Sabine Jansen

Mathematical Physics and Statistics

Ludwig-Maximilians-Universität München

Mathematical Institute

Theresienstr. 39

80333 Munich

+49 / 89 2180 4477

jansen[at]math.lmu.de

Research webpage

I love problems that combine physical intuition with mathematical precision and elegance and draw on different styles of reasoning. Eureka moments involve beautiful landscapes of structure emerging after long struggles in understanding, but also being able to pin down subtle nuances in models and reasoning, using mathematical language to bring clarity to interdisciplinary dialogue.

Description

My research within MCQST / MQC focuses on the interplay between probabilistic techniques and analytic tools to understand many-body systems.


Examples of fruitful interplay abound:

  • metastability and the rigorous proof of Arrhenius laws build on the semi-classical limit and WKB expansions;
  • spatial correlations are studied by investigating operators (Stein operators, infinitesimal generators) and their spectral gap, compare the exponential clustering theorem for gapped Hamilton operators;
  • unitary equivalence of Hamilton operators, e.g. based on different representations for algebras of creation and annihilation operators, is closely related to the notion of duality of Markov processes, which might prove helpful in investigating path integrals.


I am especially interested in low-temperature equilibrium states and phase transitions.

Publications

Revisiting Groeneveld's approach to the virial expansion

S. Jansen

Journal of Mathematical Physics 62 (2), 023302 (2021).

Show Abstract

A generalized version of Groeneveld's convergence criterion for the virial expansion and generating functionals for weighted two-connected graphs is proven. This criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions rho (s) (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions. The proof is based on recurrence relations for graph weights related to the Kirkwood-Salsburg integral equation for correlation functions. The proof does not use an inversion of the density-activity expansion; however, a Mobius inversion on the lattice of set partitions enters the derivation of the recurrence relations.

DOI: 10.1063/5.0030148

Lagrange Inversion and Combinatorial Species with Uncountable Color Palette

S. Jansen, T. Kuna, D. Tsagkarogiannis

Annales Henri Poincare

Show Abstract

We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.

DOI: 10.1007/s00023-020-01013-0

Thermodynamics of a hierarchical mixture of cubes

S. Jansen.

Journal of Statistical Physics (2020).

Show Abstract

We investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in Zd of sidelengths 2j, j?N0. Cubes belong to an admissible set B such that if two cubes overlap, then one is contained in the other. Cubes of sidelength 2j have activity zj and density ?j. We prove explicit formulas for the pressure and entropy, prove a van-der-Waals type equation of state, and invert the density-activity relations. In addition we explore phase transitions for parameter-dependent activities zj(?)=exp(2dj??Ej). We prove a sufficient criterion for absence of phase transition, show that constant energies Ej?? lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.

DOI: 10.1007/s10955-020-02531-1

Cluster Expansions with Renormalized Activities and Applications to Colloids

S. Jansen, D. Tsagkarogiannis

Annales Henri Poincare 21 (1), 45-79 (2020).

Show Abstract

We consider a binary system of small and large objects in the continuous space interacting via a nonnegative potential. By integrating over the small objects, the effective interaction between the large ones becomes multi-body. We prove convergence of the cluster expansion for the grand canonical ensemble of the effective system of large objects. To perform the combinatorial estimate of hypergraphs (due to the multi-body origin of the interaction), we exploit the underlying structure of the original binary system. Moreover, we obtain a sufficient condition for convergence which involves the surface of the large objects rather than their volume. This amounts to a significant improvement in comparison to a direct application of the known cluster expansion theorems. Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.

DOI: 10.1007/s00023-019-00868-2

Accept privacy?

Scroll to top