The Abelian sandpile model was first introduced by Bak, Tang and Wiesenfeld in 1987. Since then, a lot of researchers have studied this model and similar models, all related by the concept of self-organized criticality. In this thesis, we study a variant on the classical model where dissipative and anti-dissipative vertices are incorporated in the model. These have an influence on the critical behaviour of the model. We first introduce a definition of criticality in this model and investigate which levels of dissipation are required to guarantee non-critical behaviour. In studying this variant, we encounter random walks and Green's functions.

In this thesis, we study the asymptotic behaviour and the dynamics of a one-dimensional active particle model with excluded volume interactions. The model is a version of run-and-tumble motion, where a particle performs both symmetric random walks and active transport. The direction and the speed of the transport are governed by an internal state process. We show that this motion converges to Brownian motion upon diffusive scaling and determine the limiting diffusion coefficient. The internal state converges to a stationary distribution, by which it manifests itself in the diffusion coefficient. Furthermore, we prove that the active particle satisfies the large deviation principle. This allows us to derive an implicit expression for the rate by which the probability of rare events tends to zero. Numerically, we investigate the influence of excluded volume interactions on the diffusion coefficient and the average velocity. We find that the velocity converges exponentially to its theoretical value as the number of particles allowed per position increases. In addition, this exclusion number strongly influences the manner in which the velocity decreases for high particle densities. Predictions for the velocity as a function of particle density based on the model are compared to experimental data of the molecular motor kinesin-II. We find that the model is not adequate for approximating the velocity of molecular motors in crowded environments and extensions in the form of Langmuir kinetics are suggested.

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.

In this thesis, we study scaling and detailed properties of a class of conservative interacting particle systems. In particular, in the first part we derive the hydrodynamic equation for the symmetric exclusion process in presence of dynamic random environment. The second part of the thesis focuses on a detailed property of conservative particle systems and, more generally, of Markov processes, called duality. There, we study and characterize duality and self-duality relations for so-called symmetric interacting particle systems.@en

In the current thesis we argue that quantum mechanics is best understood as a classical theory in which each measurement hides a set of its states. Our aim will however not be to replace quantum mechanical theory by a classical theory of hidden states or to avoid any of its possibly unfavourable metaphysical implications by replacing the current paradigm by a different formulation of the theory. The current thesis on hidden variables is rather guided by the question of how quantum mechanical theory differs from classical mechanics than any attempt to overcome this difference. In contrast to the standard approach to hidden variables, we argue that quantum mechanics in its current formulation itself is best understood as a hidden states theory.

This thesis examines the approximation capabilities of path signatures within rough path spaces, focusing on both local and global universality. To this end, we provide a self-contained introduction to Rough Path theory, highlighting the interplay between additive and multiplicative functionals. This leads to the renowned Lyons' Extension theorem and the definition of rough path spaces. We also re-examine the concept of universality from a kernel-theoretic perspective, culminating in the classical universal approximation result for signatures over a compact subset of paths. To broaden the scope beyond compact domains, we introduce the framework of weighted spaces and elaborate on the notion of global universality. Specifically, we formally define globally universal kernels and prove sufficient conditions for their existence. The associated reproducing kernel Hilbert space is shown to approximate a wide range of functions over the entire domain, which may be non-(locally) compact. In particular, we apply these theoretical tools to rough path spaces, thereby cementing the global universality of signatures.

This thesis examines the approximation capabilities of path signatures within rough path spaces, focusing on both local and global universality. To this end, we provide a self-contained introduction to Rough Path theory, highlighting the interplay between additive and multiplicative functionals. This leads to the renowned Lyons' Extension theorem and the definition of rough path spaces. We also re-examine the concept of universality from a kernel-theoretic perspective, culminating in the classical universal approximation result for signatures over a compact subset of paths. To broaden the scope beyond compact domains, we introduce the framework of weighted spaces and elaborate on the notion of global universality. Specifically, we formally define globally universal kernels and prove sufficient conditions for their existence. The associated reproducing kernel Hilbert space is shown to approximate a wide range of functions over the entire domain, which may be non-(locally) compact. In particular, we apply these theoretical tools to rough path spaces, thereby cementing the global universality of signatures.

In this thesis the CRISPR-Cas9 mechanism, a promising mechanism for geneediting, is considered. Closed form expressions are derived for the probability and time to cleave or unbind for the associated Cas9 protein. The mechanism can be modelled mathematically by a birth and death process, therefore the expressions could be derived using Markov chains and semigroups. The expressions are compared to simulations and interpreted using the model of hybridization kinetics. Finally the moment generating function of the stopping time is derived for two special Markov processes, i.e. a random walk and a Brownian motion with drift. This was done using martingales.

In this paper a two agent wealth distribution model for a closed economic system developed in [2] is presented and extended. We first extend the model by randomly distributing the propensity to save of the agents. We derive a closed form of the stationary relative wealth measure of an agent. We also see that if we take both the propensity to save and the redistribution measure to be uniformly distributed, then the stationary wealth distribution of agent 1 cannot be Beta distributed. Furthermore we conjecture that given a uniform redistribution measure and Beta distributed propensity to save, the resulting wealth distribution cannot be Beta distributed either. The absence of Beta distributions in the wealth distribution shows that there cannot be product stationary measures in these cases. We also extend the model by assuming zero propensity and that the stationary product measure of one agent is conditionally Gamma(α;β) distributed, where we condition on α be independently distributed as well. We find that the class of distributions for α defined by ψ(α) = α-k, k an integer always leads to the wealth distribution for agent 1 to be heavy-tailed. We also take steps in showing that there exists a distribution for α that solves for the wealth distribution of agent 1 to be Pareto Lomax distributed.

Quantum communication has been shown to be vastly superior to classical communication in many problems. However no general statements exist which tells us how much better quantum communication is to its classical counterpart. In this thesis it was studied the minimum amount of classical bits required to exactly simulate a quantum communication process. The quantum communication process specifically studied was a quantum prepare and measurement communication problem. It has been shown that the calculation of the amount of classical bits of communication required for simulation reduces to a minimization-maximization optimization problem. Several results have been presented for for solving this optimization problem and in addition a link was made between classical simulations of quantum communication and a recent debate on the reality of the quantum state.

In this thesis, the relation between the generator of the OrnsteinUhlenbeck process and the Hamiltonian of the quantum harmonic oscillator is used to derive a new understanding of the evolution of certain quantum states. More precisely, we transform the Hamiltonian with respect to the ground state and corresponding eigenvalue to find that it is equal to minus the generator of the Ornstein-Uhlenbeck process. Next, we use the knowledge of the evolution of distributions in the OrnsteinUhlenbeck process to obtain the time evolution of corresponding quantum states. Specifically, we derived that the evolution of normal distributions in the Ornstein-Uhlenbeck process remain normally distributed with varying mean and variance. Furthermore, the ground state of the harmonic oscillator is equal to the square root of the reversible distribution of the Ornstein-Uhlenbeck process. Combining these results gives us the evolution of quantum states with an almost Gaussian wave function. If we confine one degree of freedom in the end result, we obtain the coherent states of the quantum harmonic oscillator. These are Gaussian wave packets, which means that the probability density is Gaussian with constant variance and oscillating mean. Coherent states most closely resemble classical particles in the harmonic oscillator and minimise Heisenberg’s uncertainty principle.

In this thesis we will for a quantum Markov semi-group (Φt)t≥0 on a finite von Neumann algebra N with a trace τ , investigate the property of the semi-group being gradient-Sp for some p ∈ [1, ∞]. This property was introduced in [12] (see also [9]) and has been studied in [9, 10, 12] for quantum Markov semi-groups on compact quantum groups and on q-Gaussian algebras. Beyond these classes the property gradient-Sp has not been studied; in particular for groups and their operator algebras no (non-trivial) examples were known before this thesis. The main aim of this thesis is therefore to construct interesting examples of quantum Markov semi-groups that possess the gradient-Sp property. The reason why we are interested in constructing such semi-groups, is because they can be used to obtain properties like the Akemann-Ostrand property (AO+) and strong solidity for the underlying von Neumann algebra. Over the last decade, these properties have become a topic of interest and have been studied for several von Neumann algebras, see [3, 8, 9, 10, 12, 23, 32, 33, 37, 41]. In this thesis we shall focus on group von Neumann algebras (L(Γ), τ ) for certain discrete groups Γ that possess the Haagerup property. Namely, for such groups there exists a proper, conditionally negative definite function ψ on Γ. We can then define an unbounded operator ∆ψ on the GNS-Hilbert space L2(L(Γ), τ ) as ∆ψ(λv) = ψ(v)λv and consider the corresponding quantum Markov semi-group (e−t∆ )t≥0. For this semi-group we can investigate for what p it has the gradient-Sp property. In particular we will be considering group von Neumann algebras of Coxeter groups. Namely, a Coxeter group W possesses the Haagerup property by [4], and a proper conditionally negative function ψ on W is given by the minimal word length ψ(w) = |w| w.r.t some set of generators. We will ‘almost completely’ characterize for what types of Coxeter systems the semi-group corresponding to the word length is gradient-Sp. Moreover, in the cases that we get the gradient-S2 property, we obtain the Akemann-Ostand property (AO+) and strong solidity for L(W). Hereafter, we will also consider other quantum Markov semi-groups on L(W). We consider word lengths that arise by putting different weights on the generators, and consider the semi-groups associated to these proper, conditionally negative functions. From this we obtain (AO+) and strong solidity for L(W) for some other cases. Thereafter, we will generalize some of our results obtained for L(W) to the Hecke algebras Nq(W), which are q-deformations of L(W). For the case of group von Neumann algebras L(Γ) for general groups, we shall examine for semi-groups induced by a proper, conditionally negative function ψ, how the gradient-Sp property of the semi-group (Φt)t≥0 := (e−t∆ )t≥0 relates to the gradient-Sq property of the semi-group that is generated by the αth-root ∆α of the generator. Last, we will also show a method that allows us, for right-angled word hyperbolic Coxeter groups, to obtain (AO+) and strong solidity for L(W) without building a gradient-Sp quantum Markov semi-group, but by using a slightly different method.

The random graph is a mathematical model simulating common daily cases, such as ranking and social networks. Generally, the connection between different users in the network is established through preference, and this phenomenon leads to a power-law behaviour of the degree sequence of the random graph. Other than studying this feature, the thesis also focuses on the decay of degree sequence itself and the height of the random graph models. Researches are conducted on four kinds of random graph models, including two fully studied modes as benchmarks, the Preferential Attachment Model (PAM) and Random Recursive Tree (RRT), and another common model, the Bianconi-Barabási Model (BBM). Previous researches show that the degree sequence of PAM follows a power-law with coefficient -3, and the degree sequence of RRT has an exponential tail in contrast. In BBM, each node is assigned a fitness value from a certain distribution, once the node is generated, and researches show that the degree sequence of BBM follows a power-law with coefficient depending on the fitness distribution but close to the one of PAM. Then, the thesis aims to find a random graph model with fitness interpolating between RRT and PAM. Thus, a new Recursive Fitness Model (RFM) is developed by using a recursive formula to generate fitness values instead of the fixed distribution in BBM. By simulation we conjecture that the degree sequences of RFMs still follow a power-law with coefficient close to that of PAM. The simulation further shows an interesting phenomenon: a special linear height increment of a particular RFM, the Plus-1 model, appears, in contrast to other RFM models. Finally, the thesis invents a new Preferential Attachment Inverse Model (PAMinv) by using the expectation of degree sequence of PAM. Degree sequence of PAMinv shows a different behaviour from PAM and RRT, and it could be the tool to interpolate RRT from PAM.

Cells of the most common organisms like plants and animals are filled with polymeric networks that fulfil important functions of the cell. There is however no analytically solvable model that describes diffusion in such a cell. This thesis presents a model for diffusion in polymeric environments, and some predictions about the behaviour of the model are made and confirmed by simulations. Furthermore, the Fokker-Planck equation of this problem is studied, in order to solve the problem. Certain approximations are presented and solved, and it is investigated when the approximations are sound. Moreover, a method is described that can derive a solution to an equation with certain boundary conditions, from a solution to the same equation with different boundary conditions. This thesis also shows how this novel method can be applied to this model to find a non-approximated solution to the equation, where this was not possible without this method. Finally, it is described how a multitude of partial differential equations that are linked to each other via the boundary conditions can be solved, can be solved using the method described.

In this thesis we study criticality in the context of the dissipative Abelian sandpile model. The model is linked to a simple trapped random walk, giving a practical method to determine criticality for certain landscapes of dissipative sites. The main results concern the lifetime of the random walk, especially the divergence of its first moment for traps placed on spherical shells. For the one dimensional case the point of divergence is determined with reasonable precision. In higher dimensions the divergence is shown to be possible for an infite amount of shells. The connection between the sandpile model and a random walk is shown mathematically and further researched via simulation.

In this thesis, the diffusive limit of active particle motion in Rd is studied via a technique based on homogenisation. Thereafter, this study is extended to active particle motion on a Riemannian manifold. Furthermore, as an application of active particle motion, a connection is made with the Dirac equation. On the basis of this connection, a Monte Carlo method is developed to find the ground state of a Dirac equation with static potential. The core idea of this method is based on the Diffusion Monte Carlo method for the Schrödinger equation.

In this thesis, research was done in the area of interacting particle systems. Especially, the symmetric exclusion process with local perturbations was investigated. These perturbations, were in the form of sinks and sources, which add or take away particles at certain rates. Moreover, simulations were done for the asymmetric exclusion process. This process took place on a ring, with the addition of a source. For the symmetric exclusion process with sinks and sources, certain expressions were proven for the expected occupancy of a site. For the simulations, the main goal was to find out how the jumping rates and starting density, influenced the time to get to the fully occupied state, at different source rates. The first proof was for a source at an arbitrary site. From this expression, one could see that if the rates were recurrent, the system converged to the fully occupied state. If, however, the rates were transient, the system had a limiting density. Thereafter, it was shown that if a sink and source are placed in the same arbitrary site, the system always converged to a density, which under the Bernoulli measure, was not equal to the fully occupied state. The fact that the sink and source were in the same site, was an indispensable condition. Subsequently, the case of countably many sources was investigated. For which it was also shown that recurrent rates always yield a fully occupied state, as time tends to infinity. Whereas transient rates, once again, caused a limiting density. Moreover, the special case of a simple random walk in three dimensions or higher was investigated. If, for distances far away from the origin, the source rates could be bounded above by a certain function, then the system would not converge to the fully occupied state. Also, another proof showed that a recurrent set of source sites would always let the process converge to a fully occupied state. Lastly, similar conditions for one time dependent source at the origin were proven. Namely, it was shown for recurrent rates, that if the source dies out quick enough, as t tends to infinity, the system did not converge to the fully occupied state. For the simulations, it was found that the expected time to fill up was influenced by the starting density, both at small and large source rates, in a linear way. This conclusion could not be drawn for the rate difference: p-q. Due to the large error of the fit, a linear relation was not clearly seen.  Finally, some concluding remarks were made on both aspects of the thesis, along with some recommendations for future projects related to this topic.