Geometry in statistical mechanics
Geometry is often crucial in physical phenomena because dimensionality and topological defects determine the properties of phase transitions. One classical example is the Thomson problem, which addresses the question of how to configure charges on a sphere with minimal energy. The orientational order of liquid crystal molecules in a curved sheet provides an example of the interaction between defects and curvature. This system can be formulated as an XY spin model where the curvature term enters the Hamiltonian in a very similar way to that of the magnetic vector potential in the theory of type-II superconductors.
Collective behavior in nonequilibrium
The equilibrium properties of a statistical-physical system are often characterized by a few macroscopic degrees of freedom. As the system gets out of equilibrium, however, a huge, mostly unmanageable number of degrees of freedom come into play. The possibility of an exact formalism incorporating nonequilibrium processes has recently emerged with the discovery of the so-called fluctuation theorems and the formulation of steady-state thermodynamics.
Evolutionary game theory
A number of interactions including ecological, social, and economical ones are more complicated than those of spins in that they are usually asymmetric and history dependent. For this reason, most of these systems beyond the simple physics model cannot be described by the simple Hamiltonian approach. Even if no analytical solution is available, we can expect the system to evolve by successive local adaptations, searching for an optimal point on the fitness landscape. However, the equilibrium reached by local dynamics may not be optimal in a global sense.
