# A Non-Commutative Analog of the 2-Wasserstein Metric for which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy

 Analysis Seminar Topic: A Non-Commutative Analog of the 2-Wasserstein Metric for which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy Speaker: Eric Carlen Affiliation: Rutgers, The State University of New Jersey Date: Tuesday, November 13 Time/Room: 3:15pm - 4:15pm/S-101 Video Link: https://video.ias.edu/analysis/carlen

The Fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on the space of density matrices that we show to be a natural analog of the classical $2$-Wasserstein metric, and we show that, in analogy with the classical case, the Fermionic Fokker-Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised. This is joint work with Jann Maas.