The course introduces students to the Monte Carlo method (MC), its application in numerical mathematics, and in solving physical problems, including material simulations and calculations of target quantities (properties). Furthermore, students will become familiarized with the Molecular Dynamics method (MD) and its use in particle simulations. The course will also cover related topics and concepts from the area of MD simulations of molecular systems such as empirical force fields, spatial characteristics of the simulated systems, calculation of binding energy, diffusion, etc. The theoretical explanation is supported by solving practical applications, some of which can be implemented by students themselves by using the relevant simulation software. Monte Carlo 1. Discrete and continuous random variable, basic characteristics and their properties, the law of large numbers, characteristics of the mean of independent random variables, Gaussian distribution, the central limit theorem and its practical use, bootstrap resampling. 2. Generating discrete random variable, general methods for generating continuous random variable (Inverse function method, acceptance-rejection method, superposition method) and special methods for sampling the selected continuous distributions like the Normal or Maxwell-Boltzmann distribution. 3. Fundamental MC algorithms for numerical integration, ways how to increase computation efficiency - methods of variance reduction in numerical integration, discussion of the efficiency of methods: classical or deterministic methods vs. MC with respect to the dimension of the integration domain. 4. Application of the MC method for estimating probability distributions of parameters of complex systems (for example, complex electrical circuits). 5. Solving Laplace's equation using the MC method: theory, basic version - rectangular grid, efficiency increase - random step, comparison with the classic (deterministic) numerical approach. 6. Metropolis algorithm - discrete systems: motivation, theoretical explanation (Markov chain, transition probabilities, transition matrix, detailed balance, microscopic reversibility), demonstration of the Ising model of a ferromagnet, including the calculation of characteristic quantities, Hastings' generalization of the Metropolis method (Metropolis-Hastings algorithm). 7. Metropolis algorithm - continuous systems: - theoretic transition from discrete to continuous systems, sampling of arbitrary probability distributions using Markov chains, tuning of probability distribution parameters for proposing candidates for the next members of the Markov chain (acceptance ratio), numerical integration using the Metropolis algorithm, Simulated annealing optimization method and its application, use of MC in molecular simulations. 8. Transport problem - scattering processes, effective cross-section, mean free path, random free path, methods for efficiency enhancement or the possibility of application to inhomogeneous materials. Molecular Dynamics 1. Position vector, velocity, acceleration. Newton's second law of motion/equation of motion. Numerical methods for solving equations of motion: Verlet's algorithm, Velocity Verlet's algorithm. Methods applicable when force depends on velocity: Euler's algorithm, Euler-Richardson method. 2. Temperature and pressure control in MD simulations (thermostats, barostats). 3. Model of the potential energy of a molecular system (force field). Atom types, bond and non-bond interactions, their theoretical models, methodologies for obtaining force field parameters, partial charges and methods of their calculation. 4. Diffusion, calculation of the diffusion constant using MD simulation. 5. Simulations with periodic boundary conditions. Motivation, basic idea, particle motion in periodic conditions, calculation of short-range and long-range interactions. 6. Simulation of molecular systems in explicit and implicit solvents .......
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