This paper is a review of results which have been recently obtained by applying
mathematical concepts drawn, in particular, from differential geometry and topology, to the …

Boltzmann''s formula S= In [W (E)] defines the microcanonical ensemble. The usual
textbooks on statistical mechanics start with the microensemble but rather quickly switch to …

Describing and, to a certain extent, understanding the concept of complexity has been
investigated in a variety of research fields. Various ad hoc formal definitions and …

Phase transitions are among the most impressive phenomena occurring in nature. They are
an example of emergent behavior, ie, of collective properties having no direct counterpart in …

We explore the topology of configuration spaces of hard disks experimentally and show that
several changes in the topology can already be observed with a small number of particles …

In this paper, we review modern nonlinear dynamical methods used in neuroscience and
complex data analysis. We start with the general description of nonlinear dynamics, its …

The stationary points of the potential energy function V are studied for the ϕ 4 model on a
two-dimensional square lattice with nearest-neighbor interactions. On the basis of analytical …

The relation between thermodynamic phase transitions in classical systems and topological
changes in their configuration space is discussed for two physical models and contains the …

In this second paper, we prove a necessity theorem about the topological origin of phase
transitions. We consider physical systems described by smooth microscopic interaction …

In this first paper, we demonstrate a theorem that establishes a first step toward proving a
necessary topological condition for the occurrence of first-or second-order phase transitions …