The essence of the second law is concavity

The second law is based on the property that, whatever function is chosen for the entropy, it must be concave. The relationship between fluctuations from the mean value and the monotonic tendency for the entropy to increase is elucidated. The putative expression for the entropy of a black-hole violates this fundamental tenet.1. Work supported in part by contributions from the Ministry of Public Education (M.P.I.) and the National Science Council (C.N.R.).     

Detecting nonclassicality of light via Lieb's concavity

Nonclassical light states are important for both conceptual and practical reasons: they are basic ingredients in testing and exploring quantum foundations, and are crucial resources in quantum technologies. Various useful criteria have been developed to detect nonclassicality in the literature, and several meaningful measures of nonclassicality have been introduced and measured experimentally. In this work, by use of a non-Hermitian generalization of the Wigner-Yanase-Dyson skew information and playing with operator ordering in evaluating average photon number, we develop a novel family of criteria for detecting nonclassicality of light based on Lieb's concavity, which is a deep and powerful result concerning interaction between quantum states and observables. We elucidate the information-theoretic as well as the physical meaning of the criteria, and illustrate their effectiveness in capturing and quantifying nonclassicality of various important light states.     

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

We revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is

$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) = \left({\rm Tr}\left[\left( \, {\sum\limits_{j=1}^m A_j^p } \, \right) ^{q/p} \right] \right)^{1/q} $

for m positive definite operators A _j . In our earlier paper, we only considered the case q = 1 and proved the concavity of Φ _p,1 for 0 < p ≤ 1 and the convexity for p = 2. We conjectured the convexity of Φ _p,1 for 1 < p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q ≥ 1 and still retain the convexity. Among other things this leads to a definition of an L ^q (L ^p ) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces – which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

     

超对称准经典近似和本征值问题

运用超对称准经典近似方法给出三维谐振子、氢原子的能谱,进而将该方法用于含角坐标的二阶微分方程,得到角动量平方L2的本征值和非中心势的角向本征值.     

Concavity,superadditivity and the second law

Callen's discussion of the entropy maximum principle is extended to cover the case of negative absolute temperatures, and the consequences for the stability conditions of systems exhibiting such temperatures are examined. Further, a physical example is considered to illustrate the fact that the essence of the second law of thermodynamics must be concavity rather than superadditivity.     

Convexity properties of the surface tension and equilibrium crystals

We study the thermodynamic limit of the orientation-dependent surface tension. Under general conditions, which we show to hold true for a large class of lattice systems, we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by R. L. Dobrushin and S. B. Shlosman⁽¹⁾. We discuss some consequences of these results for the equilibrium crystal shape.     

Statistical entropy of a Schwarzschild black hole

A model describing the internal microstates of particles is used to calculate the statistical entropy of a Schwarzschild black hole. The state of the system is described by a nonextensive entropy function which is superadditive and so fails to be concave. A strict maximum of the entropy does not exist; nonetheless, the entropy increases on merging two such systems.     

Limit theorem for the distribution of eigenvalues of the operator of energy

We prove the central limit theorem for the distribution of eigenvalues of the energy operator of a continuous quantum mechanical system. We consider one-dimensional and multidimensional systems.     

Operator method and eigenvalues of the triple-well anharmonic oscillator

Using an algebraic technique based on the Lie algebra of the group SO(2, 1) the eigenvalues of the triple-well anharmonic oscillator are calculated for all ranges of the coupling constant. Approximate analytic formulas for the eigenvalues up to second order are given. The results obtained are compared with those of other authors.     

Analytic continuation of harmonic sums

We present a method for calculating any (nested) harmonic sum to arbitrary accuracy for all complex values of the argument. The method utilizes the relation between harmonic sums and (derivatives of) Hurwitz zeta functions, which allows a harmonic sum to be calculated as an expansion valid for large values of its argument. A program for implementing this method is also provided.     

NMR implementations of Gauss sums

I describe the use of NMR experiments which implement Gauss sums as a method for factoring numbers and discuss whether this approach can be computationally useful.     

Evaluation of certain statistical sums

Summary Certain statistical ensembles,e.g. open chemical systems with randomly varying number of particles, are characterized by partition functions of the type ,n being a natural number anda _j ’s generalized temperatures. The state of the system is well defined if one knows the dependence ofa _j ’s on ensemble averages 〈n ^j 〉. For making the equations 〈n ^j 〉=〈n ^j 〉 (a ₁, ...,a _s) at least more accessible for numerical calculations a transformation of the partition function to a series of Fourier integrals is proposed. In the special case of the integrals can be calculated analytically transforming the statistical sum into a series of error functions.     

Metric-Adjusted Skew Information: Convexity and Restricted Forms of Superadditivity

We give a truly elementary proof of the convexity of metric-adjusted skew information following an idea of Effros. We extend earlier results of weak forms of superadditivity to general metric-adjusted skew information. Recently, Luo and Zhang introduced the notion of semi-quantum states on a bipartite system and proved superadditivity of the Wigner–Yanase–Dyson skew informations for such states. We extend this result to the general metric-adjusted skew information. We finally show that a recently introduced extension to parameter values 1 < p ≤ 2 of the WYD-information is a special case of (unbounded) metric-adjusted skew information.     

Network and eigenvalue analysis of financial transaction networks

We study a dataset containing all financial transactions between the accounts of practically all major financial players within Austria over one year. We empirically analyze transaction networks of money (in and out) flows and report the characteristic network parameters. We observe a significant dependence of network topology on the time scales of observation, and remarkably low correlation between node degrees and transaction volume. We further use transaction timeseries of the financial agents to compute covariance matrices and their eigenvalue spectra. Eigenvectors corresponding to eigenvalues deviating from the Marcenko-Pastur law are analyzed in detail. The potential for practical use as an automated detection mechanism for abnormal behavior of financial players is discussed. The opinion expressed in this paper is that of the authors and does not necessarily reflect the opinion of the OeNB or the ESCB.     

The Concavity of Entropy and Extremum Principles in Thermodynamics

We revisit the concavity property of the thermodynamic entropy in order to formulate a general proof of the minimum energy principle as well as of other equivalent extremum principles that are valid for thermodynamic potentials and corresponding Massieu functions under different constraints. The current derivation aims at providing a coherent formal framework for such principles which may be also pedagogically useful as it fully exploits and highlights the equivalence between different schemes. We also elucidate the consequences of the extremum principles for the general shape of thermodynamic potentials in relation to first-order phase transitions.     

Mean Field Dilute Ferromagnet: High Temperature and Zero Temperature Behavior

We study the mean field dilute model of a ferromagnet. We find and prove an expression for the free energy density at high temperature, and at temperature zero. We find the critical line of the model, separating the phase with zero magnetization from the phase with symmetry breaking. We also compute exactly the entropy at temperature zero, which is strictly positive. The physical behavior at temperature zero is very interesting and related to infinite dimensional percolation, and suggests possible behaviors at generic low temperatures. Lastly, we provide a complete solution for a (partially) annealed model. Our results hold both for the Poisson and the Bernoulli versions of the model.     

Homomorphisms and direct sums of nested Hilbert spaces

We prove some elementary facts about homomorphisms and infinite direct sums of nested Hilbert spaces.     

Rapid evaluation of oscillating lattice sums

We have generalized a method for the rapid evaluation of oscillating lattice sums for layered geometries, based on the Ewald's summation technique. Different types of long range interactions as well as different lattice symmetries can be considered, in particular modulated magnetic structures, periodic domain patterns, or periodic lattice distortions. As examples the demagnetizing energies for different thin films systems and magnetic excitations in a single atomic layer are calculated.     

二维耦合量子谐振子的本征值和本征函数

运用广义线性量子变换理论,给出一类二维耦合量子谐振子的能量本征值、本征函数、坐标和动量算符在能量表象中的矩阵元及演化算符.