Gibbs measures for nonlinear Schrodinger equations and many-body quantum mechanics

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

The nonlinear Schrödinger equation (NLS) is a nonlinear PDE that arises in the dynamics of many-body quantum systems. An instance of this correspondence can be seen in the phenomenon of Bose-Einstein condensation. The solution of the NLS corresponds to the Bose-Einstein condensate. This in general gives us a correspondence between a nonlinear PDE and a quantum problem. The latter is linear, albeit non-commutative. It is posed on the bosonic Fock space, in which the number of particles is not fixed.

The NLS possesses a Hamiltonian structure, that allows us to (at least formally) define a Gibbs measure, which is invariant under the flow. The construction of such a measure dates from the constructive quantum field theory in the 1970s (the work of Nelson, Glimm-Jaffe, Simon), and later work of Lebowitz-Rose-Speer and McKean-Vaninsky. Its invariance was first rigorously shown in the pioneering work of Bourgain in the 1990s. Today, Gibbs measures are used as a fundamental tool in the study of probabilistic low-regularity well-posedness theory. This is due to the fact that Gibbs measures are typically supported on low-regularity Sobolev spaces.

The main goal of my proposal is to understand how Gibbs measures arise in the correspondence between the NLS and many-body quantum theory. In the quantum problem, one works with quantum Gibbs states. These are equilibrium states on Fock space corresponding to the many-body Hamiltonian at a fixed (positive) temperature. By using the (classical) Gibbs measure, one can similarly construct the classical Gibbs states. The correspondence that we want to verify is the convergence of correlation functions of the quantum Gibbs state to those of the classical Gibbs state in an appropriately defined mean-field limit.
When working in higher dimensions, one should take special care to eliminate the divergences that arise in the problem. This is done by applying the procedure of Wick-ordering. This procedure is well-known in the classical theory and it has a clear quantum analogue.

Earlier results on this problem were obtained by Lewin-Nam-Rougerie, by the author in collaboration with Fröhlich-Knowles-Schlein, and by the author himself. The methods used to study the problem came from analysis, but also from probability, and statistical mechanics. There is still a substantial gap with what is known in this problem and what is known in the classical theory. Namely, in the classical theory it is possible to construct Gibbs measures for the NLS with very singular interaction potentials. A major challenge in the quantum problem is the lack of commutativity.

In this proposal, I aim to tackle this problem. The techniques come from different aspects of analysis, probability, and statistical mechanics. One goal would be to understand connections between techniques from the analysis of the NLS (which are primarily based on harmonic analysis) and the methods of quantum field theory.

Planned Impact

This is a proposal dealing with fundamental research in theoretical mathematics. As such, the short-term impact is of an academic nature.
The proposed research deals primarily with mathematical analysis. It also has close connections to probability theory and statistical mechanics. There already is a lot of activity in mathematical analysis in the UK, and it is expected to expand in the future. This was clearly reflected in the results of the EPSRC's 2013 CDT exercise, according to which `Mathematical analysis and its applications' should be reinforced.
One of the main themes of my proposed research is to find connections between problems and techniques from nonlinear dispersive PDEs and those from quantum field theory.
I believe that translating techniques from one set of problems to another (or at least finding suitable analogies) is bound to be beneficial for researchers working both sides. In particular, many analysts studying the nonlinear Schrödinger equation are interested in problems arising from quantum field theory and will be enthusiastic to see papers that make these connections clear. I believe that this will motivate new (interdisciplinary) research directions.

All of my results obtained as part of the proposed grant will be freely available in preprint form from the arXiv or my website. When necessary, I will post additional materials on my website. Furthermore, I will present the obtained results at seminars and conferences, both in the UK and abroad.

Publications

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Ammari Z (2022) Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs in Revista Matemática Iberoamericana

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Fröhlich J (2021) The mean-field limit of quantum Bose gases at positive temperature in Journal of the American Mathematical Society

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Fröhlich J (2022) Interacting Loop Ensembles and Bose Gases in Annales Henri Poincaré

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Fröhlich J (2020) A Path-Integral Analysis of Interacting Bose Gases and Loop Gases in Journal of Statistical Physics

 
Description The general question that was addressed was the derivation of Gibbs measures for nonlinear Schrödinger equations (NLS) from realistic physical models of interacting bosonic systems in appropriate scaling limits. Gibbs measures for the NLS are important objects used in the study of the dynamics of the equation at low regularity for randomly chosen initial data. Our methods help explain their physical origin from many-body quantum mechanics. Prior to this grant, I had worked on this project jointly with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein and we had obtained a number of partial results in the challenging higher-dimensional problem (d=2 and 3). At the beginning, we needed to add suitable modifications to the quantum many-body problem in order for the method to work.

In joint work with Fröhlich, Knowles, and Schlein done over the course of the grant, we developed a new method based on a functional integral formulation of the problem. As a result, we could solve the full problem in dimensions 2 and 3 without the aforementioned modifications. In our work, published in the Journal of the American Mathematical Society, we could give a derivation of the Gibbs measure for a nonlocal NLS with continuous interaction potential of positive type.
This corresponds to a nonlocal \Phi^4 theory. A similar result was independently obtained by Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie at the same time, using different methods. When d=2, we could also study the local \Phi^4_2 theory which corresponds to taking the interaction potential to be a delta function. This result, accepted in the Journal of the European Mathematical Society give the first derivation of such a field theory from a realistic interacting physical model of statistical mechanics.

In further joint work with Fröhlich, Knowles, and Schlein, we studied the corresponding problem on the lattice, which we reformulated in terms convergence of ensembles of random loops. This was inspired by work Ginibre and Symanzik in the 1960s. We rigorously showed the convergence of the Ginibre ensemble to the Symanzik ensemble in the mean-field (classical field) limit. In addition, we could study the classical particle limit in which all of the paths reduce to points. This result will be published in the Annales Henri Poincaré. Most of the above work was summarised in an expository work that we published in the Journal of Statistical Physics in honour of Joel Lebowitz's 90th birthday.

During the grant, I had supervised Andrew Rout as a PhD student at the University of Warwick. We studied a related problem to the one above. Our setting was in one dimension, without any positivity of the interaction. In particular, we can study a one-dimensional focusing NLS. In our work, we obtained the first derivation of a Gibbs measure for a focusing NLS from many-body quantum Gibbs states. A preprint is available on the arxiv at https://arxiv.org/abs/2206.03392.

In joint work with Zied Ammari, we studied classical Gibbs measures in the context of Kubo-Martin-Schwinger (KMS) equilibrium states. We showed a correspondence between the two for a large class of (realistic) Hamiltonian PDEs. Our result was accepted for publication in Revista Matemática Iberoamericana.
Exploitation Route As a result of our work, we have a better understanding of Gibbs measures for the nonlinear Schrödinger equation as arising from mean-field limits of quantum many-body Gibbs states. One could implement these methods to study more general P(\Phi)_2 Quantum field theories. It would be interesting to see how the methods we developed combine with more probabilistic methods.

The study of Gibbs measures as Kubo-Martin-Schwinger states yields connections to the Liouville equation and methods from optimal transport theory. This can give us a new perspective on Gibbs measures associated to a more general setting. Such an approach could also be tied into more sophisticated probabilistic methods.
Sectors Other

 
Description Analysis of Gibbs measures for nonlinear Schrödinger equations and many-body quantum Gibbs states, jointly with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein 
Organisation ETH Zurich
Department Department of Physics
Country Switzerland 
Sector Academic/University 
PI Contribution This collaboration started during my postdoctoral work under the guidance of Antti Knowles and Benjamin Schlein. The background from mathematical physics came from the work of my colleagues. The applications to nonlinear PDEs was motivated by my PhD and earlier postdoctoral work. We have continued this collaboration after I finsihed my postdoctoral posts.
Collaborator Contribution My colleagues have introduced me to many ideas and methods from mathematical physics and probability theory. We have since worked on adapting them to our projects.
Impact My collaborators are Jürg Fröhlich (ETH Zurich, Physics Department), Antti Knowles (University of Geneva, Mathematics Department), and Benjamin Schlein (University of Zurich, Mathematics Department). Both publications listed in the 'Publications' section have resulted from this collaboration. In addition to this, there have been two new preprints that have been obtained in this collaboration. They are the following. 1. The mean-field limit of quantum Bose gases at positive temperature. The arXiv link is: https://arxiv.org/abs/2001.01546. This publication has been accepted in the Journal of the American Mathematical Society (with DOI number pending). 2. Interacting loop ensembles and Bose gases. The arXiv link is: https://arxiv.org/abs/2012.05110. This preprint has been submitted. 3. The Euclidean \Phi^4_2 theory as a limit of an interaction Bose gas. The arXiv link is: 2201.07632. This preprint has been submitted. The collaboration is multi-disciplinary in the sense that it combines colleagues working both in Maths and Theoretical Physics departments.
Start Year 2015
 
Description Analysis of Gibbs measures for nonlinear Schrödinger equations and many-body quantum Gibbs states, jointly with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein 
Organisation University of Geneva
Department Section of Mathematics
Country Switzerland 
Sector Academic/University 
PI Contribution This collaboration started during my postdoctoral work under the guidance of Antti Knowles and Benjamin Schlein. The background from mathematical physics came from the work of my colleagues. The applications to nonlinear PDEs was motivated by my PhD and earlier postdoctoral work. We have continued this collaboration after I finsihed my postdoctoral posts.
Collaborator Contribution My colleagues have introduced me to many ideas and methods from mathematical physics and probability theory. We have since worked on adapting them to our projects.
Impact My collaborators are Jürg Fröhlich (ETH Zurich, Physics Department), Antti Knowles (University of Geneva, Mathematics Department), and Benjamin Schlein (University of Zurich, Mathematics Department). Both publications listed in the 'Publications' section have resulted from this collaboration. In addition to this, there have been two new preprints that have been obtained in this collaboration. They are the following. 1. The mean-field limit of quantum Bose gases at positive temperature. The arXiv link is: https://arxiv.org/abs/2001.01546. This publication has been accepted in the Journal of the American Mathematical Society (with DOI number pending). 2. Interacting loop ensembles and Bose gases. The arXiv link is: https://arxiv.org/abs/2012.05110. This preprint has been submitted. 3. The Euclidean \Phi^4_2 theory as a limit of an interaction Bose gas. The arXiv link is: 2201.07632. This preprint has been submitted. The collaboration is multi-disciplinary in the sense that it combines colleagues working both in Maths and Theoretical Physics departments.
Start Year 2015
 
Description Analysis of Gibbs measures for nonlinear Schrödinger equations and many-body quantum Gibbs states, jointly with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein 
Organisation University of Zurich
Country Switzerland 
Sector Academic/University 
PI Contribution This collaboration started during my postdoctoral work under the guidance of Antti Knowles and Benjamin Schlein. The background from mathematical physics came from the work of my colleagues. The applications to nonlinear PDEs was motivated by my PhD and earlier postdoctoral work. We have continued this collaboration after I finsihed my postdoctoral posts.
Collaborator Contribution My colleagues have introduced me to many ideas and methods from mathematical physics and probability theory. We have since worked on adapting them to our projects.
Impact My collaborators are Jürg Fröhlich (ETH Zurich, Physics Department), Antti Knowles (University of Geneva, Mathematics Department), and Benjamin Schlein (University of Zurich, Mathematics Department). Both publications listed in the 'Publications' section have resulted from this collaboration. In addition to this, there have been two new preprints that have been obtained in this collaboration. They are the following. 1. The mean-field limit of quantum Bose gases at positive temperature. The arXiv link is: https://arxiv.org/abs/2001.01546. This publication has been accepted in the Journal of the American Mathematical Society (with DOI number pending). 2. Interacting loop ensembles and Bose gases. The arXiv link is: https://arxiv.org/abs/2012.05110. This preprint has been submitted. 3. The Euclidean \Phi^4_2 theory as a limit of an interaction Bose gas. The arXiv link is: 2201.07632. This preprint has been submitted. The collaboration is multi-disciplinary in the sense that it combines colleagues working both in Maths and Theoretical Physics departments.
Start Year 2015
 
Description Analysis of lace expansions in the study of models of statistical mechanics with Grega Saksida and Daniel Ueltschi. 
Organisation University of Warwick
Department Warwick Mathematics Institute
Country United Kingdom 
Sector Academic/University 
PI Contribution I have organised a reading group in which we are studying the paper "The continuous-time lace expansion" by David Brydges, Tyler Helmuth, and Mark Holmes.
Collaborator Contribution Since October 2023, my colleague Daniel Ueltschi and I are co-advising Grega Saksida, a PhD student whose research will involve lace expansions applied to models of statistical mechanics. So far, we have been doing the background reading in the reading group. The main result that we have been studying is the aforementioned work by Brydges, Helmuth, and Holmes.
Impact We are currently laying the groundwork for the PhD research project.
Start Year 2023
 
Description Analysis of nonlocal \Phi^4_3 measures and their convergence to the \Phi^4_3 measure (jointly with Jürg Fröhlich, Antti Knowles, Trishen Gunaratnam, and Benjamin Schlein) 
Organisation ETH Zurich
Department Department of Physics
Country Switzerland 
Sector Academic/University 
PI Contribution The goal is to continue the analysis of the nonlocal \Phi^4_d measures that were considered in our previous work when d=2. The analysis when d=3 is more involved. Some of the techniques that I helped develop in our earlier work have to be applied further in this context.
Collaborator Contribution Trishen Gunaratnam has explained to us the methods from from from stochastic analysis that are used in this framework. We hope to unify them with methods from perturbation theory.
Impact We hope to produce a publication as a result of this collaboration.
Start Year 2021
 
Description Analysis of nonlocal \Phi^4_3 measures and their convergence to the \Phi^4_3 measure (jointly with Jürg Fröhlich, Antti Knowles, Trishen Gunaratnam, and Benjamin Schlein) 
Organisation University of Geneva
Department Section of Mathematics
Country Switzerland 
Sector Academic/University 
PI Contribution The goal is to continue the analysis of the nonlocal \Phi^4_d measures that were considered in our previous work when d=2. The analysis when d=3 is more involved. Some of the techniques that I helped develop in our earlier work have to be applied further in this context.
Collaborator Contribution Trishen Gunaratnam has explained to us the methods from from from stochastic analysis that are used in this framework. We hope to unify them with methods from perturbation theory.
Impact We hope to produce a publication as a result of this collaboration.
Start Year 2021
 
Description Analysis of nonlocal \Phi^4_3 measures and their convergence to the \Phi^4_3 measure (jointly with Jürg Fröhlich, Antti Knowles, Trishen Gunaratnam, and Benjamin Schlein) 
Organisation University of Zurich
Country Switzerland 
Sector Academic/University 
PI Contribution The goal is to continue the analysis of the nonlocal \Phi^4_d measures that were considered in our previous work when d=2. The analysis when d=3 is more involved. Some of the techniques that I helped develop in our earlier work have to be applied further in this context.
Collaborator Contribution Trishen Gunaratnam has explained to us the methods from from from stochastic analysis that are used in this framework. We hope to unify them with methods from perturbation theory.
Impact We hope to produce a publication as a result of this collaboration.
Start Year 2021
 
Description Gibbs measures for the nonlinear Schrödinger equation and KMS states, with Zied Ammari, Shahnaz Farhat, and Andrew Rout 
Organisation University of Rennes 1
Country France 
Sector Academic/University 
PI Contribution My colleague Zied Ammari invited me to give a seminar at the University of Rennes about the work on Gibbs measures that I did jointly with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein. In later discussion, he explained to me the connection with KMS states. This seems to be a promising new direction that gives us new insight into the problem. I have tried to connect our earlier work to the theory that I have learned from my collaborator.
Collaborator Contribution Zied Ammari introduced me to the study of KMS states. It is quite a powerful theory. We plan to continue our collaboration, involving our PhD students Shahnaz Farhat and Andrew Rout. In particular, part of the grant will be used to finance the visit of Shahnaz Farhat to the University of Warwick in May-June, 2023.
Impact We have one publication so far, titled "Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs", which has now been accepted in Revista Matemática Iberoamericana (see URL above).
Start Year 2020
 
Description PhD supervision of Andrew Rout 
Organisation University of Warwick
Department Warwick Mathematics Institute
Country United Kingdom 
Sector Academic/University 
PI Contribution I am acting as the main PhD supervisor of Andrew Rout, who is a second-year PhD student in maths at the University of Warwick. I have suggested a problem for his PhD and I have been meeting with him for weekly discussions.
Collaborator Contribution Andrew has worked through substantial amounts of literature. He has suggested several key ideas for our project.
Impact We have some partial results for our project and we hope to have a preprint at some point in the next year.
Start Year 2019
 
Description PhD supervision of Andrew Rout 
Organisation University of Warwick
Department Warwick Mathematics Institute
Country United Kingdom 
Sector Academic/University 
PI Contribution I am acting as the main PhD supervisor of Andrew Rout, who is a second-year PhD student in maths at the University of Warwick. I have suggested a problem for his PhD and I have been meeting with him for weekly discussions.
Collaborator Contribution Andrew has worked through substantial amounts of literature. He has suggested several key ideas for our project.
Impact We have some partial results for our project and we hope to have a preprint at some point in the next year.
Start Year 2019
 
Description 15th Meeting Groupe de Recherche, DynQua Quantum Dynamics, University of Rennes, February 3, 2023. 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was invited as a plenary speaker to the 15th annual Groupe de Recherche DynQua meeting in Rennes, France.
Year(s) Of Engagement Activity 2023
URL https://perso.univ-rennes1.fr/zied.ammari/gdr-dynqua-rennes/
 
Description Basque Center for Applied Mathematics and University of the Basque Country, Joint Analysis and PDE Seminar, Bilbao, Spain, June 9, 2022. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was invited to give a seminar at the BCAM (Basque Centre for Applied Mathematics) in Bilbao.
Year(s) Of Engagement Activity 2022
URL http://www.bcamath.org/en/seminars/jointbcam-upvehuanalysisandpdeseminar20220609
 
Description Oberwolfach workshop on Deterministic Dynamics and Randomness in PDE, May 22, 2022. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was selected to give a presentation at the workshop Deterministic Dynamics and Randomness in PDE, organised at the MFO Institute in Oberwolfach, Germany
Year(s) Of Engagement Activity 2022
URL https://www.mfo.de/occasion/2221/www_view
 
Description Organisation of conference on many-body quantum mechanics and dispersive PDEs 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I am planning to use my EPSRC New Investigator Award grant to organise an international workshop on many-body quantum mechanics and dispersive PDEs at the University of Warwick, June 12-16, 2023.
Year(s) Of Engagement Activity 2023
 
Description Quantissima in the Serenissima IV, Venice, Italy, August 22--26, 2022. 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave several presentations at the conference "Quantissima in the Serenissima" organised in Venice in August 2022.
Year(s) Of Engagement Activity 2022
URL http://www.mathphys.org/Venice22/
 
Description Texas A&M University, Nonlinear Partial Differential Equations Seminar (via Zoom) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was invited to present the work 'Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs' (arXiv link https://arxiv.org/abs/2102.12202), written jointly with Zied Ammari in the seminar at Texas A&M University. The talk was followed by an interesting discussion with Professors Jonas Lührmann, Edriss Titi, and Anna Mazzucato.
Year(s) Of Engagement Activity 2021
URL https://www.math.tamu.edu/Calendar/listday/index.php?date=20211130
 
Description University of Bielefeld, Oberseminar Analysis, January 18, 2023 (via Zoom). 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was invited to present in the Analysis Seminar at the University of Bielefeld (via Zoom).
Year(s) Of Engagement Activity 2023
 
Description University of Loughborough, Analysis Seminar, December 7, 2022. 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact I was invited to give the Analysis Seminar at the University of Loughborough.
Year(s) Of Engagement Activity 2022
URL https://www.lboro.ac.uk/departments/maths/events/seminars/analysis-seminar/2022/seminar-analysis-202...
 
Description University of Warsaw, Faculty of Physics, Zoom Seminar 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact I gave a seminar talk on the work 'Interacting loop ensembles and Bose gases' (arXiv preprint https://arxiv.org/abs/2012.05110), written jointly with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein. This was a two-hour talk which resulted in a lively and useful discussion with Professors Marcin Napiorkowski and Jan Derezinski.
Year(s) Of Engagement Activity 2021
 
Description University of Warwick, Colloquium (via Zoom). 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other audiences
Results and Impact I gave a presentation to my colleagues in the weekly departmental colloquium. In the talk, I presented results related to the EPSRC proposal. The talk was geared to a general maths audience. Due to the lockdown restrictions, the seminar was given on zoom. The colloquium generated a lively discussion afterwards in which I could discuss my research with colleagues from different fields of maths.
Year(s) Of Engagement Activity 2020
URL https://warwick.ac.uk/fac/sci/maths/research/events/colloquium/2021/abstracts/#Sohinger