Particle Swarms in Statistical Physics

Abstract

Statistical physics is the area of physics which studies the properties of systems composed of many microscopic particles (like atoms and molecules). When combined, the interactions between these particles produce the macroscopic features of the systems. The systems are usually characterized by a very large number of variables and the limited possibilities for observing the properties of the components of the system. For these reasons, solving problems arisen in Statistical Physics with analytical approaches is usually ineffective and sometimes impossible. However, statistical approaches (such as Monte Carlo simulation) can provide acceptable approximations for solutions of these problems. Moreover, recent studies showed that nature inspired metaheuristics (like Genetic Algorithms, Evolutionary Strategies, Particle Swarm Optimization, etc) can also be used to simulate, analyse, and optimize such systems, providing fast and accurate results. Apart from physical implications, problems from Statistical Physics are also important in fields like biology, chemistry, mathematics, communications, economy, sociology, etc. We will present two important problems from Statistical Physics and discuss how one can use Particle Swarm Optimization (PSO) to tackle them. First, we will discuss how the realvalued version of PSO can be used to mini mize the energy of a system composed of repulsive point charges confined on a sphere. This is known as the Thomson problem and it is included in Stephen Smale's famous list of 18 unsolved mathematical problems to be solved in the 21st century. This problem also arises in biology, chemistry, communications, economy, etc. Latter on, we will discuss how the binary version of PSO can be used to search ground states of Ising spin glasses. Spin glasses are materials that simultaneously present ferromagnetic and anti-ferromagnetic interactions among their atoms. A ground state of a spin glass is a configuration of the system in which this has the lowest energy possible. Besides its importance for Statistical Physics, this problem has applications in neural network theory, computer science, biology, etc.