We propose a general non-linear analytical framework to study the evolution of the opinion state of a population of moving individuals. This novel scheme allows us to study a broad range of social phenomena, like, for example, the influence of agent interaction dynamics in the opinion formation or the inclusion of different individual's idiosyncrasies. We consider societies composed by agents who adopt one of the n possible opinions or internal states. The opinion state may only be modified while the agent keeps contact with another one. In general, this framework could be solved numerically, and, for some special perturbative cases, it is possible to find analytical steady states. In order to check our scheme for different social conventions, we implement computational simulations of an ensemble of self-propelled agents, finding a good agreement between theory and simulation results. We found, for slow society kinetics in all the cases studied, that there exist a shift of the opinion populations towards the moderate opinion states. This suggest that active speed can be understood as a parameter measuring the social temperature of the community.

The one-dimensional motion of a chain of N beads is studied to determine its diffusion coefficient. We found an exact analytical expression for all through two methods by resorting to the Einstein relation. Results are tested with the help of Monte Carlo simulations.

An extension of a recently introduced one-dimensional model, the necklace model, is used to study the reptation of a chain of N particles in a two-dimensional square lattice. The mobilities of end and middle particles of a chain are governed by three free parameters. This new model mimics the behavior of a long linear and flexible polymer in a gel. Noninteracting and self-avoiding chains are considered. For both cases, analytical approximations for the diffusion coefficient of the center of mass of the chain, for all values of N , are proposed. The validity of these approximations for different values of the free parameters is verified by means of Monte Carlo simulations. Extensions to higher dimensions are also discussed.