Fachbereich Mathematik

# Mathematical Statistical Physics

``Thou turn thy mind the more unto these bodies
Which here are witnessed tumbling in the light:
Namely, because such tumblings are a sign
That motions also of the primal stuff
Secret and viewless lurk beneath, behind.
For thou wilt mark here many a speck, impelled
By viewless blows, to change its little course,
And beaten backwards to return again,
Hither and thither in all directions round.
Lo, all their shifting movement is of old,
From the primeval atoms; for the same
Primordial seeds of things first move of self,
And then those bodies built of unions small
And nearest, as it were, unto the powers
Of the primeval atoms, are stirred up
By impulse of those atoms' unseen blows,
And these thereafter goad the next in size;
Thus motion ascends from the primevals on,
And stage by stage emerges to our sense,
Until those objects also move which we
Can mark in sunbeams, though it not appears
What blows do urge them.
Herein wonder not
How 'it is that, while the seeds of things are all
Moving forever, the sum yet seems to stand
Supremely still, except in cases where
A thing shows motion of its frame as whole.''

(Lucretius, around 100 B.C.)

Place and time: Tue and Thu, 8-10, N14 / M1. The basics of maths and physics will be taught in the first lectures. In the second and third week of the course, the foundations of classical mechanics and thermodynamics will be taught in class, basic concepts of probability (axioms of Kolmogoroff, random variables, expectation and variance, law of large numbers) will be taught in S6.

Registration: Registration procedure for the exam will be explained in the first lectures. No action from your side before the first meeting is required.

Exam: The written exam takes place on Thursday, July 23 rd in the regular classroom at 8:20 until 9:50. You are allowed to have a handwritten piece of paper (A4) with you during the exam. The exam will be graded the same day, you can check your results and the corrections on Friday  the 24 the of July between 10 and 12 in Peter's office. The exam will be similar to the mock exam you find under "homework assignments".

Prerequisites: Prerequisites for the course are the foundational math courses (analysis, linear algebra). Knowledge in thermodynamic and probability is helpful but not required.

Course description: Around the beginning of the twentieth century, three groundbreaking ideas manifestly changed the physical world view: relativity, quantum mechanics and kinetic gas theory. The latter will be topic of this class. Kinetic gas theory is based on the idea that the physics of macroscopic objects, in particular gases, can be explained by the motion of microscopic entities (atoms). It gives a fundamental understanding of thermodynamics.

The idea to derive the behavior of large objects from the behavior of its components has many applications that reach into many different areas, also outside of physics. So in class we will focus on the main ideas and mathematical features of kinetic gas theory, both starting from classical as well as quantum systems.

Content: At the beginning basic concepts of probability will be introduced for later use. We then discuss on very simple toy models, how to derive gas laws from the Newtonian motion of many particles. Following Boltzmann, the microscopic understanding of entropy and the second law of thermodynamics will be introduced and other concepts of entropy will be discussed. Poincare recurrence and the ergodic theorem will be proven for general dynamical systems and their role related to thermodynamics discussed. A heuristic derivation of the Boltzmann equation will be given and the H-theorem, i.e. the fact that entropy for solutions of the Boltzmann equation never decreases, will be proven. Brownian motion will be another topic of this class.

Then we will discuss the derivation of effective equations for many-body systems, in particular Vlasov and Boltzmann. Later quantum systems will be treated. Bose-Einstein statistics as well as Fermi-Dirac and their physical implications will be discussed and other interesting features of quantum systems.

Exercise Classes: Friday 10:15-11:45 in S8

Stochastic lecture: Tuesday 8:15-10:45 and Thursday 8:15-10:45 in S6