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Abstract: Abstract We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase in entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase in the specific (mean) energy and entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest. PubDate: 2023-03-18

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Abstract: Abstract In a recent work Dappiaggi (Commun Contemp Math 24:2150075, 2022), a novel framework aimed at studying at a perturbative level a large class of nonlinear, scalar, real, stochastic PDEs has been developed and inspired by the algebraic approach to quantum field theory. The main advantage is the possibility of computing the expectation value and the correlation functions of the underlying solutions accounting for renormalization intrinsically and without resorting to any specific regularization scheme. In this work, we prove that it is possible to extend the range of applicability of this framework to cover also the stochastic nonlinear Schrödinger equation in which randomness is codified by an additive, Gaussian, complex white noise. PubDate: 2023-03-17

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Abstract: Abstract Ideas and results in the quantum theory of experiments are reviewed. To fix ideas, a concrete example of indirect measurements, an experiment devised by Guerlin et al. (Nature 448(7156):889–893, 2007), and theoretical interpretations thereof (Bauer and Bernard, Phys Rev A 84(4):044103, 2011; Bauer et al., Ann H Poincaré 14:639–679, 2013) are recalled. Subsequently two important elements of the Copenhagen interpretation of quantum mechanics, viz. the von Neumann- and the Lüders measurement postulates, are recalled and rendered more precise. Next, a model originally proposed by Gisin et al. (Phys Rev Lett 52:1657–1660, 1984) is described and shown to imply these postulates. It is then used to provide a theoretical description of the experiment in Guerlin et al. (Nature 448(7156):889–893, 2007) involving a “Heisenberg cut” differing from the one invoked in (Bauer and Bernard, Phys Rev A 84(4):044103, 2011; Bauer et al., Ann H Poincaré 14:639–679, 2013; J Stat Phys 162:924–958, 2016). Some technical issues in the analysis of Gisin’s model are elaborated upon. The paper concludes with remarks on a general principle that implies a universal law governing the stochastic time evolution of states of individual physical systems featuring events and leading to a solution of the so-called measurement problem in quantum mechanics. PubDate: 2023-03-16

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Abstract: Abstract The anomaly cancellation is a basic property of the Standard Model, crucial for its consistence. We consider a lattice chiral gauge theory of massless Wilson fermions interacting with a non-compact massive U(1) field coupled with left- and right-handed fermions in four dimensions. We prove in the infinite volume limit, for weak coupling and inverse lattice step of the order of boson mass, that the anomaly vanishes up to subleading corrections and under the same condition as in the continuum. The proof is based on a combination of exact Renormalization Group, non-perturbative decay bounds of correlations and lattice symmetries. PubDate: 2023-03-13

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Abstract: Abstract We study the slowly varying, non-autonomous quantum dynamics of a translation-invariant spin or fermion system on the lattice \(\mathbb {Z}^d\) . This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in Jakšić et al. (Approach to equilibrium in translation-invariant quantum systems: some structural results, in the present issue of Ann. H. Poincaré, 2023), we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work. PubDate: 2023-03-13

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Abstract: Abstract We provide a new construction of infraparticle states in the massless Nelson model. The approximating sequence of our infraparticle state does not involve any infrared cut-offs. Its derivative w.r.t. the time parameter t is given by a simple explicit formula. The convergence of this sequence as \(t\rightarrow \infty \) to a nonzero limit is then obtained by the Cook method combined with stationary phase estimates. To apply the latter technique, we exploit recent results on regularity of ground states in the massless Nelson model, which hold in the low coupling regime. PubDate: 2023-03-08

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Abstract: Abstract In this paper, we study the localization length of the \(1+1\) continuum directed polymer, defined as the distance between the endpoints of two paths sampled independently from the quenched polymer measure. We show that the localization length converges in distribution in the thermodynamic limit, and derive an explicit density formula of the limiting distribution. As a consequence, we prove the \(\tfrac{3}{2}\) -power law decay of the density, confirming the physics prediction of Hwa and Fisher (Phys Rev B 49(5):3136, 1994). Our proof uses the recent result of Das and Zhu (Localization of the continuum directed random polymer, 2022). PubDate: 2023-03-02

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Abstract: Abstract We give the asymptotic growth of the number of primitive periodic trajectories of a two-dimensional dispersive billiard, when we prescribe their number of bounces on one of the obstacles. PubDate: 2023-03-01

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Abstract: Abstract We establish some local and global well-posedness for Hartree–Fock equations of N particles (HFP) with Cauchy data in Lebesgue spaces \(L^p \cap L^2 \) for \(1\le p \le \infty \) . Similar results are proven for fractional HFP in Fourier–Lebesgue spaces \( \widehat{L}^p \cap L^2 \ (1\le p \le \infty ).\) On the other hand, we show that the Cauchy problem for HFP is mildly ill-posed if we simply work in \(\widehat{L}^p \ (2<p\le \infty )\) . Analogue results hold for reduced HFP. In the process, we prove the boundedeness of various trilinear estimates for Hartree type non linearity in these spaces which may be of independent interest. As a consequence, we get natural \(L^p\) and \(\widehat{L}^p\) extension of classical well-posedness theories of Hartree and Hartree–Fock equations with Cauchy data in just \(L^2-\) based Sobolev spaces. PubDate: 2023-03-01

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Abstract: Abstract We consider the 1d Schrödinger operator with decaying random potential and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions, which is based on the formulation by Rifkind and Virág (Geom Funct Anal 28:1394–1419, 2018). As a result, we have completely different behavior depending on the decaying rate \(\alpha > 0\) of the potential: The limiting measure is equal to (1) Lebesgue measure for the supercritical case ( \(\alpha > 1/2\) ), (2) a measure of which the density has power-law decay with Brownian fluctuation for critical case ( \(\alpha =1/2\) ), and (3) the delta measure with its atom being uniformly distributed for the subcritical case ( \(\alpha <1/2\) ). This result is consistent with the previous study on spectral and statistical properties. PubDate: 2023-03-01

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Abstract: Abstract We study the ground states of the pieces’ model in the Fermi-Dirac statistics in the thermodynamic limit. In other words, we consider the minimizing configurations of n interacting fermions in an interval [0, L] divided into pieces by a Poisson point process, when \( \frac{n}{L}\rightarrow \rho >0 \) as \( L \rightarrow \infty \) . We notice that a decomposition into groups of pieces arises from the hypothesis of finite-range pairwise interaction. Under assumptions of convexity and non-degeneracy of the subsystems, we get an almost complete factorization of any ground state. This method applies at least for groups comprising one or two particles. It improves the expansion of the thermodynamic limit of the ground state energy per particle up to the error \( O(\rho ^{2-\delta }) \) , with \( 0<\delta <1 \) [see Klopp and Veniaminov (in: Frontiers in Analysis and Probability, Springer, Cham, 2020)]. It also provides an approximate ground state for the pieces’ model. PubDate: 2023-03-01

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Abstract: Abstract Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups. PubDate: 2023-03-01

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Abstract: Abstract We investigate the relation between the localization of generalized Wannier bases and the topological properties of two-dimensional gapped quantum systems of independent electrons in a disordered background, including magnetic fields, as in the case of Chern insulators and quantum Hall systems. We prove that the existence of a well-localized generalized Wannier basis for the Fermi projection implies the vanishing of the Chern character, which is proportional to the Hall conductivity in the linear response regime. Moreover, we state a localization dichotomy conjecture for general non-periodic gapped quantum systems. PubDate: 2023-03-01

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Abstract: Abstract It is well-known that for usual Schrödinger operators weakly coupled bound states exist in dimensions one and two, whereas in higher dimensions the famous Cwikel–Lieb–Rozenblum bound holds. We show for a large class of Schrödinger-type operators with general kinetic energies that these two phenomena are complementary. We explicitly get a natural semi-classical type bound on the number of bound states precisely in the situation when weakly coupled bound states exist not. PubDate: 2023-03-01

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Abstract: Abstract We consider a model of branching Brownian motion with self-repulsion. Self-repulsion is introduced via a change of measure that penalises particles spending time in an \(\epsilon \) -neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable, and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time-dependent reaction term. PubDate: 2023-03-01

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Abstract: Abstract It is a fundamental problem in mathematical physics to derive macroscopic transport equations from microscopic models. In this paper, we derive the linear Boltzmann equation in the low-density limit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer configurations. Previously this result was known only for the single-scatterer problem on the flat torus, and for uniformly random scatterer configurations where no damping is required. The damping is critical in establishing convergence—in the absence of damping the limiting behaviour depends on the exact configuration under consideration, and indeed, the linear Boltzmann equation is not expected to appear for periodic and other highly ordered configurations. PubDate: 2023-03-01

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Abstract: Abstract A finite quantum system \(\mathrm{S}\) is coupled to a thermal, bosonic reservoir \(\mathrm{R}\) . Initial \(\mathrm{S}\mathrm{R}\) states are possibly correlated, obtained by applying a quantum operation taken from a large class, to the uncoupled equilibrium state. We show that the full system–reservoir dynamics is given by a Markovian term plus a correlation term, plus a remainder small in the coupling constant \(\lambda \) uniformly for all times \(t\ge 0\) . The correlation term decays polynomially in time, at a speed independent of \(\lambda \) . After this, the Markovian term becomes dominant, where the system evolves according to the completely positive, trace-preserving semigroup generated by the Davies generator, while the reservoir stays stationary in equilibrium. This shows that (a) after initial \(\mathrm{S}\mathrm{R}\) correlations decay, the \(\mathrm{S}\mathrm{R}\) dynamics enters a regime where both the Born and Markov approximations are valid, and (b) the reduced system dynamics is Markovian for all times, even for correlated \(\mathrm{S}\mathrm{R}\) initial states. PubDate: 2023-03-01

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Abstract: Abstract We treat the ultraviolet problem for polaron-type models in nonrelativistic quantum field theory. Assuming that the dispersion relations of particles and the field have the same growth at infinity, we cover all subcritical (superrenormalisable) interactions. The Hamiltonian without cutoff is exhibited as an explicit self-adjoint operator obtained by a finite iteration procedure. The cutoff Hamiltonians converge to this operator in the strong resolvent sense after subtraction of a perturbative approximation for the ground-state energy. PubDate: 2023-02-28

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Abstract: Abstract Surface quasi-geostrophy (SQG) describes the two-dimensional active transport of a temperature field in a strongly stratified and rotating environment. Besides its relevance to geophysics, SQG bears formal resemblance with various flows of interest for turbulence studies, from passive scalar and Burgers to incompressible fluids in two and three dimensions. This analogy is here substantiated by considering the turbulent SQG regime emerging from deterministic and smooth initial data prescribed by the superposition of a few Fourier modes. While still unsettled in the inviscid case, the initial value problem is known to be mathematically well-posed when regularised by a small viscosity. In practice, numerics reveal that in the presence of viscosity, a turbulent regime appears in finite time, which features three of the distinctive anomalies usually observed in three-dimensional developed turbulence: (i) dissipative anomaly, (ii) multifractal scaling, and (iii) super-diffusive separation of fluid particles, both backward and forward in time. These three anomalies point towards three spontaneously broken symmetries in the vanishing viscosity limit: scale invariance, time reversal, and uniqueness of the Lagrangian flow, a fascinating phenomenon that Krzysztof Gawȩdzki dubbed spontaneous stochasticity. In the light of Gawȩdzki’s work on the passive scalar problem, we argue that spontaneous stochasticity and irreversibility are intertwined in SQG and provide numerical evidence for this connection. Our numerics, though, reveal that the deterministic SQG setting only features a tempered version of spontaneous stochasticity, characterised in particular by non-universal statistics. PubDate: 2023-02-23

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Abstract: Abstract The study of physical effects of quatum fields in black hole spacetimes, which is related to questions such as the validity of the strong cosmic censorship conjecture, requires a Hadamard state describing the physical situation. Here, we consider the theory of a free scalar field on a Kerr-de Sitter spacetime, focussing on spacetimes with sufficiently small angular momentum of the black hole and sufficiently small cosmological constant. We demonstrate that an extension of the Unruh state, which describes the expected late-time behaviour in spherically symmetric gravitational collapse, can be rigorously constructed for the free scalar field on such Kerr-de Sitter spacetimes. In addition, we show that this extension of the Unruh state is a Hadamard state in the black hole exterior and in the black hole interior up to the inner horizon. This provides a physically motivated Hadamard state for the study of free scalar fields in rotating black hole spacetimes. PubDate: 2023-02-21