AbstractIn this thesis, we investigate a family of fast-slow Hamiltonian systems, which is parametrised by a small scale parameter ε, indicating the typical timescale ratio of the fast and slow degrees of freedom. It is known that the family of mechanical systems converges for ε→0 to a homogenised system. In this thesis, we are interested in the situation where ε is small but positive.
A significant part of this thesis is devoted to the rigorous derivation of the second-order corrections to the homogenised degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that constitute the average evolution of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations.
Furthermore, based on the second-order asymptotic expansion, we investigate the energy of the fast degrees of freedom from a thermodynamic point of view. More specifically, we define and expand a temperature, an entropy, and an external force and show that they satisfy to leading- as well as on average to second-order thermodynamic energy relations similar to the first and second law of thermodynamics.
Finally, we analyse the second-order asymptotic expansion of the slow degrees of freedom for a specific test model from a numerical point of view. Supported by simulations, we examine their approximation error on short and long time intervals and compare their total computation time with that of the slow solution to the original system of similar accuracy.
|Date of Award||14 Feb 2022|
|Supervisor||Johannes Zimmer (Supervisor) & Karsten Matthies (Supervisor)|
- Two-scale Hamiltonian
- Asymptotic expansion