KMS States, Entropy and the Variational Principle¶in Full C*-Dynamical Systems

To any periodic and full C ^*-dynamical system , an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius type theorem asserts the existence of KMS states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Unital completely positive maps implemented by partitions of unity {x _j } of grade 1 are considered, resembling the “canonical endomorphism” of the Cuntz algebras. The relationship between the Voiculescu topological entropy of and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary. The measure-theoretic entropy of , in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure-theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy is obtained for the “canonical endomorphism” of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz–Krieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done. Received: 1 February 2000 / Accepted: 23 February 2000     

Information and Entropy Flow in the Kalman–Bucy Filter

We investigate the information theoretic properties of Kalman–Bucy filters in continuous time, developing notions of information supply, storage and dissipation. Introducing a concept of energy, we develop a physical analogy in which the unobserved signal describes a statistical mechanical system interacting with a heat bath. The abstract universe comprising the signal and the heat bath obeys a non-increase law of entropy; however, with the introduction of partial observations, this law can be violated. The Kalman–Bucy filter behaves like a Maxwellian demon in this analogy, returning signal energy to the heat bath without causing entropy increase. This is made possible by the steady supply of new information. In a second analogy the signal and filter interact, setting up a stationary non-equilibrium state, in which energy flows between the heat bath, the signal and the filter without causing any overall entropy increase. We introduce a rate of interactive entropy flow that isolates the statistical mechanics of this flow from marginal effects. Both analogies provide quantitative examples of Landauers Principle.     

Probability Description and Entropy of Classical and Quantum Systems

Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered.     

Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs

We investigate a class of anharmonic crystals in d dimensions, d≥1, coupled to both external and internal heat baths of the Ornstein-Uhlenbeck type. The external heat baths, applied at the boundaries in the 1-direction, are at specified, unequal, temperatures T _l and T _r . The temperatures of the internal baths are determined in a self-consistent way by the requirement that there be no net energy exchange with the system in the non-equilibrium stationary state (NESS). We prove the existence of such a stationary self-consistent profile of temperatures for a finite system and show that it minimizes the entropy production to leading order in (T _l ?T _r ). In the NESS the heat conductivity κ is defined as the heat flux per unit area divided by the length of the system and (T _l ?T _r ). In the limit when the temperatures of the external reservoirs go to the same temperature T, κ(T) is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This κ(T) remains bounded as the size of the system goes to infinity. We also show that the corresponding infinite system Green-Kubo formula yields a finite result. Stronger results are obtained under the assumption that the self-consistent profile remains bounded.     

Δ—scaling and Information Entropy in Ultra—Relativistic Nucleus—Nucleus Collisions

The Δ-scaling method has been applied to ultra-relativistic p p,C C and Pb Pb collision data simulated using a high-energy Monte Carlo package,LUCIAE 3.0.The Δ-scaling is found to be valid for some physical variables,such as charged particle multiplicity,strange particle multiplicity and number of binary nucleon-nucleon-nucleon collisions from these simulated nucleus-nucleus collisions over an extended energy ranging from E1ab=20 to 200A GeV.In addition we derive the information entropy from the multiplicity distribution as a function of beam energy for these collisions.     

Discrete Dynamical Systems¶Associated with Root Systems of Indefinite Type

A geometric charactrization of the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but which exhibits chaotic behavior, is presented. It is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a Cremona isometry on the Picard group of the surface. It is also shown that the group of Cremona isometries is isomorphic to an extended Weyl group of indefinite type. A method to construct the mappings associated with some root systems of indefinite type is also presented. Received: 19 March 2001 / Accepted: 11 July 2001     

Dynamical Equations for Quantum Information and Application in Information Channel

We establish several dynamical equations for quantum information density. It is demonstrated that quantum information density shares the same formalism of the Liouville equation, subdynamics kinetic equation and Fokker-Planck equation as the density operator and also possesses the superposition property. These allow one to use quantum information density directly to model quantum information. The kinetic equations for quantum information density reveal that the dynamical process of quantum information may be related to dissipative, Markovian, or diffusional information flows, together causing irreversibility. Finally, we discuss superposition of quantum information density, which allows us to construct a quantum information channel in the coherent state representation using harmonic oscillator based encoded quantum information, and obtain a formula for quantum dynamical mutual information.     

How does the Entropy/Information Bound Work?

According to the universal entropy bound, the entropy (and hence information capacity) of a complete weakly self-gravitating physical system can be bounded exclusively in terms of its circumscribing radius and total gravitating energy. The bound’s correctness is supported by explicit statistical calculations of entropy, gedanken experiments involving the generalized second law, and Bousso’s covariant holographic bound. On the other hand, it is not always obvious in a particular example how the system avoids having too many states for given energy, and hence violating the bound. We analyze in detail several purported counterexamples of this type, and exhibit in each case the mechanism behind the bound’s efficacy. In memoriam Asher Peres.     

Properties of Segal’s Entropy for Quantum Systems

Properties of Segal’s entropy for semifinite and finite von Neumann algebras are investigated. In particular, its invariance with respect to a trace-preserving normal *-homomorphism is studied, as well as norm-continuity in the trace norm on the set of bounded in the operator norm density matrices.     

Extended Diffeomorphism Algebras¶and Trajectories in Jet Space

Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra?DRO(N) be the extension of diff(N)⊕ diff(1) by its four inequivalent Virasoro-like cocycles. Here diff(N) is the diffeomorphism algebra in N-dimensional spacetime and diff(1) describes reparametrizations of trajectories in the space of tensor-valued p-jets. DRO(N) has a Fock module for each p and each representation of gl(N). Analogous representations for gauge algebras (higher-dimensional Kac–Moody algebras) are also given. The reparametrization symmetry can be eliminated by a gauge fixing procedure, resulting in previously discovered modules. In this process, two DRO(N) cocycles transmute into anisotropic cocycles for diff(N). Thus the Fock modules of toroidal Lie algebras and their derivation algebras are geometrically explained. Received: 29 October 1998 / Accepted: 2 May 2000     

Random Homogenization and Singular Perturbations¶in Perforated Domains

The paper considers the singularly perturbed Dirichlet problem −ɛΔu ^ɛ+u ^ɛ=f in a randomly perforated domain Ω^ɛ, which is obtained from a bounded open set Ω in R ^N after removing many holes of size ɛ ^q . The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Imposing certain conditions on the domain, the behaviour of u ^ɛ when ɛ→ 0 in Lebesgue spaces L ⁿ (Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size ɛ ^p with the intensity λɛ^{− r} is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach. Received: 15 December 1999 / Accepted: 14 April 2000     

Strange Attractors Across the Boundary¶of Hyperbolic Systems

We study a bifurcation of Axiom A (hyperbolic) vector fields in dimension three leading to robust strange attractors with singularities. The Axiom A vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift. The attractors arising from this kind of bifurcation are not equivalent to the geometric Lorenz attractors. Received: 4 December 1998 / Accepted: 5 October 1999     

Low Temperature Phase Diagrams¶of Fermionic Lattice Systems

We consider fermionic lattice systems with Hamiltonian H=H ^{(0)}+λH _Q , where H ^{(0)} is diagonal in the occupation number basis, while H _Q is a suitable “quantum perturbation”. We assume that H ^{(0)} is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitations, while H _Q is a finite range or exponentially decaying Hamiltonian that can be written as a sum of even monomials in the fermionic creation and annihilation operators. Mapping the d dimensional quantum system onto a classical contour system on a d+1 dimensional lattice, we use standard Pirogov–Sinai theory to show that the low temperature phase diagram of the quantum system is a small perturbation of the zero temperature phase diagram of the classical system, provided λ is sufficiently small. Particular attention is paid to the sign problems arising from the fermionic nature of the quantum particles. As a simple application of our methods, we consider the Hubbard model with an additional nearest neighbor repulsion. For this model, we rigorously establish the existence of a paramagnetic phase with commensurate staggered charge order for the narrow band case at sufficiently low temperatures. Received: 23 December 1996/ Accepted: 7 April 1999     

Bootstrap Multiscale Analysis and Localization¶in Random Media

We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e ^{− L ζ}, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 − (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions. Received: 29 November 2000 / Accepted: 21 June 2001     

On Lieb's Conjecture for the Wehrl Entropy¶of Bloch Coherent States

Lieb@s conjecture for the Wehrl entropy of Bloch coherent states is proved for spin 1 and spin 3/2. Using a geometric representation we solve the entropy integrals for states of arbitrary spin and evaluate them explicitly in the cases of spin 1, 3/2, and 2. We also give a group theoretic proof for all spin of a related inequality. Received: 2 March 1999 / Accepted: 7 May 1999     

A Geometric Criterion for Positive Topological Entropy¶II: Homoclinic Tangencies

In a series of important papers [GS1,GS2] Gavrilov and Shilnikov established a topological conjugacy between a surface diffeomorphism having a dissipative hyperbolic periodic point with certain types of quadratic homoclinic tangencies and the full shift on two symbols, thus exhibiting horseshoes near a tangential homoclinic point. In this note, which should be viewed of as an addendum to [BW] we extend this result by showing that such a diffeomorphism with a one-sided isolated homoclinic tangency having any order contact, possible with infinite order contact, possesses a horseshoe near the homoclinic point. Received: 2 March 1999 / Accepted: 14 May 1999     

Information Erasure and Recovery in Quantum Memory

We show that information in quantum memory can be erased and recovered perfectly if it is necessary. The fact that the final states of environment are completely determined by the initial states of the system allows an erasure operation to be realized by a swap operation between the system and an ancilla. Therefore, the erased information can be recovered. When there is an irreversible process, e.g. an irreversible operation or a decoherence process, in the erasure process, the information would be erased perpetually. We present that quantum erasure will also give heat dissipation in the environment. A classical limit of quantum erasure is given to coincide with Landauer‘s erasure principle.     

Squeezing and Entanglement in Continuous Variable Systems

Based on total variance of a pair of Einstein-Podolsky-Rosen (EPR) type operators, the generalized EPR entangled states in continuous variable systems are defined. We show that such entangled states must correspond to two-mode squeezing states whether these states are Gaussian or not and whether they are pure or not. With help of the relation between the total variance and the entanglement, the degree of such entangIement is also defined. Through analysing some specific cases, we see that this method is very convenient and easy in practical applications. In addition, an entangled state with no squeezing is studied, which reveals that there certainly exists something unknown about entanglement in continuous variable systems.