Emergence of equilibrium thermodynamic properties in quantum pure states. II. Analysis of a spin model system

A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.Comment: 30 pages, 10 figure

Signatures of Anderson localization and delocalized random quantum states

We consider the notion of equilibration for an isolated quantum system exhibiting Anderson localization. The system is assumed to be in a pure state, i.e., described by a wave-function undergoing unitary dynamics. We focus on the simplest model of a 1D disordered chain and we analyse both the dynamics of an initially localized state and the dynamics of quantum states drawn at random from the ensemble corresponding to the minimum knowledge about the initial state. While in the former case the site distribution remains confined in a limited portion of the chain, the site distribution of random pure state fluctuates around an equilibrium average that is delocalized over the entire chain. A clear connection between the equilibration observed when the system is initialized in a fully localized state and the amplitude of dynamical fluctuations of a typical random pure state is established

Beyond quantum microcanonical statistics

Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e. the wavefunction, of an isolated system which is determined to calculate molecular properties and to consider the time evolution according to the unitary Schr\"odinger equation. On the other hand a mixed state, i.e. a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem

Pilot-wave quantum theory with a single Bohm's trajectory

The representation of a quantum system as the spatial configuration of its constituents evolving in time as a trajectory under the action of the wave-function, is the main objective of the Bohm theory. However, its standard formulation is referred to the statistical ensemble of its possible trajectories. The statistical ensemble is introduced in order to establish the exact correspondence (the Born's rule) between the probability density on the spatial configurations and the quantum distribution, that is the squared modulus of the wave-function. In this work we explore the possibility of using the pilot wave theory at the level of a single Bohm's trajectory. The pilot wave theory allows a formally self-consistent representation of quantum systems as a single Bohm's trajectory, but in this case there is no room for the Born's rule at least in its standard form. We will show that a correspondence exists between the statistical distribution of configurations along the single Bohm's trajectory and the quantum distribution for a subsystem interacting with the environment in a multicomponent system. To this aim, we present the numerical results of the single Bohm's trajectory description of the model system of six confined rotors with random interactions. We find a rather close correspondence between the coordinate distribution of one rotor along its trajectory and the time averaged marginal quantum distribution for the same rotor. This might be considered as the counterpart of the standard Born's rule. Furthermore a strongly fluctuating behavior with a fast loss of correlation is found for the evolution of each rotor coordinate. This suggests that a Markov process might well approximate the evolution of the Bohm's coordinate of a single rotor and it is shown that the correspondence between coordinate distribution and quantum distribution of the rotor is exactly verified

Emergence of equilibrium thermodynamic properties in quantum pure states. I. Theory

Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schr\"odinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system.Comment: 30 pages, 1 figur

Typicality in Ensembles of Quantum States: Monte Carlo Sampling versus Analytical Approximations

Random Quantum States are presently of interest in the fields of quantum information theory and quantum chaos. Moreover, a detailed study of their properties can shed light on some foundational issues of the quantum statistical mechanics such as the emergence of well defined thermal properties from the pure quantum mechanical description of large many body systems. When dealing with an ensemble of pure quantum states, two questions naturally arise: what is the probability density function on the parameters which specify the state of the system in a given ensemble? And, does there exist a most typical value of a function of interest in the considered ensemble? Here two different ensembles are considered: the Random Pure State Ensemble (RPSE) and the Fixed Expectation Energy Ensemble (FEEE). By means of a suitable parameterization of the wave function in terms of populations and phases, we focus on the probability distribution of the populations in such ensembles. A comparison is made between the distribution induced by the inherent geometry of the Hilbert Space and an approximate distribution derived by means of the minimization of the informational functional. While the latter can be analytically handled, the exact geometrical distribution is sampled by a Metropolis-Hastings algorithm. The analysis is made for an ensemble of wavefunctions describing an ideal system composed of n spins 1/2 and reveals the salient differences between the geometrical and the approximate distributions. The analytical approximations are proven to be useful tools in order to obtain ensemble averaged quantity. In particular we focus on the distribution of the Shannon entropy by providing an explanation of the emergence of a typical value of this quantity in the ensembles.Comment: 27 pages, 7 figure

NuSTAR Spectroscopy of Multi-Component X-ray Reflection from NGC 1068

We report on observations of NGC1068 with NuSTAR, which provide the best constraints to date on its $>10$~keV spectral shape. We find no strong variability over the past two decades, consistent with its Compton-thick AGN classification. The combined NuSTAR, Chandra, XMM-Newton, and Swift-BAT spectral dataset offers new insights into the complex reflected emission. The critical combination of the high signal-to-noise NuSTAR data and a spatial decomposition with Chandra allow us to break several model degeneracies and greatly aid physical interpretation. When modeled as a monolithic (i.e., a single N_H) reflector, none of the common Compton-reflection models are able to match the neutral fluorescence lines and broad spectral shape of the Compton reflection. A multi-component reflector with three distinct column densities (e.g., N_H~1.5e23, 5e24, and 1e25 cm^{-2}) provides a more reasonable fit to the spectral lines and Compton hump, with near-solar Fe abundances. In this model, the higher N_H components provide the bulk of the Compton hump flux while the lower N_H component produces much of the line emission, effectively decoupling two key features of Compton reflection. We note that ~30% of the neutral Fe Kalpha line flux arises from >2" (~140 pc), implying that a significant fraction of the <10 keV reflected component arises from regions well outside of a parsec-scale torus. These results likely have ramifications for the interpretation of poorer signal-to-noise observations and/or more distant objects [Abridged].Comment: Submitted to ApJ; 23 pages (ApJ format); 11 figures and 3 tables; Comments welcomed

Oncologic Outcomes of Incidental Versus Biopsy-diagnosed Grade Group 1 Prostate Cancer:A Multi-institutional Study

Background and objective: Patients diagnosed with grade group (GG) 1 prostate cancer (PCa) following treatment for benign disease (“incidental” PCa) are typically managed with active surveillance (AS). It is not known how their outcomes compare with those observed in patients diagnosed with GG1 on biopsy. We aimed at determining whether long-term oncologic outcomes of AS for patients with GG1 PCa differ according to the type of diagnosis: incidental versus biopsy detected. Methods: A retrospective, multi-institutional analysis of PCa patients with GG1 on AS at eight institutions was conducted. Competing risk analyses estimated the incidence of metastases, PCa mortality, and conversion to treatment. As a secondary analysis, we estimated the risk of GG ≥2 on the first follow-up biopsy according to the type of initial diagnosis. Key findings and limitations: A total of 213 versus 1900 patients with incidental versus biopsy-diagnosed GG1 were identified. Patients with incidental cancers were followed with repeated biopsies and multiparametric magnetic resonance imaging less frequently than those diagnosed on biopsy. The 10-yr incidence of treatment was 22% for incidental cancers versus 53% for biopsy (subdistribution hazard ratio [sHR] 0.34, 95% confidence interval [CI] 0.26–0.46, p &lt; 0.001). Distant metastases developed in one patient with incidental cancer versus 17 diagnosed on biopsy and were diagnosed with molecular imaging in 13 (72%) patients. The 10-yr incidence of metastases was 0.8% for patients with incidental PCa and 2% for those diagnosed on biopsy (sHR 0.35, 95% CI 0.05–2.54, p = 0.3). The risk of GG ≥2 on the first follow-up biopsy was low if the initial diagnosis was incidental (7% vs 22%, p &lt; 0.001). Conclusions and clinical implications: Patients with GG1 incidental PCa should be evaluated further to exclude aggressive disease, preferably with a biopsy. If no cancer is found on biopsy, then they should receive the same follow-up of a patient with a negative biopsy. Further research should confirm whether imaging and biopsies can be avoided if postoperative prostate-specific antigen is low (&lt;1–2 ng/ml). Patient summary: We compared the outcomes of patients with low-grade prostate cancer on active surveillance according to the type of their initial diagnosis. Patients who have low-grade cancer diagnosed on a procedure to relieve urinary symptoms (incidental prostate cancer) are followed less intensively and undergo curative-intended treatment less frequently. We also found that patients with incidental prostate cancer are more likely to have no cancer on their first follow-up biopsy than patients who have low-grade cancer initially diagnosed on a biopsy. These patients have a more favorable prognosis than their biopsy-detected counterparts and should be managed the same way as patients with negative biopsies if they undergo a subsequent biopsy that shows no cancer.</p