Periodic Reporting for period 1 - RAMBAS (Rigorous Approximations for Many-Body Quantum Systems)
Periodo di rendicontazione: 2022-10-01 al 2025-03-31
From first principles of quantum mechanics, physical properties of many-body quantum systems are usually encoded into Schr\"odinger equations. However, since the complexity of the Schr\"odinger equations grows so fast with the number of particles, it is generally impossible to solve them by current numerical techniques. Therefore, in practice approximate theories are often applied, which focus only on some collective behaviors of the systems in question.
The corroboration of such effective models largely depends on mathematical methods. The overall goal of RAMBAS is to justify key effective approximations used in many-body quantum physics, including the mean-field, quasi-free, and random-phase approximations, as well as to derive subtle corrections in critical regimes.
Building on my unique expertise in mathematical physics, I will 1) develop general techniques to understand corrections to the mean-field and Bogoliubov approximations for dilute Bose gases, 2) introduce rigorous bosonization methods and combine them with existing techniques from the theory of Bose gases to understand Fermi gases, and 3) employ the bosonization structure of Fermi gases to study the many-body quantum dynamics in long time scales, thus deriving quantum kinetic equations.
By applying and suitably inventing mathematical techniques from functional analysis, spectral theory, calculus of variations and partial differential equations, RAMBAS will take standard approximations of quantum systems to the next level, with special focus on those particularly challenging situations where the particle correlation plays a central role but is yet not adequately addressed. RAMBAS will thereby provide the physics community with crucial mathematical tools, which are at the same time rigorous and applicable.
The corroboration of such effective models largely depends on mathematical methods. The overall goal of RAMBAS is to justify key effective approximations used in many-body quantum physics, including the mean-field, quasi-free, and random-phase approximations, as well as to derive subtle corrections in critical regimes.
Building on my unique expertise in mathematical physics, I will 1) develop general techniques to understand corrections to the mean-field and Bogoliubov approximations for dilute Bose gases, 2) introduce rigorous bosonization methods and combine them with existing techniques from the theory of Bose gases to understand Fermi gases, and 3) employ the bosonization structure of Fermi gases to study the many-body quantum dynamics in long time scales, thus deriving quantum kinetic equations.
By applying and suitably inventing mathematical techniques from functional analysis, spectral theory, calculus of variations and partial differential equations, RAMBAS will take standard approximations of quantum systems to the next level, with special focus on those particularly challenging situations where the particle correlation plays a central role but is yet not adequately addressed. RAMBAS will thereby provide the physics community with crucial mathematical tools, which are at the same time rigorous and applicable.
The project is divided into three work packages (WP). Over the first two years, several key goals have been achieved, as detailed below.
WP1 is devoted to studying the excitation spectrum of dilute Bose gases. The general goal is to obtain a rigorous understanding of the Hartree and Bogoliubov approximations. Concerning the first task of WP1, an asymptotic formula for the free energy of dilute Bose gases at low temperatures has been established in the PI's joint paper with Haberberger, Hainzl, Seiringer, and Triay. This offers a rigorous justification of the celebrated 1957 Lee-Huang-Yang prediction. Moreover, the correlation structure of the low-lying eigenfunctions in the Gross-Pitaevskii regime has been investigated in the PI's joint work with Rademacher, which provides a strong justification for Bose-Einstein condensation (BEC), and in the PI's joint work with Brennecke and Lee, which determines the correction to the BEC. Further results in this direction include Visconti’s papers on dilute Bose gases with three-body interactions.
WP2 is devoted to understanding the correlation energy of Fermi gases via the bosonization method. Concerning the first task of WP2, the derivation of the Gell-Mann–Brueckner formula for Coulomb gas in the mean-field regime has been obtained in the PI's joint work with Christiansen and Hainzl. The derivation of the full Gell-Mann–Brueckner conjecture in the thermodynamic limit remains a major open question. In the second task of WP2, the Huang–Yang formula in the low-density limit has been investigated in the PI's joint work with Giacomelli, Hainzl, and Seiringer.
WP3 is devoted to the derivation of quantum kinetic equations. So far, the PI has focused on the first task concerning the derivation of the fermionic kinetic equation at equilibrium. Some progress has been made jointly with Madsen, Spohn, and Tran. The question of long-time dynamics far from equilibrium remains open.
WP1 is devoted to studying the excitation spectrum of dilute Bose gases. The general goal is to obtain a rigorous understanding of the Hartree and Bogoliubov approximations. Concerning the first task of WP1, an asymptotic formula for the free energy of dilute Bose gases at low temperatures has been established in the PI's joint paper with Haberberger, Hainzl, Seiringer, and Triay. This offers a rigorous justification of the celebrated 1957 Lee-Huang-Yang prediction. Moreover, the correlation structure of the low-lying eigenfunctions in the Gross-Pitaevskii regime has been investigated in the PI's joint work with Rademacher, which provides a strong justification for Bose-Einstein condensation (BEC), and in the PI's joint work with Brennecke and Lee, which determines the correction to the BEC. Further results in this direction include Visconti’s papers on dilute Bose gases with three-body interactions.
WP2 is devoted to understanding the correlation energy of Fermi gases via the bosonization method. Concerning the first task of WP2, the derivation of the Gell-Mann–Brueckner formula for Coulomb gas in the mean-field regime has been obtained in the PI's joint work with Christiansen and Hainzl. The derivation of the full Gell-Mann–Brueckner conjecture in the thermodynamic limit remains a major open question. In the second task of WP2, the Huang–Yang formula in the low-density limit has been investigated in the PI's joint work with Giacomelli, Hainzl, and Seiringer.
WP3 is devoted to the derivation of quantum kinetic equations. So far, the PI has focused on the first task concerning the derivation of the fermionic kinetic equation at equilibrium. Some progress has been made jointly with Madsen, Spohn, and Tran. The question of long-time dynamics far from equilibrium remains open.
Over the first two years, we have provided a satisfactory explanation for the Lee-Huang-Yang conjecture concerning the excitation spectrum of the dilute Bose gas, the Huang-Yang conjecture concerning the correlation energy of the dilute Fermi gas, and the Gell-Mann–Brueckner conjecture concerning the correlation energy of the electron gas in the mean-field regime. All of these are well-known challenging problems in mathematical physics. The new methods and techniques emerging from these works will provide general tools to justify fundamental approximations in many-body quantum physics, bringing them to a mathematically rigorous level.