| Statistical mechanics provides the framework for understanding systems with macroscopically large numbers of particles (degrees of freedom). The key is a statistical, probabilistic description of macroscopic systems in terms of only a few parameters. Of particular interest are situations where an infinitesimal change in one of these parameters (e.g., temperature, magnetic field, number of defects, etc) results in a phase transition between different states of matter (phases) – liquid and gaseous, para‐ and ferromagnetic, conducting and superconducting, metallic and insulating, etc.
In this course we will focus at the continuous phase transitions where ‘most visible’ properties of matter – density, magnetisation, conductivity, etc, exhibit no abrupt changes. On the contrary, their derivatives (compressibility, susceptibility, heat capacity, etc) are discontinuous or divergent at a certain critical point. The characteristic features of critical phenomena in the vicinity of the critical point are their scale invariance and universality. The scale invariance (related to fractal geometry of the critical state) allows us to characterise critical phenomena by a small number of mutually related critical exponents. Universality means the existence of wide classes of very different physical systems exhibiting identical critical behaviour and, in particular, having the same critical exponents.
Critical phenomena are ubiquitous in nature, from transitions between different phases of matter to self‐organised criticality to biological evolution to financial markets to – most probably – the Big Bang that was at the origin of everything. We will focus mostly on those firmly based on the Gibbs distribution (which is the underlying principle of statistical physics) but also consider percolating systems – a simpler class of scale-invariant system exhibiting critical behaviour. | 