Large deviations for the local particle densities

We analyze the relations between the large deviation principle of the “local” particle densities of the x− and k−spaces respectively. Here the k−space means the space of momentums (the Fourier transform counterpart of the x− space). This study gives new insights on the results of papers [2], where the authors have found the corresponding large deviation principle of the local particle density in the x− space. In particular, for a very large class of stable Hamiltonians we show that the “local” particle densities (x− and k−spaces) are equal to each other from the point of view of the large deviation principle. In other words, the “local” particle densities in the x− and k−spaces are in this case exponentially equivalent [1]. Applying this result to the specific case of the Perfect Bose Gas, we found an alternative proof to the one done in [2]

A New Microscopic Theory of Superfluidity at all Temperatures

Following the program suggested in [1], we get a new microscopic theory of superfluidity for all temperatures and densities. In particular, the corresponding phase diagram of this theory exhibits: (i) a thermodynamic behavior corresponding to the Mean-Field Gas for small densities or high temperatures, (ii) the ”Landau-type” excitation spectrum in the presence of non-conventional Bose condensation for high densities or small temperatures, (iii) a coexistence of particles inside and outside the condensate with the formation of “Cooper pairs”, even at zero-temperature (experimentally, an estimate of the fraction of condensate in liquid 4 He at T=0 K is 9 %, see [2, 3]). In contrast to Bogoliubov’s last approach and with the caveat that the full interacting Hamiltonian is truncated, the analysis performed here is rigorous by involving for the first time a complete thermodynamic analysis of a non-trivial continuous gas in the canonical ensemble

Critical Analysis of the Bogoliubov Theory of Superfluidity

The microscopic theory of superfluidity [1—3] was proposed by Bogoliubov in 1947 to explain the Landau-type excitation spectrum of He-4. An analysis of the Bogoliubov theory has already been performed in the recent review [4]. Here we add some new critical analyses of this theory. This leads us to consider the superstable Bogoliubov model [5]. It gives rise to an improvement of the previous theory which will be explained with more details in a next paper [6]: coexistence in the superfluid liquid of particles inside and outside the Bose condensate (even at zero temperature), Bose/Bogoliubov statistics, “Cooper pairs” in the Bose condensate, Landau-type excitation spectrum..

A new theory of superfluidity

The understanding of superfluidity represents one of the most challenging problems in modern physics. From the observations of [1-3], in various respects the Bogoliubov theory [4—8] is not appropriate as the model of superfluidity for Helium 4. His outstanding achievement, i.e., the derivation of the Landau-type excitation spectrum [9, 10] from the full interacting Hamiltonian, is based on a series of recipes or approximations, which were shown to be wrong, even from their starting point [11—14]. We therefore present some very promising new results performed in [15]. In particular, we explain a new theory of superfluidity at all temperatures. At this point we then touch one of the most fascinating problems of contemporary mathematical physics the proof of the existence of superfluidity in interacting (non-dilute) systems

Large Deviations in the Superstable Weakly Imperfect Bose Gas

The superstable Weakly Imperfect Bose Gas {(WIBG)} was originally derived to solve the inconsistency of the Bogoliubov theory of superfluidity. Its grand-canonical thermodynamics was recently solved but not at {point of} the {(first order)} phase transition. This paper proposes to close this gap by using the large deviations formalism and in particular the analysis of the Kac distribution function. It turns out that, as a function of the chemical potential, the discontinuity of the Bose condensate density at the phase transition {point} disappears as a function of the particle density. Indeed, the Bose condensate continuously starts at the first critical particle density and progressively grows but the free-energy per particle stays constant until the second critical density is reached. At higher particle densities, the Bose condensate density as well as the free-energy per particle both increase {monotonously}

Lieb–Robinson Bounds for Multi–Commutators and Applications to Response Theory

We generalize to multi-commutators the usual Lieb–Robinson bounds for commutators. In the spirit of constructive QFT, this is done so as to allow the use of combinatorics of minimally connected graphs (tree expansions) in order to estimate time-dependent multi-commutators for interacting fermions. Lieb–Robinson bounds for multi-commutators are effective mathematical tools to handle analytic aspects of the dynamics of quantum particles with interactions which are non-vanishing in the whole space and possibly time-dependent. To illustrate this, we prove that the bounds for multi-commutators of order three yield existence of fundamental solutions for the corresponding non-autonomous initial value problems for observables of interacting fermions on lattices. We further show how bounds for multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting fermions to external perturbations. All results also apply to quantum spin systems, with obvious modifications. However, we only explain the fermionic case in detail, in view of applications to microscopic quantum theory of electrical conduction discussed here and because this case is technically more involved.FAPESP under Grant 2013/13215-5 Basque Government through the grant IT641-13 SEV-2013-0323 MTM2014-5385

Universal Bounds for Large Determinants from Non–Commutative Ho ̈lder Inequalities in Fermionic Constructive Quantum Field Theory

Efficiently bounding large determinants is an essential step in non–relati- vistic fermionic constructive quantum field theory, because, together with the summability of the interaction and the covariance, it implies the abso- lute convergence of the perturbation expansion of all correlation functions in terms of powers of the strength u ∈ R of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one–particle Hamiltonians. We find the smallest universal determinant bound to be exactly 1. In particular, the convergence of pertur- bation series at u = 0 of any fermionic quantum field theory is ensured by the decay properties of the covariance and the interparticle interaction alone. Our proofs use Ho ̈lder inequalities for general non–commutative Lp–spaces derived by Araki and Masuda [AM]