Around the turn of the 20th century, Einstein discovered an interesting connection between the flow of heat and the random movement of particles. At this time, in order to attract popularity to his work, Einstein showed how computing certain diffusion coefficients in this scenario, one may be able to compute the size of molecules. This drew attention to the idea, which in many ways was more formal rather than rigorous. In spite of its heuristic nature, Einstein's idea helped to inspire major developments in probability theory and analysis, including the Central Limit Theorem, which may be seen as the missing rigorous mathematical link connecting random movement of particles to the heat equation.
This thesis project is suitable for students who have enjoyed analysis courses, such as Fourier analysis. There will be some elements of random walks and probability theory, but it is not necessary to know these topics in advance. Students are welcome from all programs at Chalmers and GU. What is most important is a good mathematical background together with motivation and interest in connections between mathematics, physics, and chemistry. Students will learn how to derive the heat equation using the laws of thermodynamics and compute the fundamental solution in n-dimensional Euclidean space in the spirit of Fourier. We shall then show how the partial differential equation which describes the large-scale flow of heat corresponds to Brownian motion of particles at the small, molecular scale. We will investigate Einstein's method for computing the size of molecules and make the heuristic notions a bit more rigorous and precise. There is the possibility to investigate various aspects of the heat equation and its solution, so there will be some amount of flexibility as to the precise content of the thesis based on the interests of the students.
Obs! För GU-studenter räknas projektet som ett projekt i Matematik (MMG900/MMG910).
Förkunskapskrav Fourier analysis, analysis in one and several variables, complex analysis.
Examinator Maria Roginskaya, Marina Axelson-Fisk
Institution Matematiska vetenskaper