Quantum technology is evolving faster than ever. Currently, all eyes are on the quantum computer, the promising computer that can solve problems which are unsolvable for regular computers. In order to understand this new technology, it is necessary to understand the qubit, the basic unit of quantum information. This can be done by means of the density matrix: a mathematical representation of the state of a system. The aim of this thesis is to find the density matrix of a single qubit in closed (isolated) and open quantum systems. In the case of a closed system, an alternating sequence of two processes with different Hamiltonians is considered, which both last a fixed amount of time after one another. These systems have been solved using a direct formula for a 2 × 2 matrix with distinct eigenvalues raised to the power n for any n ∈ N. In the case of an open system, dissipation is taken into account compared to a single process in a closed system. These open systems have been solved numerically using the Lindblad master equation and the spin-boson model to model the environment as a bath of bosons. However, the use of multiple approximations and assumptions questions the validity of the results for ’strong’ interactions. Suggestions for further research include investigating quantitatively when the Lindblad equation is valid to use and solve open quantum systems using different models for the environment.

Knowledge of particle deposition is important in clinical settings or when discussing environmental effects of aerosols on humans. Particle deposition in the human respiratory tract is determined by breathing patterns and lung morphology, as well as particle properties and deposition mechanisms. In this study we develop a 1-dimensional model that numerically solves the general dynamic aerosol equation in the human respiratory tract. The model can be used to calculate deposition fractions for a range of initial parameters. We use Weibel's morphometric model to describe the lung geometry. The model is validated by comparing it with previous numerical results, and running sensitivity tests to examine its consistency with parametric variations. The model proved to be computationally efficient, requiring just seconds to run a simulation. We use this to perform a number of parametric studies, most notably changing the tidal volume and the breathing rate. For both of these, an increase in either the volume or the rate decreased the deposition fraction across the spectrum of particle sizes, apart from at the tails of the distribution. We also examine the effect of particle density on the deposition fraction, which increases with an increasing density. The source code is published along with this thesis, allowing anyone to perform arbitrary parametric studies of their own.

Level-set percolation on the Discrete Gaussian Free Field (DGFF) turned out to be a hot topic within mathematical physics over the last couple of years. In particular, the DGFF on Z^d , with homogeneously weighted nearest-neighbour interactions, i.e. all conductances equal to 1, has been studied in detail. These models can be simulated with great efficiency. In this research, we abandon the homogeneity requirement and look at three-dimensional DGFFs with arbitrary conductances. Our goal is to find a quick and reliable method to simulate such DGFFs on a finite lattice. Since this is, in essence, a high-dimensional Gaussian sampling problem, we investigated this problem using the Conjugate Gradients (CG) linear solver as a Gaussian sampler. To see how it performed, we compared our implementation of the CG sampler with known methods for DGFFs in the unit conductance case. Finally, as a showcase of our implementation, we studied level-set percolation on a DGFF with a simple checkerboard conductance pattern. Our main conclusion is that the CG algorithm is very suitable for simulating Discrete Gaussian Free Fields. Since it does not make any assumptions on the conductances, it can be used to generate DGFFs with arbitrary conductances. However, there are still a number of issues with our implementation. The biggest one is concerning the stopping tolerance of the CG sampler. Once the tolerance is set smaller than some lattice size-dependent threshold, the percolative behaviour of the resulting sample changes drastically. We have not been able to explain this. Moreover, we would recommend making the implementation usable for parallel computing. We have been limited to relatively small lattice sizes during this project. Consequently, the use of certain finite-size scaling arguments when analysing level-set percolation might not always have been as justified. Finally, based on our study of the DGFF on a lattice with checkerboard conductances a and b (a < b), we conjectured that, in its percolative behaviour, this DGFF resembles a DGFF defined on a lattice with constant conductance c, where c is a weighted average of a and b. The weight of a is expected to be larger than the weight of b.

In this thesis the hydrodynamic behaviour of an idealised relatively short tidal estuary is studied using a cross-sectionally averaged model, i.e. a one dimensional model. The geometry of the estuary is simplified and only the hydrodynamic processes are considered. This model is used to gain insight in the behaviour of currents and water levels influenced by a movable barrier situated in the estuary. It is investigated whether shifting the motion of the barrier in time influences the sedimentation in the embayment.

Economic engineering models individual agents as inertia elements and can be viewed as a microeconomic theory based on analogs with classical mechanics. In this thesis the economic engineering concept of modeling individual agents is used to model economic systems consisting of many agents, e.g. an entire country. This is done using classical statistical physics. In statistical physics the microscopic movement of individual gas particles as described by classical mechanics and the macroscopic properties of gases as described by thermodynamics are linked. Using this insight, microeconomics and macroeconomics are linked within the economic engineering framework. The critical contribution of this thesis is finding the analog of Gibbs' interpretation of entropy, calling it the amount of diversification. This is done as follows. A thermodynamic system in equilibrium is seen as the analog of a Pareto optimal economy or macroeconomic equilibrium. The 2nd law of thermodynamics guarantees the existence of thermodynamic equilibrium and it follows that the entropy is maximized in equilibrium. Clausius interpreted the entropy as an "arrow of time" that pushes the system towards equilibrium. In this thesis Adam Smith's invisible hand is then viewed as an economic analog of Clausius' entropy. Gibbs gives a statistical interpretation to entropy. By calling the amount of diversification the analog of the Gibbs entropy, it follows that a Pareto optimal economy is fully diversified. The amount of diversification contains both the distribution of economic rent over agents and the portfolio diversification of agents for different goods. Based on the diversification analog, the thesis develops several further analogs. The analog of the partition function is called the opportunity function and gives the opportunities for extracting profits from an economic system by trading. From this the economic engineering analog of the free energy follows. The temperature and chemical potential are given economic engineering analogs as well, namely the level of welfare and the disposable income per capita respectively. The thesis is finalized with applications of the theory developed.

In this thesis the behaviour of a Bose Einstein condensate is explored that consists of bosons that annihilate. In order to do this a system where bose einstein condensation occurs is modeled as a Zero Range process which is a special case of a Markov process. First we made a single slow site Zero range process with uniform rates and see how this would be influenced by annihilating particles. For this system it is shown that, in the large system limit, the number of particles in the slow site is given by a differential equation. A series of realization of different sizes of this system are done to support that the fraction of the particles in the ground state converges to the differential equation. In order to relate this mathematical approach to physical Bose Einstein condensates it is established that the equilibrium distributions of multiple bosons in one system is the same as the detailed balance of a zero range process where the transition rates depend on the occupation of a site, with an indication function, that is equal to 0 when there are no particles at that site and some constant 𝑐 if there is one or more particles at that site. In statistical physics we need to take the energy of the state of the system into account. If the particles do not have interaction with each other this is the sum over all particles of the energy of the particle site. As a zero range model only has rates with one particle hop we can see that the difference in energy of the system is equal to the difference in energy of the sites of hop. In order to stay in detailed balance the rate for the specific hop must be exp(BΔ𝐸) times the opposite rate, where B = 1 / 𝑘_b 𝑇. Now we can go from any system with a set of spin orbitals and energies of those spin orbitals and design a zero range model where the spin orbitals correspond to sites of the zero range model with rates between those sites, such that the systems detailed balance distribution is the same distribution as the equilibrium of the physical system. In this zero range model an annihilation term is introduced. This can be any function of occupation for the site, but in this report a simple relation that decreases when Δ𝐸 increases is used. In order to see the effect on such systems the system is numerically approached for a 100 particles system in a 3 dimensional harmonic potential. This system is too complicated to get an differential equation that can be easily solved. Therefore it is approximated. This numerical approximation appears to have similar behaviour to the numerical approximation done in [3]. However the results are not identical. The advantage of the model in this thesis is that an annihilation therm can be implemented. This approximation of a 3 dimensional harmonic oscillator with annihilating particles is done. If annihilation is slow enough, the exited sites form a different equilibrium compared to a Bose Einstein condensate of non annihilating particles. This happens as long as there is a condensate in the ground state.

As long as a wave has a large enough wavelength, it should reflect off of smooth surfaces specularly: that is what the law of reflection states. This phenomenon is widely known, and used in for instance sonars. Electrons that reflect off of boundaries within conductors should abide this law as well, due to the particle-wave duality. However, it was recently discovered that for disordered graphene boundaries electrons scatter diffusively even when their wavelengths should be long enough not to. Graphene is a 2-dimensional semiconductor with linear dispersion at low energies, making it a widely researched topic as its applications in future electronic parts are promising. Its bilayer counterpart is made by connecting two layers on top of each other, which in contrast has a quadratic dispersion at low energies. In this work, the specularity of electron reflections within bilayer graphene were studied. This was done by constructing a tight-binding model, where random potentials on the outermost atoms represented imperfections of the lattice. Subsequently the variance of the scattering angle and the scattering phase was analysed, which can be derived using the scattering matrix of themodel. This was done numerically for both, and analytically for the phase as well. It was discovered that the variance of the scattering angle increases cubically with the Fermi wavelength of the electrons. This happens no matter what the distance between the Fermi energy and the disorder mean is, as long as the disorder strength is nonzero. The behaviour of these reflections is the exact opposite of what the law of reflection states. Furthermore, the variance of the scattering phase remains constant when the Fermi energy is close to the disorder mean, and changes inversely proportional to the boundary periodicity if the Fermi energy is further away. Consequently, the variance of the scattering phase versus the Fermi wavelength and the variance of the scattering phase versus the boundary periodicity do not share the same trend in bilayer graphene. More research is needed to verify this phenomenon, which could be done using the continuum description or a magnetic focusing device.

This work is a bachelor thesis featuring a collection of explorations of the Kuramoto model in the presence of noise in one and two dimensions. The model was explored analytically and numerically. It contains a study of literature and new analytical derivations. An attempt to link the fraction of oscillators to the order parameter resulted in experience through the process of deriving the result, but a plot of the first few terms of this expansion showed it not being close. The assumption on which the expansion was based however was shown to be valid in simulations. More successfully an upper bound on the probability of synchronization was found. This probability was found on a ring consisting of N oscillators, and verified of a ring of 3 oscillators. The results was a bound better than one that was already calculated, though the method of verification used was not ideal. An analysis of the stability of solutions on this 3-ring resulted a visual representation of entrainment and an amount of stable solutions possible, which was 1. This result was in accordance with the more general result following from the investigation of a system of coupled rings with arbitrary sizes of either ring. We then expanded this system to conclude that the stability of the entire system could be split into the stability of either ring and the possibility of having a solution for the linking phase difference. We elaborated on general practices when numerically analyzing SDEs and integrations methods. The Kuramoto model proved to be kind in the sense that the Euler-forward and Milstein scheme coincided. In a more physics-centered approach we studied the Ising and XY model. This prompted literature study on the Kosterlitz Thouless transition, through analysing the spin-wave Hamiltonian and the energy of free vortex of a pair of vortices. We also then explored conservation of the vortex-pair behaviour in a noise including Kuramoto model, which verified a duality in partition functions as described in ref. [1].

Chiral induced spin selectivity (CISS) is an unexplained phenomenon in physics, in which the interaction within chiral molecules brings about a selectivity in the spin of the electrons. The Breit interaction has been suggested as a possible explanation, but has not yet been researched in the context of CISS. In this thesis, first a derivation of the Breit equation is given, which in the second part is being used to construct a simulation of electron transport through a simple chiral molecule. The spin-polarization for the transmission including the Breit interaction was compared to the case excluding the Breit interaction, and the two were found to be substantially different. From that it followed that the influence of the Breit interaction on the transport of an electron through a chiral molecule is significant and it should therefore not be disregarded in calculations on CISS.

As quantum computers are developing, they are beginning to become useful for practical applications, for example in the field of quantum metrology. In this work, a variational quantum algorithm is used to find an optimal probe state for measuring parameters in a noisy environment. This is achieved by optimizing a cost on a quantum computer, based on the Fisher information of the parameters to be estimated. These parameters are then estimated using maximum likelihood estimators. In a simulation, a probe state was found that performed better than the best possible state for noiseless measurements, although this could not be reproduced on an actual quantum computer.

Non-photochemical laser-induced nucleation (NPLIN) is a process where a crystalline phase is formed out of solution by exposure to a laser beam. In NPLIN, the nucleation probability is strongly dependent on the beam intensity and weakly dependent on the wavelength. NPLIN offers a feasible alternative to energy-intensive industrial crystallisation methods. Although several mechanisms have been proposed, little is known about NPLIN at the molecular level. Some theories suggest that nucleation rates are enhanced through the heating of nanoparticles by absorption of electromagnetic energy. In this work, molecular dynamics simulations are performed on the clustering of ions in the vicinity of a heated nanoparticle in an aqueous supersaturated KCl solution. The spherical symmetry of a spherical nanoparticle in solution is exploited by modelling a laterally periodic water column of an initial length of 500 Angstrom enclosed between a planar iron(III) oxide nanoparticle surface and a graphene piston. A cavitation bubble is formed after nanoparticle heating, leading to an increase of clustered ions according to a bond order criterion. The clustering should correspond to nucleation in experimental systems because the clusters satisfy Delta G < 0 for locally valid NPT ensembles. The results corroborate concentration based NPLIN mechanisms as the clustering is visibly induced by local solute evaporation. Pressure based mechanisms appear ineffective because no effects of pressure waves are observed. Thermostatting the graphene sheet does not yield observable dissipation of the thermal energy generated through the nanoparticle heating and at the ion clustering sites and, obstructing completion of the cavitation cycle at a feasible simulation duration. It is suggested to repeat the simulations using a conically shaped system and a piston of higher transverse thermal conductivity.

We consider adaptive network models with discrete and continuous state sets obeying dynamical rules that enable application to swarming systems. The 2-state adaptive network contains a supercritical pitchfork bifurcation in the transition between ordered and disordered stationary solutions. We derive an adaptive network model that works on a continuous state set and apply it to swarming motion in both a mean field and a moment closure approximation. In numerical solutions of the mean field approximation the relation between the variance of the ordered stationary distributions and the system parameters is given by a square root function. Cauchy distributions form a good fit to these steady state distributions, although they are not the analytic stationary solutions. We show that in numerical solutions of the moment closure approximation a bistable region is formed, in which the initial condition determines if the system ends up in an ordered or a disordered state. Further research could focus on finding the exact details of the corresponding subcritical pitchfork and saddle-node bifurcations and comparing the derived models to real-life swarming systems.

This report has been made to gain more insight and knowledge of the Keplerian method for collision detection simulations by simplifying N-body systems to a list or particles with orbital parameters. Collision detection is an essential part in modeling the evolution of protoplanetary disk becoming planetary systems by the merging of planetesimal objects. Presumably Kep- lerian systems allow for a computationally efficient algorithm, having its computational time influenced by orbital parameters, unlike the efficient octree method which only scales with the number of particles. The aim of this research was to create a collision detection model for a simplified proto- planetary disk system using Kepler systems and analyse the results of the model to check the reliability of this simulation method, i.e. if the quality of the results are trustworthy, perform- ing consistently well, and are accurate. In addition, the method was tested on its computational efficiency by investigating the run time of separate parts of the algorithm. In the initialisation of the algorithm a particle list is created and sorted by increasing apoap- sis. Then by a sweep and prune filter the list is checked for possible collision pairs. For pairs that have a small enough minimal orbital intersection distance that they can collide, the colli- sion time is calculated and added to a time-ranked collision list. The algorithm keeps merging the pairs on top op the list, creating a new particle with new possible collision pairs and new collision times, and updates the time-ranked collision list after each merge. An empty collision list implies the end of the simulation. Various parameters as well as length of the lists are saved within the algorithm for subsequent examination. The results of the Keplerian model found showed similarities to other simulation studies, and include the perceived critical eccentricity of 0.02, preservation of shape of body radius distribution, and the formation of large bodies. Simulations showed three stages of collisions, characterised by collisions occurring homogeneously throughout the disk, large body colli- sions, a final stage of small body collisions. The performance of the algorithm could be sum- marised by the number of checks by the sweep and prune filter #checks ∝ N 2 and #checks ∝ ε for eccentricity ε < 0.1, the number of pairs in the collision pair list was found to be propor- tional to the number of starting particles N, the body radius s, the inverse of the inclination 1/I, and the eccentricity ε, so #pairs ∝ N2, s, 1/I, ε. For N < 1000 the computation time of the algorithm appeared to be proportional to N2, but for N < 10.000 proportional to to a higher power. For ε < 10−9 and N < 10.000 the computation time was found to be runtime ∝ s. The conclusion of this research is that the Keplerian model could be used to model N-body systems for collision detection because of its behavioural resemblance to observable astrophys- ical systems. The computation time of the Keplerian system scales with a higher order per N than for instance the three code algorithm, but has showed to be influenced by orbital parame- ters in which this method differentiates itself from other simulation methods, with a remarkable influence on the total computation time by the body radius s of the system. More research could be done for higher orders of N to better model reality as well as providing more insight on the proportionality of the computation time. Recommendations to the algorithm consist of adding planetary spin as a particle parameter, and methods to incorporate apsidal precession in a com- petent manner.

This thesis contains a rigorous derivation of the path integral formulation of the Isingmodel with multiple original proofs. Besides that, thesis also contains various resultsof simulations of the 2D square lattice Ising Model with nearest-neighbour interactionsusing the Swendsen-Wang algorithm. Using finite size scaling to find critical exponentγ, we reported a value of γ = 1.748 ± 0.004. After calculating the relaxation times τ forvarious thermodynamic variables, we found the value for the dynamic critical exponentz to be in the range of z = 0.180 ± 0.004 and z = 0.282 ± 0.005.

In this report, we have taken a look at three different types of trebuchets: the seesaw, the traction trebuchet or mangonel, and the counterweight trebuchet. We have created models to describe their motion by making use of Lagrangian mechanics and have compared them based on their maximum ranges. Moreover, we have also calculated a lower bound for the dimensions of themain beam needed to prevent it from breaking based on a static state. We have seen that the mangonel and trebuchet improved on the seesaw by increasing the range roughly sixfold which was expected. The maximum range of the trebuchet, however, slightly decreased compared to the mangonel. This was something we did not expect but think is due to the fact that the dimensions of the catapults were not optimised. Lastly, we found that the beam needed to be at least 18.7 cm to prevent it from breaking. However, for the real catapults, the beam would most likely need to be thicker to prevent it from breaking.

Report about how Matrix Product States can be used to analyze a sub set of problems in quantum mechanics. A exact derivation is given of the Ising Model in a Transverse field and with these solutions the TEBD algorithm using Matrix Product states is validated.