Daniel Ueltschi

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The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research.

Eugene Wigner

My research area deals with the mathematical description of quantum systems with a large number of particles. From the mathematical point of view, I am at the crossroad of analysis, probability theory, and functional analysis. From the physical point of view, I work in condensed matter theory and statistical mechanics. References to the article cited below can be found in my list of publications, or by downloading the pdf file of the article by clicking on it.

Prospective PhD students: Some of these research topics are suitable for a PhD thesis. To learn more about the PhD program at the University of Warwick, look here.

Bosonic systems.
I started working on bosonic systems in 2005. A popular approach involves path integrals, where bosonic particles are represented by "space-time'' Brownian trajectories. The presence of infinite cycles is believed to be related to Bose-Einstein condensation and superfluidity. The articles [U 06a], [U 06b] discuss an explicit relation between cycle lengths and the off-diagonal correlation function. It suggests that equivalence between these concepts holds for weakly interacting systems, but not for strongly interacting systems that undergo a regular condensation into a solid phase.

I am currently working on a probabilistic and geometric approach to bosonic systems using models of random spatial permutations. They should help us understand the effects of particle interactions on the critical temperature (or critical density) of the Bose-Einstein condensation. The article [U 08] reviews the topic and computes the interactions between permutation jumps that come from original particle interactions. The article [BU 08] (with V. Betz) introduces the general mathematical setting and proposes rigorous results on the occurrence of infinite cycles. The earlier article [GRU 07] (with D. Gandolfo and J. Ruiz) mainly contains numerical simulations on a related model; surprisingly, it suggests that certain general properties (such as the distribution of macroscopic cycles) are universal, i.e. they do not depend of the microscopic details of the models.

Here is a talk that I delivered in Warwick in June 2008.

Past research.
Let me mention a few topics that occupied me earlier, both in physics and in mathematics. In physics,

Quantum lattice systems. With C. Borgs and R. Kotecký, we developed a rigorous perturbation theory, the "Quantum Pirogov-Sinai Theory'' [BKU 96], [KU 99]. See also [Fröhlich Rey-Bellet U 01] and [Fröhlich Fernández U 06].
The Falicov-Kimball model. It is a variant of the Hubbard model that plays a central role in condensed matter physics.
With J. K. Freericks and E. H. Lieb, we proved that the ground state displays "segregation'' away from half-filling [FLU 02], [FLU 02a]. I subsequently extended these results to the asymmetric Hubbard model [U 04a].
Ferromagnetism. With J. Fröhlich, we considered a related 2-band model for which we could prove the existence of ferromagnetic ground states [FU 05].
Junctions and non-equilibrium physics. With J. Fröhlich and M. Merkli, we described junctions between reservoirs of electrons and proved in particular that entropy production is strictly positive, and that Onsager relations hold true [FFM 03].

And in mathematics,

The article [FLU 02] (with J. K. Freericks and E. H. Lieb) contains a lower bound on the sum of eigenvalues of the discrete Laplacian in domains of arbitrary shape. It extends earlier results of P. Li and S. T. Yau.
A central limit theorem for interacting random walks [U 02]; it allows for attractive interactions.
A favorite result of mine is a theorem on cluster expansions [U 04]. It simplifies, clarifies, and extends an important method in statistical mechanics, and I believe it will prove useful in the future.

Last modified June 2008.