We study the thermal effects on the frictional properties of atomically thin sheets. We simulate a simple model based on the Prandtl-Tomlinson model that reproduces the layer dependence of friction and strengthening effects seen in atomic force microscope experiments. We investigate sliding at a constant speed as well as reversing direction. We also investigate contact aging: the changes that occur to the contact when the sliding stops completely. We compare the numerical results to analytical calculations based on Kramers rates. We find that there is a slower than exponential contact aging that weakens the contact and that we expect will be observable in experiments. We discuss the implications for sliding as well as aging experiments.

This doctoral thesis investigates three different topics: How friction evolves in atomically thin layered materials (2D materials); How social dynamics can be used to model grand scale common-pool resource games; Benchmarking of various image reconstruction algorithms in atomic force microscopy experiments. While these topics are diverse, they share being complex out-of-equilibrium systems. Furthermore, our approach to these topics will be the same: using simple models to obtain qualitative information about a system's dynamics. In the case of atomically thin layered materials, we will be expanding on the influential Prandtl-Tomlinson model and obtain an improved model constituting a substantial improvement in the theoretical description of friction in these systems. In the context of social dynamics, we will introduce a novel model representing a new approach to consensus rates on social networks in relation to society spanning coordination problems. For the image reconstruction project, our ambition is to investigate a new method for recreating free-energy surfaces based on AFM experiment, however, for this project only preliminary results are included.

Friction is a ubiquitous phenomenon that greatly affects our everyday lives and is responsible for large amounts of energy loss in industrialised societies. Layered materials such as graphene have interesting frictional properties and are often used as (additives to) lubricants to reduce friction and protect against wear. Experimental Atomic Force Microscopy studies and detailed simulations have shown a number of intriguing effects such as frictional strengthening and dependence of friction on the number of layers covering a surface. Here, we propose a simple, fundamental, model for friction on thin sheets. We use our model to explain a variety of seemingly contradictory experimental as well as numerical results. This model can serve as a basis for understanding friction on thin sheets, and opens up new possibilities for ultimately controlling their friction and wear protection.

The scope of this licentiate thesis is twofold: 1. Investigate the frictional properties of systems with layered materials; 2. Employing image recognition algorithms to find the substrates in AFM experiments. As of today, there is a clear dichotomy separating these projects, however, it is the long term goal that they should coalesce in a not too distance future. The friction in layered materials projects is already finished, in this project we expanded the venerated Prandtl-Tomlison model to incorporate atomically thin layered materials such as graphene. This project has proved successful beyond our expectations, and a score of experimental results and conflicts in the field can be explained and resolved using our model. The image reconstruction project however, is still on a basic level. So far we have compared a standard model – Histogram Analysis Method– for image reconstruction on the nano-level with a popular image reconstruction algorithm –Lucy Richardson Deconvolution – from astronomy and shown that the latter is more suitable for these kind of systems. However, this project is far from finished, and the results in this part should be regarded as both partial and preliminary.

Stabilizer states are eigenvectors of maximal commuting sets of operators in a finite Heisenberg group. States that are far from being stabilizer states include magic states in quantum computation, MUB-balanced states, and SIC vectors. In prime dimensions the latter two fall under the umbrella of minimum uncertainty states (MUSs) in the sense of Wootters and Sussman. We study the correlation between two ways in which the notion of ` far from being a stabilizer state' can be quantified. Two theorems valid for all prime dimensions are given, as well as detailed results for low dimensions. In dimension 7 we identify the MUB-balanced states as being antipodal to the SIC vectors within the set of MUS, in a sense that we make definite. In dimension 4 we show that the states that come closest to being MUS with respect to all of the six stabilizer MUBs are the fiducial vectors for Alltop MUBs.