Extraordinary applications of statistical physics

Timetable

1. Ultra-short introduction to statistical physics

2. The Potts model and the Jones polynomials in knot theory

3. Canonical ensemble of random graphs

4. The Hopfield model and the mean-field of Ising perceptron

5. Grand-canonical catastrophe of number fluctuation in an ideal
    Bose-Einstein condensate  and the number theory

6. Mesoscopic models of polymers

Module aim (aims)

The main objective of the course is to familiarize students with unconventional applications of statistical physics and its methods used in selected problems of mathematics and biology.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

The lecture refers to fundamental methods used in statistical physics and probability theory, as well as the random graph theory, the knot theory and the number theory.

Syllabus

At the beginning we will discuss the basic techniques used in statistical physics in order to
analyse the properties of complex systems. Next, we will familiarize with the extendent
version of the Ising model, socalled the Potts model and show how to construct the
Jones polynomials which characterize the topology of knots. This procedure is based on
the graphs, the mathematical objects we will later describe in terms of statistical canonical
ensemble. In turn, the Ising model will lead us to the formulation of the Hopfield model of
the neural networks and the mean-field approach to the perceptron. On the other hand,
the grand-canonical ensemble will allow us to plunge into the meanders of the famous
Riemann hypothesis, which has not been fully proven to nowadays. Finally, we will
consider the mesoscopic models of polymers, like for instance DNA and proteins, to
to which also the knot theory is widely used.

Reading list

1. F. Y. Wu, Knot theory and statistical mechanics, RMP 64, 1099 (1992)

2. L. H. Kauffman, Knot theory and statistical mechanics, Int. J. of Mod. Phys. 11, 39 (1997)

3. H. Huang, Statistical mechanics of neural networks, Springer Singapore (2021)

4. A.-L. Barabasi, Network science - personal introduction (2014)

5. D. Schumayer, D. A. W. Hutchinson, Physics of the Riemann hypothesis, RMP 83, 307 (2011)

6. W. Sung, Statistical physics of biological matter, Springer Nature B. V. (2018)