Quantum Fluctuations and Large Deviation Principle for Microscopic Currents of Free Fermions in Disordered Media

We contribute an extension of large-deviation results obtained in [N.J.B. Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures Appl. 125 (2019) 209] on conductivity theory at atomic scale of free lattice fermions in disordered media. Disorder is modeled by (i) a random external potential, like in the celebrated Anderson model, and (ii) a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric current densities around their (classical) macroscopic value is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Here, the quantum fluctuations of linear response currents are shown to exist in the thermodynamic limit and we mathematically prove that they are related to the rate function of the large deviation principle associated with current densities. We also demonstrate that, in general, they do not vanish (in the thermodynamic limit) and the quantum uncertainty around the macroscopic current density disappears exponentially fast with an exponential rate proportional to the squared deviation of the current from its macroscopic value and the inverse current fluctuation, with respect to growing space (volume) scales.FAPESP (2017/22340-9); CNPq (309723/2020-5); by the Basque Government through the grant IT641-13; MTM2017-82160-C2-2-

Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems

It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many--fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Omega (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many--fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short--range interaction which can be arbitrarily strong, provided Omega is chosen large enough. Moreover, we give - for the first time - nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency.Comment: 29 pages LaTe

Microscopic Conductivity of Lattice Fermions at Equilibrium - Part I: Non-Interacting Particles

We consider free lattice fermions subjected to a static bounded potential and a time- and space-dependent electric field. For any bounded convex region $\mathcal{R}\subset \mathbb{R}^{d}$ ($d\geq 1$) of space, electric fields $\mathcal{E}$ within $\mathcal{R}$ drive currents. At leading order, uniformly with respect to the volume $\left| \mathcal{R}\right|$ of $\mathcal{R}$ and the particular choice of the static potential, the dependency on $\mathcal{E}$ of the current is linear and described by a conductivity distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of $\mathcal{R}$, in accordance with Ohm's law in Fourier space. This finite measure is the Fourier transform of a time-correlation function of current fluctuations, i.e., the conductivity distribution satisfies Green-Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace-Fourier transform of a so-called quantum current viscosity. The real and imaginary parts of conductivity distributions satisfy Kramers-Kronig relations. At leading order, uniformly with respect to parameters, the heat production is the classical work performed by electric fields on the system in presence of currents. The conductivity measure is uniformly bounded with respect to parameters of the system and it is never the trivial measure $0\,\mathrm{d}\nu$. Therefore, electric fields generally produce heat in such systems. In fact, the conductivity measure defines a quadratic form in the space of Schwartz functions, the Legendre-Fenchel transform of which describes the resistivity of the system. This leads to Joule's law, i.e., the heat produced by currents is proportional to the resistivity and the square of currents

Accuracy of Classical Conductivity Theory at Atomic Scales for Free Fermions in Disordered Media

The growing need for smaller electronic components has recently sparked the interest in the breakdown of the classical conductivity theory near the atomic scale, at which quantum effects should dominate. In 2012, experimental measurements of electric resistance of nanowires in Si doped with phosphorus atoms demonstrate that quantum effects on charge transport almost disappear for nanowires of lengths larger than a few nanometers, even at very low temperature (4.2K). We mathematically prove, for non-interacting lattice fermions with disorder, that quantum uncertainty of microscopic electric current density around their (classical) macroscopic values is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. This is in accordance with the above experimental observation. Disorder is modeled by a random external potential along with random, complex-valued, hopping amplitudes. The celebrated tight-binding Anderson model is one particular example of the general case considered here. Our mathematical analysis is based on Combes-Thomas estimates, the Akcoglu-Krengel ergodic theorem, and the large deviation formalism, in particular the Gärtner-Ellis theorem.This research is supported by CNPq (308337/2017-4), FAPESP (2016/02503-8, 2017/22340-9), as well as by the Basque Government through the grant IT641-13 and the BERC 2018-2022 program, and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2017-0718, MTM2017-82160-C2-2-P

Grand-canonical Thermodynamic Formalism via IFS: volume, temperature, gas pressure and grand-canonical topological pressure

We consider here a dynamic model for a gas in which a variable number of particles $N \in \mathbb{N}_0 := \mathbb{N} \cup \{0\}$ can be located at a site. This point of view leads us to the grand-canonical framework and the need for a chemical potential. The dynamics is played by the shift acting on the set of sequences $\Omega := \mathcal{A}^\mathbb{N}$, where the alphabet is $\mathcal{A} := \mathbb{N} \cup \{0\}$. Introducing new variables like the number of particles $N$ and the chemical potential $\mu$, we adapt the concept of grand-canonical partition sum of thermodynamics of gases to a symbolic dynamical setting considering a Lipschitz family of potentials % (A_N)_{N \in \mathbb{N}_0}, $A_N:\Omega \to \mathbb{R}$. Our main results will be obtained from adapting well-known %properties the results will be obtained through the use of known properties of the Thermodynamic Formalism for IFS with weights to our setting. In this direction, we introduce the grand-canonical transfer (Ruelle) operator: $\mathcal{L}_{\beta, \mu}(f)=g$, when, $\beta>0,\mu<0,$ and $$g(x)= \mathcal{L}_{\beta, \mu}(f) (x) =\sum_{N \in \mathbb{N}_0} e^{\beta \, \mu\, N }\, \sum_{j \in \mathcal{A}} e^{- \,\beta\, A_N(jx)} f(jx).$$ We show the existence of the main eigenvalue, an associated eigenfunction, and an eigenprobability for $\mathcal{L}_{\beta, \mu}$. Considering the concept of entropy for holonomic probabilities on $\mathbb{N}_0\times \Omega$% , we relate these items with the variational problem of maximizing grand-canonical pressure. In another direction, in the appendix, we briefly digress on a possible interpretation of the concept of topological pressure as related to the gas pressure of gas thermodynamics

Parameter estimation of a transformer with saturation using inrush measurements

a b s t r a c t This paper presents a method to compute the parameters of a transformer model with saturation using the voltage and current waveforms of an inrush test and a no-load test. The transformer is modeled with their electric and magnetic equivalent circuits and a single-valued function that characterizes its non-linear magnetic behavior. A 3-kVA single-phase transformer and a 5-kVA three-phase three-legged transformer have been tested in the laboratory. The method to obtain the parameters of the non-linear flux-current relation that characterize the saturation has been described in the paper. The analytical function used to adjust the experimental measurements fits them very well

Double shunt technique for hybrid palliation of hypoplastic left heart syndrome: a case report

We report a technique to palliate hypoplastic left heart syndrome, with no PDA stenting, but with double polytetrafluoroethylene shunt from pulmonary artery to ascending and descending aorta by combined thoracotomies. A 30-day-old female was operated with this technique. Five months after first operation, the child was submitted to Norwood/Glenn operation. Good hemodinamic recovery and initial clinical evolution was observed. The child was extubated in 8th post operatory day and reentubated in the next day due to pulmonary infection. Despite antibiotic treatment, the child died after systemic infectious complications

Large Deviations in Weakly Interacting Fermions - Generating Functions as Gaussian Berezin Integrals and Bounds on Large Pfaffians

We prove that the G\"{a}rtner--Ellis generating function of probability distributions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform Pfaffian bound and to be summable in general cases of interest, including systems that are \emph{not} translation invariant. The Berezin integral representation can thus be used to obtain convergent expansions of the generating function in terms of powers of its parameter. The derivation and analysis of the expansions of logarithms of Berezin integrals are the subject of the second part of the present work. Such technical results are also useful, for instance, in the context of quantum information theory, in the computation of relative entropy densities associated with fermionic Gibbs states, and in the theory of quantum normal fluctuations for weakly interacting fermion systems.FAPESP (2017/22340-9) CNPq (309723/2020-5) by the Basque Government through the grant IT641-13 MTM2017-82160-C2-2-P