limit.