Coherent Transport and Dynamical Entropy for Fermionic Systems

This paper consists of two parts. First we set up a general scheme of local traps in a homogeneous deterministic quantum system. The current of particles caught by the trap is linked to the dynamical behaviour of the trap states. In this way, transport properties in a homogeneous system are related to spectral properties of a coherent dynamics. Next we apply the scheme to a system of Fermions in the one-particle approximation. We obtain in particular lower bounds for the dynamical entropy in terms of the current induced by the trap.     

The Entropy Formula for SRB-Measures of Lattice Dynamical Systems

We give a detailed proof of the entropy formula for SRB-measures of coupled hyperbolic attractors over integer lattices. We show that the topological pressure for the potential function of the SRB-measure is zero.     

Information Entropy to Probe Revivals in Dynamical Systems

It is shown that sum of information entropies in position and momentum space, quantifies the temporal information in wave packet dynamics of a dynamical system. Quantum fractional revivals are investigated on these bases in periodically driven Fermi-Ulam accelerator. It is observed that the entropic measure provides deeper insight of the wave packet dynamics for the long time evolution as compared with conventional autocorrelation function. It is shown that these revival times are not symmetric in driven situations and may lead to a random behavior.     

Relative Entropy and Identification of Gibbs Measures in Dynamical Systems

In this work we explore the idea of using the relative entropy of ergodic measures for the identification of Gibbs measures in dynamical systems. The question we face is how to estimate the thermodynamic potential (together with a grammar) from a sample produced by the corresponding Gibbs state.     

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

We consider a generic classical many particle system described by an autonomous Hamiltonian H(x ¹,…,x ^N+2) which, in addition, has a conserved quantity V(x ¹,…,x ^N+2)=v, so that the Poisson bracket {H,V} vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over sub-manifolds given by the intersection of the constant energy hyper-surfaces with those defined by V(x ¹,…,x ^N+2)=v. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.     

Hausdorff Dimensions of Zero-Entropy Sets of Dynamical Systems with Positive Entropy

Suppose that (X,T) is a compact positive entropy dynamical system which we mean that X is a compact metric space and T: X→ X is a continuous transformation of X and the topological entropy h(T)>0. A point x ∈ X is called a zero-entropy point provided , where is the forward orbit of x under T and Orb₊(x) is the closure. Let ε⁰(X, T) denote the set of all zero-entropy points. Naturally, one would like to ask the following important question: How big is ε⁰(X, T) for a dynamical system? In this paper, we answer this question. More precisely, we prove that if, furthermore, (X, T) is locally expanding, then the Hausdorff dimension of ε⁰(X, T) vanishes.     

Quantifying Information without Entropy: Identifying Intermittent Disturbances in Dynamical Systems

A system’s response to disturbances in an internal or external driving signal can be characterized as performing an implicit computation, where the dynamics of the system are a manifestation of its new state holding some memory about those disturbances. Identifying small disturbances in the response signal requires detailed information about the dynamics of the inputs, which can be challenging. This paper presents a new method called the Information Impulse Function (IIF) for detecting and time-localizing small disturbances in system response data. The novelty of IIF is its ability to measure relative information content without using Boltzmann’s equation by modeling signal transmission as a series of dissipative steps. Since a detailed expression of the informational structure in the signal is achieved with IIF, it is ideal for detecting disturbances in the response signal, i.e., the system dynamics. Those findings are based on numerical studies of the topological structure of the dynamics of a nonlinear system due to perturbated driving signals. The IIF is compared to both the Permutation entropy and Shannon entropy to demonstrate its entropy-like relationship with system state and its degree of sensitivity to perturbations in a driving signal.     

Modeling Thermostating, Entropy Currents, and Cross Effects by Dynamical Systems

A generalized multibaker map with periodic boundary conditions is shown to model boundary-driven transport, when the driving is applied by a perturbation of the dynamics localized in a macroscopically small region. In this case there are sustained density gradients in the steady state. A non-uniform stationary temperature profile can be maintained by incorporating a heat source into the dynamics, which deviates from the one of a bulk system only in a (macroscopically small) localized region such that a heat (or entropy) flux can enter an attached thermostat only in that region. For these settings the relation between the average phase-space contraction, the entropy flux to the thermostat and irreversible entropy production is clarified for stationary and non-stationary states. In addition, thermoelectric cross effects are described by a multibaker chain consisting of two parts with different transport properties, modeling a junction between two metals.     

Quantum Chaos and Dynamical Entropy

We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More specifically, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may reduce the dynamical entropy of the quantum gas in comparison to free motion. Received: 26 June 1997 / Accepted: 13 April 1998     

Complexity for Extended Dynamical Systems

We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, ϵ-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.     

Geometrical Dissipation for Dynamical Systems

On a Riemannian manifold (M, g) we consider the k?+?1 functions F ₁, . . . , F _k , G and construct the vector fields that conserve F ₁, . . . , F _k and dissipate G with a prescribed rate. We study the geometry of these vector fields and prove that they are of gradient type on regular leaves corresponding to F ₁, . . . , F _k . By using these constructions we show that the cubic Morrison dissipation and the Landau-Lifschitz equation can be formulated in a unitary form.     

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.Research supported in part by NSF Vigre Grant DMS-0091946Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158Acknowledgement The authors are grateful for the support of the Banff International Research Station, at which we formulated and proved Theorem 3.     

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy hs（Ф） of a quantum dynamical systems Ф = （ L, s, Ф）, where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy hs（ Ф, A） of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism Ф, we prove a few results on that, define the entropy of a dynamical system hs（Ф）, and show its invariance. The concept of sufficient families is also given and we establish that hs （Ф） comes out to be equal to the supremum of hs （Ф,A）, where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system （ L, s, Ф）, which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system （B, s0, Ф）, where B is a Boolean algebra and so is a state on B.     

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h_s(φ,A) of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism φ, we prove a few results on that, define the entropy of a dynamical system h_s(Φ), and show its invariance. The concept of sufficient families is also given and we establish that h_s(Φ) comes out to be equal to the supremum of h_s(φ,A), where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,φ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B, s₀,φ), where B is a Boolean algebra and s₀ is a state on B.     

Entropy, Superposition and Dynamical Maps

Aspects of quantum entropy and relative quantum entropy are discussed in the Hilbert model. It is shown that finite values of the relative entropy of states implies a superposition relation between the states. The property is studied in case of tensor product of states and for state reductions. A “Schmidt-like” state, derived from the reduced states, is considered. It is shown that its entropy, relative to the product of the reduced states, is not smaller than the entropy of the reduced states. The main existing results concerning the changement of superposition and entropy under dynamical map are recalled in a uniform way. A class of possible dynamical maps, not necessarily linear, is proposed that do not decrease the entropy.     

Continuous Phase Transitions for Dynamical Systems

We study the asymptotic expansion of the topological pressure of one–parameter families of potentials at a point of non-analyticity. The singularity is related qualitatively and quantitatively to non–Gaussian limit laws and to slow decay of correlations with respect to the equilibrium measure.This work was partially supported by NSF grant DMS–0400687.Dedicated to Y. Pesin on the occasion of his 60th birthday     

Optimal Concentration Inequalities for Dynamical Systems

For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.