We present an implementation of the Wisdom-Holman integrator for simulating gravitational dynamics in planetary systems: systems with one dominant central mass and \(N\) orbiting bodies, such as the Solar System. The Wisdom-Holman integrator models the motion of non-central bodies as unperturbed Kepler orbits and integrates gravitational interactions between orbiting planets as weak perturbations. Two methods to advance a body along its orbits are investigated: one using coordinate transformations (Method A) and one based on the \(f\) and \(g\) functions (Method B). Method B is shown to be significantly faster than Method A, without a significant loss of accuracy, making it the preferred method for most simulations. Simulations of the Solar System using large time steps, including time steps exceeding the orbital period of some planets, are explored to determine whether the increase in computational speed justifies the loss in accuracy. Simulations with a fixed Sun and with a dynamic Sun are considered. Results indicate that for fixed Sun simulations, while accurate simulations require small time steps, larger steps still capture the qualitative behaviour of the system. The step size of simulations with a dynamic Sun is limited by \(\Dt\st{max} = T_\mercury / 6\). However, for small step sizes, dynamic Sun simulations accurately describe the Solar System and the restricted three-body problem in which resonance occurs. This is achieved without the use of Jacobian coordinates, which are commonly used in implementations of the Wisdom-Holman integrator. The fixed Sun and dynamic Sun simulations are shown to conserve energy, suggesting an accurate description of the system simulated. ... Read more

Chirality Induced Spin Selectivity is the phenomenon where the chirality of certain molecules favours the transmission of electrons based on their spin. Among many examples, this long-studied phenomenon appears in two-terminal transport experiments, where different magnetisations of the leads can give different current-voltage characteristics. In previous research by Rikken [20], the chiral geometry of the device was determined as a necessary condition for antisymmetric IV curves.

In this thesis, we implemented a Büttiker probe (BP) in a 6-helicene model based on the previous work of Geyer [9]. The probe mimics the decoherence in a two-terminal CISS experiment. Moreover, this enables us to magnetise both leads independently. By altering the magnetisation of the leads and the orientation of the Büttiker probe, we were able to analyse many possible experimental setups. This enabled us to express the current difference in terms of bias voltage, magnetisation and BP orientation, where the latter was the research objective.

Isotropic Büttiker probes lead to a CISS effect, which is absent in a coherent electron transport model. Further research is needed to determine the exact nature of the numerical errors in our isotropic BP experiments. The results of anisotropic BPs can be explained assuming that the current difference is linear in the anisotropy of the BP. Further research is needed to strengthen this conjecture. These results are supported for lead magnetisation along an axis, perpendicular to the helical axis of the molecule, as well as magnetisation along this helical axis. ... Read more

The central theme of this dissertation is the experimental investigation of electrical transport through single-molecule junctions using the mechanically controlled break-junction (MCBJ) technique at room temperature. Of particular interest is the phenomenon of quantum interference, wherein the wave-like nature of the electron plays a crucial role in the electrical conductance of various molecules. Chemical design and the influence of external stimuli such as mechanical manipulation or applying strong electric fields can impact the interference, of which the latter is also used to investigate conductance switching due to conformational changes.

Chapter 1 serves as an introduction to the field of molecular electronics, discussing the origin of the field and landmarks, and provides theoretical considerations concerning charge transport in metal-molecule-metal junctions. Finally, a general introduction on quantuminterference is given.

Chapter 2 broadly discusses the MCBJ experimental method, where the mechanical and electrical equipment for two set-ups is discussed alongside the modus operandi for doing fast-breaking experiments. Subsequently, data analysis using machine learning methods will be discussed. Ultimately, reference measurements on bare gold electrodes and an oligo(phenylene-ethynylene) (OPE) molecule were performed as a benchmark for roomtemperature high-bias and current-voltage characterization on other molecules. For high-bias experiments we note that measurements can be performed reliably and shows in the case of the OPE molecule that its conductance increases as a function of applied bias voltage in agreement with the single-level model.

Chapter 3 revolves around the study of the electrical properties of molecules containing a cyclophane core. For molecules containing a paracyclophane core we observe that ortho-connections suppress the conductance more so than a para-connection as compared to a meta-connection. Additionally, we find para-connections to cyclophane units to be the common denominator in showing mechanosensitive behaviour, i.e., in which themolecule changes its conductance strongly due to mechanical deformation.

In Chapter 4 we have developed a method to reconstruct the observed destructive quantum interference dip in a molecule with a naphthalenophane core, opting for establishing a closer link between theory and experiment. Two complementary techniques at room temperature were used for this study: (i) the MCBJ technique, which allows for large statistical sampling fortifying the robustness of the dip reconstructionmethod; (ii) the alternating-current scanning tunneling microscopy break-junction technique (ACSTM- BJ) allowing for the continuous simultaneous measurements of the conductance and the corresponding thermopower, providing additional information on the destructive quantum interference dip. We find a sinusoidal response of the thermopower across the conductance dip without a sign change. Theoretical calculations on conductance and thermopower including electrode distance and energy alignment variations emphasize the crucial role of thermal fluctuations at roomtemperature.

For Chapter 5 we change pace and shift towards molecular switches. Three differently anchored norbordaniene molecules were investigated under high-bias circumstances. For all compounds, we find two conductance states. We find no full switching between two conductance states, as the two states are present across a wide-range of applied bias voltages and no clear population differences between the states are found. Alternatively to the explanation of the switching within the molecule itself, one can argue that either we observe two different configurations of the molecular junction or that interactions of the short linkers of the molecules, by interactions with the gold surface, quench the switching between the states unlike previous published results using a molecule with the same backbone but with longer linkers.

Chapter 6 investigates the effect of chemical design on the conductance of macrocyclic structures, studying them with different substituents. We observe a clear difference in conductance between para- and meta-connections in the core using thiophene and benzene substituents, consecutively. Here, the created para- connected path shows a higher conductance than its meta counterpart. Different connections, para and meta, in molecules with the same backbone show less of an effect on the conductance of the molecular junction. Additionally, preliminary results of one of the compounds using room temperature current-voltage characteristics shows a negative differential conductance and hysteretic behaviour.

Lastly, in Chapter 7 we conclude the obtained results from this dissertation and place them in a broader perspective.
... Read more

Chirality-induced spin selectivity (CISS) is a general term denoting the interplay between the chiral structure of molecules and the electron spin. CISS has been studied for over two decades, leading to a consistent picture of experimental phenomena. In this thesis, we focus on transport experiments exhibiting CISS. In spite of the efforts of many scientists, there is no theoretical explanation for the high degrees of CISS that have been measured in electron transport experiments. In general, it is agreed upon by theorists that the effect is caused by the interplay between the electronic spin-orbit interaction and the helical molecule geometry, in combination with phase-breaking effects such as electron-electron or electron-phonon interactions. As of yet, no theoretical model has achieved realistic degrees of SOC without drastic inflation of the spin-orbit interaction strength.
In this thesis we explore the possibility of using matrix product states (MPS) to study spin-selectivity in boundary-driven electron transport through tight-binding models of chiral molecules, a novel approach in the field of CISS. To this end, we use a model proposed by Fransson in 2019, which considers interacting electrons in a Hubbard model with a spin-orbit interaction adapted from the Kane-Mele model. In this thesis work, the fermionic Hubbard model is mapped to a double spin chain using the Jordan-Wigner transformation. The state of the system is described by a matrix product density operator (MPDO) which is vectorised to a matrix product state (MPS). The system dynamics are described by a vectorised Lindblad equation. The advantage of this approach lies in the fact that it does not require the use of any systematic approximations to the Hamiltonian, in contrast to previous studies. The developed method is validated against the results of previous works studying the boundary-driven Heisenberg-XXZ model and the boundary-driven Hubbard model.
The method is shown to be capable of reproducing chirality-induced spin-selective effects for short chains. The results of this study show a finite magnetocurrent that is odd in bias voltage with an associated magnetoresistance of less than 1%. This is in line with previous studies of this model, but two orders of magnitude lower than experimentally measured values. However, these results are obtained using highly inflated values for the spin-orbit interaction strength. In multiple cases, the results do not satisfy the Onsager-Casimir and Büttiker reciprocity principles, which state that the magnetocurrent should vanish in the low-driving and in the non-interacting regimes. Moreover, the continuity of the current in the steady state was not fully satisfied.
We provide evidence that indicates these problems result from the time-integration error introduced by the Suzuki-Trotter decomposition. We expect that these can be mitigated using higher order time-integration schemes. From the results of this study we can conclude that matrix product states are a viable tool to study CISS in bound-electron transport. However, the method presented in this thesis suffer from numerical errors. We present several suggestions for improvement which address these shortcomings.
... Read more

In this thesis we implement the Lindblad equation in the matrix product state (MPS) formalism using an operator splitting method. We developed a second-order method based on a Trotter approximation and a third-order high-dimensional midpoint method and we proposed a new fourth-order method based on Duhamel’s principle and a nested RK4 method, all of which preserve positivity and Hermiticity of the density operator. We simulated spin transport through an XXZ-Hamiltonian Heisenberg chain, for which we found the magnetisation profile and measured a spin current of 0.04-0.05. The results obtained are consistent with the existing literature. The extensive error analysis shows that the time step ∆t is the main contributor to the error, if the bond dimension χ is set to at least 15. The third-order method is in general preferred to the second-order method, as only this method preserves trace. We also analysed the Hubbard model, including a spin orbit coupling, in order to propose a method for simulating the chiral induced spin selectivity (CISS) effect. ... Read more

In this thesis the hydrodynamic limit of the Freezing Model is studied. The model consists of an integer line on which particles can get frozen to different degrees, analogous to jumping to another integer line, with certain rates and can get unfrozen with certain rates per frozen layer. The main result of the thesis is a proof that the hydrodynamic limit for the Freezing model converges to a system of PDE’s describing the particle density for each layer, either the ground layer or a frozen one. Firstly, it is proven that the position of a random walker in the Freezing Model, appropriately scaled, converges to a so-called Switching Brownian Motion. This together with duality is used in the rest of the proof. Secondly, it is proven that the expectation of the empirical field densities of the layers converges towards the solutions of the aforementioned PDE’s. Lastly, it is proven that the variance of the empirical field density converges to 0. Under an additional diffusive scaling of the system of PDE’s for the particle densities, a condition for diffusive behaviour is set up involving the ratio of the freezing and unfreezing rates. It is shown that the PDE’s collapse into the heat equation if this condition is satisfied. Finally, the case where the condition is not satisfied is investigated. The model is then no longer memoryless like a Markov process and shows sub-diffusive behaviour. The rescaled position of a single particle then no longer converges to Brownian motion, but to Brownian motion on the time scale on which the particle occupies the ground layer. This time scale is t β−1 with 1 < β < 2. This grows slower than t for large enough t. Additionally, simulations in Python were built to show that the model exhibits diffusive or nondiffusive behaviour depending on the jump rates of the process. Before everything, some mathematical preliminaries about probability theory, Markov theory, random walks and duality are given. ... Read more

In this thesis, we use methods offered by state of the art nanofabrication and single-molecule measurement techniques to capture polycyclic aromatic hydrocarbon (PAH) all-organic, di-radical molecules in solid-state devices, and study their electronic transport properties in different conditions with a focus on their magnetic properties. ... Read more

Quantum random walks are the quantum analogs of classical random walks and appear to be promising tools to design fast quantum algorithms. Therefore it is important to study their time-related features and see how these differ compared to the classical case. For the discrete-time quantum walk on the line it has been shown that the probability to be absorbed by an absorbing boundary equals 2/π in contrast to the classical case where this probability equals 1, hence a quantum walk may continue forever without getting absorbed. It is also shown that mixing times of discrete-time quantum walks on the hypercube scale with n, the dimension of the hypercube, which is faster than O(n log(n)) for the classical random walk. So the quantum walk might offer a slight speed-up compared to the classical case. Finally, it is shown that the mixing times of the continuous-time quantum walk on a 2-layer multiplex graph depend on the eigenvalue gaps of the corresponding Laplacian matrix L. When the strength of the connections between the layers of a multiplex graph becomes very large, the eigenvalues of the Laplacian matrix converge. Thus the mixing times of the continuous-time quantum walk on 2-layer multiplex graphs converge. ... Read more

Humans are efficient at moving due to their exceptional mastery of bipedal locomotion. Several models have been made that attempt to model the motion of the centre of mass with a spring-mass system with various degree of success. For example, a two dimensional model tracks the height of the centre of mass well and a three dimensional model explains the normal force exerted on the ground.

In this report a three dimensional model is constructed to describe the motion of the centre of mass, with the purpose to determine the spring constant and the rest length in a simple prosthetic leg. The research question is: can a three dimensional spring-mass model accurately track the centre of mass and can this be used to determine the optimal properties of the spring of a simple prosthetic leg?

The spring mass model consists of two springs connected to the ground and the centre of mass. The springs do not exert a force on the centre of mass if they are extended. The model incorporates steps by moving the ground connection points instantaneously over fixed points on a rail. The model can be used in two distinct ways. The first generates a periodic trajectory by finding the optimal initial positions y₀ and z₀ (PPFA) and the second finds the optimal spring parameters k and u to fit a data set (DFA). The optimal conditions are found by discretizing the parameter space and using an iterative process. The space around the best parameters becomes the parameter space for the next iteration.

The PPFA shows that the initial lateral displacement is nearly constant with respect to the spring constant when the rest length is chosen such that the centre of mass is at constant height if the model is stationary. The PPFA also shows that there are two distinct y-trajectories possible for walking. Running, however, has only one trajectory. Several versions of the model are fit to the data: only discretised step widths, discretised step widths and a x₀-offset, and distinct legs. There is not much difference between these models. The x₀-offset is only Δs = -0.02m confirming that the position of the feet does align with the extreme values of the y- and z-position. The model fits the general characteristics well but fine details in the motion are lost. Future research should investigate how cumbersome these deviations of the data are perceived by people using a simple prosthetic. ... Read more

This research investigates the impact of gravitational scatterings caused by close encounters between particles in an N-body Kepler system, addressing three main questions: (1) the influence of scatterings on system evolution, (2) the correspondence between simulated and expected average times between scatterings, and (3) the effect of increasing different parameters individually on the average scattering time. Simulations demonstrate an average scattering angle of 15.2 degrees for particles involved in the top 10 percent of scatterings. This would indicate a non-negligible impact of gravitational scatterings, especially for systems with heavier bodies. The results indicate that the simulated average time between scatterings is higher than the expected average, necessitating further research for accurate estimation. Moreover, the time between scatterings decreases over time, before reaching a stationary state after roughly 300 scatterings. On this domain, the correlation coefficient between the scattering time and the scattering counter was found to be  -0.08. By varying the test domains for different parameters, a new expression for the expected time between two scatterings is proposed based on simulation data. A clear connection was found between the scattering time and the number of particles, the maximum orbital radius and the maximum inclination angle. The study acknowledges limitations, including the non-stationary initialization state and linear approximations to most computations, suggesting avenues for future improvement. Overall, this research aims to find the role of gravitational scatterings in Kepler systems and underscores the need to consider these interactions, which are now often considered to be negligible. ... Read more

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. ... Read more

In an attempt to find alternatives for solving partial differential equations (PDEs)
with traditional numerical methods, a new field has emerged which incorporates
the residual of a PDE into the loss function of an Artificial Neural Network. This
method is called Physics-Informed Neural Network (PINN). In this thesis, we study dense neural networks (DNNs), including codes developed in the context of this bachelor project. We derive the backpropagation equations necessary for training and use different configurations in a DNN to test its interpolating accuracy. We distinguish between a-PINNs which use automatic differentiation to evaluate a PDE, and n-PINNs which approximate differential operators in a PDE with numerical differentiation. We compare both PINNs on the harmonic oscillator, the 1D heat equation and the 1-soliton and 2-soliton solutions of the Korteweg-De Vries (KdV) equation. Both PINNs could accurately converge to the solution, except to the 2-soliton solution, where the a-PINN outperformed the n-PINN. Furthermore, we tested a highly nonlinear problem of the KdV equation, which can be described by a train of solitons. We observed that PINNs are inaccurate if insufficient training samples are used for training. Adding training samples on the interior from a numerical solution leads to a good qualitative agreement, though more effort is required to find a better network configuration to obtain more accurate predictions.
Additionally, PINNs were used for inverse problems to derive an unknown coefficient in a PDE and proved to be highly accurate for noiseless data. When we
generated training samples with 10% noise from a uniform distribution, the PINN
results’ relative error stayed within a margin of under 2%. However, inverse PINNs are much more inefficient compared to nonlinear least squares methods like the Levenberg–Marquardt algorithm.
As of now, PINNs are still very early in development and stand no match against
traditional numerical methods to a known PDE. They may, however, provide a
useful alternative in the future as they are constantly being improved. ... Read more

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. ... Read more

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. ... Read more

This thesis contains the development of continuous Kepler orbit- and a discrete numerical integration-based collision detection algorithms in a system of LEO satellites, which in combination with collision algorithm form a simplified space debris evolution model. This model is then used to study the Kessler syndrome. The continuous and discrete algorithms get their names from the solutions of the Two Body Problem (TBP) and the methods for collision detection that they are based on; the analytical and continuous time solution of TBP resulting in the Kepler orbits and the numerical, discrete time Velocity Verlet integration of the TBP. The collision model consists of an algorithm for fragmentation collisions largely based on the NASA Standard Breakup Model and a method for elastic, random scattering collisions. Comparison between the continuous and discrete algorithms shows that on average both predict the same time to the first collision in a system of homogeneously distributed satellites. The algorithms differ in their efficiency depending on the number and the radius of the satellites in and the geometry of the system. For relatively small satellite numbers in large systems, the continuous algorithm is computationally more efficient. However, as more satellites or fragments result from previous collision, the continuous algorithm is outperformed by the discrete algorithm. Consequentially, its time complexity appears to be O(N²). Armed with this knowledge, the continuous algorithm is used to show that an initially small system of satellites is able to evolve into a large population of debris particles within several decades. Similarly, the discrete algorithm is used to show that an ordered collection of satellites in an homogeneously distributed system of debris-like particles exhibits the effect that a collision early on in the simulation can cause a cascade of collisions at a later stage. Hence Both the discrete and continuous algorithms predict a Kessler Syndrome and mimic predictions made by more advanced models from leading space agencies like NASA’s LEGEND, ESA’s DELTA and JAXA’s LEODEEM [Lio+13].Future research could focus on including atmospheric drag and gravitational perturbations to the continuous algorithm, thereby lengthening the time frame during which it can realistically simulate a system of satellites in LEO. To achieve this, it is suggested that one execute the calculations inherent to the algorithm in parallel on a GPU, as these are independent of each other.  ... Read more

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. ... Read more

The behaviour of the dilute Curie-Weiss model has been analysed for Gaussian, bounded and subGaussian couplings. This model is an abstraction of the Curie-Weiss model which is a model of a spin configuration where instead of a constant coupling between spins the spin coupling is a realisation of a random variable. To analyse this behaviour first the behaviour of the standard Curie-Weiss model is depicted. A notable part from this behaviour is the presence of a phase transition., indicating that this model can be used for a study of phase transitions. After having analysed the behaviour of the standard Curie-Weiss model theorems, stating the closeness between the standard Curie-Weiss model and the randomly dilute Curie-Weiss model for Gaussian, bounded and sub-Gaussian couplings, will be proven. Indicating that the models behave approximately the same way. These theorems will be in the form of a bound for the on randomly dilute Curie-Weiss model as an exponential multiplied by the standard Curie-Weiss model with the probability of this bound being a sub-Gaussian distribution. ... Read more

The aim of this thesis is to simulate quantum spin chains at a finite temperature. This has been achieved by using purification to write the incoherent thermal states of the spin chain as pure states, after which the matrix product state (MPS) formalism to simulate the spin chain. The influence of temperature on the spin chain has been tested by comparing chain magnetization to the strength of the external magnetic field at different temperatures. Real time evolution has also been applied under the assumption that the time evolution is adiabatic. The results have been compared to the analytical solution found by directly calculating the density matrix. The model is very accurate for imaginary time evolution, which is required to evolve the system to the desired temperature. During real time evolution the results oscillate slightly around the analytical solution, which remains constant. This oscillation is a consequence of model truncation and the Suzuki-Trotter approximation. The effect of allocating more numerical resources to limit this phenomenon is explored, and a method to increase reachable timescales by decreasing entanglement growth in the chain over time is tested and verified. ... Read more

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. ... Read more

In this thesis, the diffusive limit of active particle motion in R^d 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. ... Read more