Seminar Mathematical Statistical Physics

Winter semester 2025/26

Talks start: week of November 2, 2025.
Time and day: not fixed yet, will be determined with the participants in the week of October 20, 2025.
Taught by: Prof. Dr. Roderich Tumulka and Lukas Nullmeier
Language of presentations: English
Studiengänge: This seminar is offered primarily for the master's program in mathematical physics but is open to all study programs.
Prerequisites: Basic knowledge in quantum mechanics, analysis, linear algebra, and probability theory is expected.
Talks: 90 minutes, one talk per week
Maximum number of participants: 14
Some possible topics (each with a suggested title and a short description of the goals of the talk) [and some literature]:
- Random matrix theory, in particular the Gaussian unitary ensemble
  "Gaussian unitary ensemble" is a name for the distribution of a random nxn matrix H whose entries are independent complex Gaussian random variables, except for the symmetries needed for making H Hermitian. To discuss the properties of H (Wigner semicircle law, uniform eigenbasis) and relevance to quantum statistical mechanics.
- Approach to macroscopic thermal equilibrium
  [0911.1724,1307.0572] To prove a theorem saying that for most Hamiltonians, every wave function sooner or later reaches macroscopic thermal equilibrium. To discuss the unrealistic trait of this theorem that the thermalization occurs ultrafast.
- Canonical typicality
  [cond-mat/0511091,quant-ph/0511225] Canonical typicality is the fact that for most wave functions of a system coupled to a large but finite heat bath, the reduced density matrix of the system is approximately canonical. To elucidate this fact and prove the theorem behind it.
- The GAP measure
  [quant-ph/0309021,1104.5482] The GAP measures are a family of probability measures on the unit sphere of a Hilbert space that arise as the thermal equilibrium distribution of conditional wave functions. To define these measures and the concept of conditional wave function, discuss their properties, and outline the proof that for a system coupled to a large but finite heat bath, the asymptotic distribution of the system's conditional wave function is a GAP measure.
- Smoothness of GAP-typical functions
  [math-ph/0509028] To present and discuss several senses in which the GAP measure associated with a canonical density matrix for H = - Laplacian is concentrated on smooth functions, with proofs for selected results.
- Dynamical typicality
  [2210.10018,2303.13242] To prove a theorem saying that most initial wave functions psi from a high-dimensional Hilbert subspace, the time evolution of psi looks macroscopically the same.
- Limitations to measuring the wave function of the universe
  [2410.16860] For a quantum world, the past hypothesis requires that the initial wave function of the universe is typical in a certain high-dimensional Hilbert subspace. To explain a theorem saying that it is impossible to measure a wave function with high precision and, in more depth, another theorem (related to dynamical typicality) saying that it is impossible to measure a wave function from a high-dimensional subspace even with low precision.
- Maxwell's demon
  James C. Maxwell illustrated the fact that, according to statistical mechanics, entropy decrease is not impossible but only very, very unlikely, by pointing out that an intelligent being able to observe and compute the motion of every atom in a gas (a "demon") could transport heat from a cooler gas container to a hotter one by opening a door between the containers whenever a fast molecule arrives from the cooler side but closing it whenever a slow molecule arrives. To review arguments about why real beings cannot do the same thing. This topic refers to classical physics only.
- Normal typicality
  [0907.0108,2210.10018] To outline the proofs of two theorems, the first saying that a macroscopic quantum system will have, most of the time, approximately the same superposition weights of all macro states, and second, that for most Hamiltonians these weights will be proportional to the dimension of the subspaces associated with each macro state.
- Landauer's principle in classical and quantum mechanics
  [2007.11748] Landauer's principle concerns the thermodynamic properties of computers; it provides a lower bound on the entropy increase caused by a computation, more specifically by erasing the memory. To derive the principle from Boltzmann's definition of entropy.
- The debate about the past hypothesis
  [1202.1818,1903.11870,philsci-archive.pitt.edu/17253/,philsci-archive.pitt.edu/22619/] The past hypothesis as an explanation of the thermodynamic arrow of time is a subject of current scientific debate. To present and evaluate the arguments of both sides.
- Carroll and Chen's explanation of the arrow of time without a past hypothesis
  [gr-qc/0505037,hep-th/0410270,1602.05601] To elucidate how this proposed explanation of the arrow of time works and how it compares to the past hypothesis and Boltzmann's fluctuation hypothesis.
- Boltzmann brains in different interpretations of quantum mechanics
  [1508.01017,1812.01909] A Boltzmann brain is a set of atoms that came together by pure chance in just the same state as your brain is presently in. This is extremely unlikely, but it should occur in our universe if it has infinite volume or will continue to exist forever. To discuss what to think of Boltzmann brains, and what the consequences for physical theories of the universe are.
- What is logical and what is not in the movie "Tenet"
  The 2020 science fiction movie "Tenet" explores what would happen if some physical objects or people had an arrow of time opposite to the usual one. To analyze this idea scientifically.
Literature: will be recommended individually by topic of the presentation.
Register: We still have open spots. If you are interested in taking this seminar, please email Prof. Tumulka.