In this thesis, Boltzmann's H-theorem is studied and applied to prove a general convergence to equilibrium for the adiabatic piston paradox, governed by a specially-derived kinetic Fokker-Planck equation. We review general results in kinetic theory on the Boltzmann collision operator, and rates of convergence to equilibrium. Furthermore, we turn our attention to simulations of the piston paradox, and apply the algorithm of Sigureirsson et al. for particle-particle collision dynamics to the piston setting, determining empirically the optimal parameters.

In this thesis, wealth distribution in a closed economic system is examined by studying the simple inclusion process (SIP). The simple inclusion process is a model coming from statistical physics that models the jumping of particles in a graph. In the model particles have attraction among each other and each site also has a characteristic attraction parameter, denoted by the variable α. We regard the simple inclusion process as an agent based model for the economy for which sites represent agents and the particles represent wealth transferring from agent to agent. For the model, invariant measures are found that represent possible long term distributions of wealth in the system. We extend the model by looking at α as a parameter drawn from its own underlying probability distribution, ψ(α). In the view of wealth distribution, α can be seen as the wealth attraction an agent has. We set out to look for conditions on the distribution of α for which we obtain asymptotic power law behaviour of the resulting wealth distribution. It is shown that if a higher-order moment of ψ(α) diverges, the resulting wealth distribution will have a weak asymptotic power law lower bound. By setting stricter conditions on the distribution of α, we obtain that the wealth distribution will be asymptotically equal to a power law.

The Leslie model considers age-specific birth and survival rates to describe how a population size and age distribution changes over time. This thesis investigates the long-term dynamics of the Leslie model for population growth, utilizing mathematical theorems such as Perron-Frobenius, Doeblin’s theorem, and Branching processes.

The Perron-Frobenius theorem guarantees the existence of a dominant eigenvalue. This dominant eigenvalue and its corresponding eigenvector represent the long-term behaviour of a population; The dominant eigenvalue indicates the long-term population growth, and the corresponding eigenvector indicates the long-term age distribution. Furthermore, the Perron-Frobenius theorem implies that a population asymptotically reaches a stable age distribution that is independent of its initial age structure. Once this stable age distribution is reached, the population continues to grow exponentially, exhibiting Malthusian behaviour. Doeblin’s theorem, although not directly applicable to the complete Leslie model, provides valuable insights into the long-term behaviour of Markov chains. As Doeblin‘s theorem can not be applied to the complete Leslie model, the reproduction process of the Leslie model is formulated as a Branching process. Introducing the Leslie model as a Branching process allows for the consideration of demographic stochasticity. Simulations reveal that for larger populations, the Branching process closely mirrors the Leslie model, while disparities become more pronounced in smaller populations. These results illustrate the impact of probabilistic factors in population dynamics, as well as the strength of the Leslie model for larger populations.