**Prof. Eric Carlen**

**“Entropy and quantum non-equilibrium statistical mechanics”**

**Rutgers University**

**Lecture 1: Monday, November 21, 2022 at 15:30**

**Title**: Monotonicity, convexity and some mathematical problems from quantum mechanics

**Abstract**: Monotonicity and convexity theorems have long played a fundamental role in science. Boltzmann’s famous H-Theorem on the steady increase of entropy is one example, and the convexity property of entropy is crucial to equilibrium statistical mechanics. Moreover, there is a close connection between monotonicity and convexity theorems; one often proves a monotonicity theorem starting from a convexity theorem.

The mathematical structure of quantum mechanics is quite different from that of classical mechanics, and classical strategies of proof are no longer applicable. However, new approaches have been developed in which quantum monotonicity results are the source of quantum convexity results, and this has led to recent progress, with contribution by myself and co-authors that will be surveyed in this lecture. The lecture will not assume prior familiarity with the topic.

**Lecture 2: Wednesday, November 23, 2022 at 15:30**

**Title**: Reversible quantum Markov semigroups as gradient flow for quantum relative entropy

**Abstract**: According to the Data Processing Inequality, for any semigroup of completely positive trace preserving operators; that is a quantum semigroup, the relative entropy of a state evolving under the action of this semigroup with respect to an invariant state is monotone decreasing. When can the semigroup be viewed as gradient flow with respect to some metric and the relative entropy? This is not always the case. We discuss the necessary conditions and the known sufficient conditions, giving details of the construction of the metrics, and we shall explain why this is useful for proving inequalities, especially entropy production inequalities. Many people have contributed to this subject, and while other approaches will be discussed, the focus will be on results obtained together with Jan Maas.

**Lecture 3: Thursday, November 24, 2022 at 15:30**

**Title**: Uniform in N entropy production bounds for some quantum Kac models

**Abstract**: We prove a logarithmic Sobolev inequality for a family of quantum Kac models with constants that are uniform in the number of particles. A key role is played by recent progress in understanding the rate of return to equilibrium of a classical random transposition process on the multislice.

This is joint work with Michael Loss.

Light refreshments will be given before the talks in the faculty lounge on the 8th floor.