- The student understands the principles of the Monte Carlo method and their mathematical justification.
- The student can implement algorithms for the generation of pseudo-random numbers with desired distribution.
- The student can apply Monte Carlo integration and Markov Chain Monte Carlo simulations to a variety of problems in physics.
- The student can collect, analyse and interpret the numerical output of simulations and perform error analysis.
- The student can formulate a research question and select and implement the appropriate Monte Carlo techniques for the problem.
- The student can effectively report and communicate project results to fellow students and teachers.
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This course gives an introduction to Monte Carlo methods, or, in other words, to the use of random numbers to solve numerical problems in physics. Many computational problems in physics can be formulated in terms of an integration (and/or summation) of some function over a space of configurations, think of partition functions in statistical physics or path integrals in field theory. More often than not an analytic solution is missing and one has to resort to numerical methods. Brute-force numerical integration becomes impractical if the configuration space has more than a couple of dimensions, while for instance the phase spaces of many-particle systems can have enormous dimensionality. This is where Monte Carlo methods come to the rescue, by approximating integrals via an appropriate random sampling of configurations.
In this course students will not just learn the mathematical principles of the Monte Carlo method and their applicability, but they will also put them to practice addressing a variety of problems in physics. The course can roughly be divided in three parts. The first part of the course will consist of lectures introducing the principles and computer classes with assignments that predominantly involve Python programming and data analysis related to the study material. The second part of the course will be more focused on applications, including topics like: statistical physics of spin systems; lattice field theory; toy models in quantum gravity. Based on these introductions students will select a topic for their final research project. In the last part of the course students will complete this project under guidance of the course assistant(s) and teacher. This will involve preparing a research plan, implementing the necessary algorithms, data gathering, data analysis and reporting.
Although some applications will focus on topics in Particle and Astrophysics, the course can be followed by students from other specializations well.
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Bachelor courses on: Probability theory; Programming in Python; some elements of Statistical Mechanics.
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In order to pass the course the student is required to complete a final Monte Carlo research project. The grade for the course will be determined based on
- Assignments during the first half of the course.
- The final project report and presentation.
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