@article{13225, abstract = {Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only |⋅|V^∈ℓ1(Z3). Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.}, author = {Benedikter, Niels P and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {4}, publisher = {Springer Nature}, title = {{Correlation energy of a weakly interacting Fermi gas with large interaction potential}}, doi = {10.1007/s00205-023-01893-6}, volume = {247}, year = {2023}, } @article{10551, abstract = {The Dean–Kawasaki equation—a strongly singular SPDE—is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N≫1. The singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean–Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean–Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in N−1 (in suitable weak metrics). In other words, the Dean–Kawasaki equation may be interpreted as a “recipe” for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.}, author = {Cornalba, Federico and Fischer, Julian L}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {5}, publisher = {Springer Nature}, title = {{The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles}}, doi = {10.1007/s00205-023-01903-7}, volume = {247}, year = {2023}, } @article{10224, abstract = {We investigate the Fröhlich polaron model on a three-dimensional torus, and give a proof of the second-order quantum corrections to its ground-state energy in the strong-coupling limit. Compared to previous work in the confined case, the translational symmetry (and its breaking in the Pekar approximation) makes the analysis substantially more challenging.}, author = {Feliciangeli, Dario and Seiringer, Robert}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {3}, pages = {1835–1906}, publisher = {Springer Nature}, title = {{The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics}}, doi = {10.1007/s00205-021-01715-7}, volume = {242}, year = {2021}, } @article{10549, abstract = {We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on \mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale \varepsilon >0, we establish homogenization error estimates of the order \varepsilon in case d\geqq 3, and of the order \varepsilon |\log \varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence \varepsilon ^\delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/\varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,\alpha } regularity theory is available.}, author = {Fischer, Julian L and Neukamm, Stefan}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, keywords = {Mechanical Engineering, Mathematics (miscellaneous), Analysis}, number = {1}, pages = {343--452}, publisher = {Springer Nature}, title = {{Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems}}, doi = {10.1007/s00205-021-01686-9}, volume = {242}, year = {2021}, } @article{7650, abstract = {We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa(2ϱ2−ϱ20). Here ϱ denotes the density of the system and ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.}, author = {Deuchert, Andreas and Seiringer, Robert}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {6}, pages = {1217--1271}, publisher = {Springer Nature}, title = {{Gross-Pitaevskii limit of a homogeneous Bose gas at positive temperature}}, doi = {10.1007/s00205-020-01489-4}, volume = {236}, year = {2020}, } @article{8130, abstract = {We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ε. The interaction is non-negative and scaled in such a way that its scattering length is of order ε/N, while its range is proportional to (ε/N)β with scaling parameter β∈(0,1]. We consider the simultaneous limit (N,ε)→(∞,0) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter β. For β∈(0,1), we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β=1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.}, author = {Bossmann, Lea}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {11}, pages = {541--606}, publisher = {Springer Nature}, title = {{Derivation of the 2d Gross–Pitaevskii equation for strongly confined 3d Bosons}}, doi = {10.1007/s00205-020-01548-w}, volume = {238}, year = {2020}, } @article{6617, abstract = {The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: a sample of the random material is chosen—the representative volume—and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well; for example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in materials science, Le Bris, Legoll and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide a rigorous analysis of this selection approach for representative volumes in the context of stochastic homogenization of linear elliptic equations. In particular, we prove that the method essentially never performs worse than a random selection of the material sample and may perform much better if the selection criterion for the material samples is chosen suitably.}, author = {Fischer, Julian L}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {2}, pages = {635–726}, publisher = {Springer}, title = {{The choice of representative volumes in the approximation of effective properties of random materials}}, doi = {10.1007/s00205-019-01400-w}, volume = {234}, year = {2019}, } @article{6002, abstract = {The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove the existence of a phase transition in this model and provide its phase diagram.}, author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan Philip}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {3}, pages = {1037--1090}, publisher = {Springer Nature}, title = {{The Bogoliubov free energy functional I: Existence of minimizers and phase diagram}}, doi = {10.1007/s00205-018-1232-6}, volume = {229}, year = {2018}, }