Entropy of a fibered dynamical system

The entropy of the geodesic flow associated to a fibered dynamical system is shown to be zero; in particular the entropy of a quantizable dynamical system is zero. An ergodic dynamical system which defines a quantizable dynamical system is outlined.This research was supported in part by NSF GP-20856 A 1     

Dynamical entropy of generalized quantum Markov chains

We compute the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of shift automorphism of generalized quantum Markov chains as defined by Accardi and Frigerio. For any generalized quantum Markov chain defined via a finite set of conditional density amplitudes, we show that the dynamical entropy is equal to the mean entropy.Research supported in part by the Basic Science Research Program, Korean Ministry of Education, 1993–1994.     

Thermodynamic arrow for a mixing system

The purpose of this paper is to study the appearance of time asymmetry in dynamical systems. The systems are harmonic oscillators and a certain mixing flow on the torus. The asymmetry is a kind of frictional force, but we emphasize that the boundary conditions, a usual source of asymmetry in studies of this sort, are taken to be time symmetric. For the mixing flow the response of the system, as reflected in its entropy as a function of time, occurs only subsequent to the friction, while for the oscillators the effects are both before and after. Some general discussion also takes up the question of which of the foregoing systems is a better model of the physical world for purposes of correlating arrows of time.Work supported in part by U.S. Army Research Office (Durham, N.C.).     

Quantum transmission processes and their entropies

In classical information theory, one of the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov-Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes-Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy are done [10, 3, 8, 39, 15, 44, 9, 27, 28, 2, 19, 45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S- mixing entropy by Ohya [26, 27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S- mixing entropy. In this paper, we discuss the efficiency of information transmission to calculate the mean mutual entropy with respect to the modulated initial states and the connected channel for the quantum dynamical systems.     

Dynamical entropy and the third law of thermodynamics

We show that the third law of thermodynamics holds if the equilibrium states cluster appropriately in space and time uniformly for small temperatures. In this case, the entropy density and the dynamical entropy of the shift coincide. As an example in a one-dimensional lattice system with short-range interactions these cluster conditions are verified.Work supported in part by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P6077P.     

Space-time ergodic properties of systems of infinitely many independent particles

We investigate the ergodic properties of the equilibrium states of systems of infinitely many particles with respect to the group generated by space translations and time evolution. The particles are assumed to move independently in a periodic external field. We show that insofar as good thermodynamic behavior is concerned these properties provide much sharper discrimination than the ergodic properties of the time evolution alone. In particular, we show that though the infinite ideal gas is mixing in the space-time framework, it has vanishing space-time entropy and fails to be a space-timeK-system. On the other hand, if the particles interact with fixed convex scatterers (the Lorentz gas) the system forms a space-timeK-system. Also, the space-time entropy of a system of the type we consider is shown to equal its time entropy per unit volume.Research supported in part by the National Science Foundation Grant No. GP-16147 A No. 1.     

Entropy of self-gravitating radiation

We examine the entropy of self-gravitating radiation confined to a spherical box of radiusR in the context of general relativity. We expect that configurations (i.e., initial data) which extremize total entropy will be spherically symmetric, time symmetric distributions of radiation in local thermodynamic equilibrium. Assuming this is the case, we prove that extrema ofS coincide precisely with static equilibrium configurations of the radiation fluid. Furthermore, dynamically stable equilibrium configurations are shown to coincide with local maxima ofS. The equilibrium configurations and their entropies are calculated and their properties are discussed. However, it is shown that entropies higher than these local extrema can be achieved and, indeed, arbitrarily high entropies can be attained by configurations inside of or outside but arbitrarily near their own Schwarzschild radius. However, if we limit consideration to configurations which are outside their own Schwarzschild radius by at least one radiation wavelength, then the entropy is bounded and we find S_max MR, whereM is the total mass. This supports the validity for self-gravitating systems of the Bekenstein upper limit on the entropy to energy ratio of material bodies.     

Entropy production and time asymmetry in nonequilibrium fluctuations

The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.     

A class of nonmixing dynamical systems with monotonic semigroup property

In order to illustrate the class of conservative dynamical systems for which a Boltzmann entropy can be obtained under finite coarse-graining [2], we consider dynamical systems defined by the shift transformation on K , where K is any finite set of integers. We give a class of non-Markovian invariant measures that verify the Chapman-Kolmogorov equation (equivalent to a Boltzmann entropy) for any positive stochastic matrix and that are ergodic but not weakly mixing.     

Monte Carlo renormalization group Calculations for polymers

A simple method based on Wilson's renormalization group ideas is applied to calculate the dynamical critical exponentz for polymer chains in different dynamical regimes. It is shown that the Doi-Edwards reptating chain does not belong to the same dynamical universality class as the Rouse chain. The earlier results based on (4 –d, d space dimensionality) expansion for chains with excluded volume effect are recovered without any expansion. When combined with the Monte Carlo techniques, this method results in a simple scheme for calculating the static and dynamic exponents for a polymer chain with a prescribed dynamics. Numerical results suggest that the slithering snake model of Wall and Mandel for the dynamics is in a different dynamic universality class than the Rouse chain.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.Research supported in part by the National Science Foundation (Grant No. DMR-8112968) and the Petroleum Research Fund, administered by the American Chemical Society.     

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.     

Entropy inequalities

Some inequalities and relations among entropies of reduced quantum mechanical density matrices are discussed and proved. While these are not as strong as those available for classical systems they are nonetheless powerful enough to establish the existence of the limiting mean entropy for translationally invariant states of quantum continuous systems.Work supported by National Science Foundation Grant GP-9414.     

A Half-space Problem for the Boltzmann Equation with Specular Reflection Boundary Condition

There are many open problems on the stability of nonlinear wave patterns to the Boltzmann equation even though the corresponding stability theory has been comparatively well-established for the gas dynamical systems. In this paper, we study the nonlinear stability of a rarefaction wave profile to the Boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut-off. The analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection, and the decomposition on both the solution and the Boltzmann equation introduced in [24, 26] for energy method.Research supported by the RGC Competitive Earmarked Research Grant, CityU 1142/01P.Research supported by the JSPS Research Fellowship for Foreign Researchers, the National Natural Science Foundation of China (10329101, 10431060), the National Key Program for Basic Research of China under grant 2002CCA03700, and the grant from the Chinese Academy of Sciences entitled Yin Jin Guo Wai Jie Chu Ren Cai Ji Jin.     

On Rényi Permutation Entropy

Among various modifications of the permutation entropy defined as the Shannon entropy of the ordinal pattern distribution underlying a system, a variant based on Rényi entropies was considered in a few papers. This paper discusses the relatively new concept of Rényi permutation entropies in dependence of non-negative real number q parameterizing the family of Rényi entropies and providing the Shannon entropy for q=1. Its relationship to Kolmogorov–Sinai entropy and, for q=2, to the recently introduced symbolic correlation integral are touched.     

The distribution of Lyapunov exponents: Exact results for random matrices

Simple exact expressions are derived for all the Lyapunov exponents of certainN-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution asN. In the case of the time-ordered product integral of exp[N ^–1/2 dW], where the entries of theN×N matrixW(t) are independent standard Wiener processes, the exponents are equally spaced for fixedN and thus have a uniform distribution as N.John S. Guggenheim Memorial Fellow. Research supported in part by NSF Grant MCS 80-19384     

Dynamical Yang-Baxter Equation and Quantum Vector Bundles

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, etc. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and its Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups.The research is supported in part by the Israel Academy of Sciences grant no. 8007/99-03, the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center Group Theoretic Methods in the study of Algebraic Varieties of the Israel Science foundation, and by Russian Foundation for Basic Research grant no. 03-01-00593.Deceased January 2004Acknowledgement We are grateful to J. Bernstein, V. Ostapenko, and S. Shnider for stimulating discussions within the Quantum groups seminar at the Department of Mathematics, Bar Ilan University. We appreciate useful remarks by M. Gorelik, V. Hinich, and A. Joseph during a talk at the Weizmann Institute. Our special thanks to P. Etingof for his comments on various aspects of the subject.     

A nonequilibrium entropy for dynamical systems

It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the timet. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy ast+ and the fine-grained entropy ast–. In particular, the conditions can always be satisfied if the system is aK-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.Research supported in part by NSF grants PHY78-03816 and PHY78-15920.     

The four laws of black hole mechanics

Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area A of the event horizon and the surface gravity of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which correspond to and in some ways transcend the four laws of thermodynamics.Research supported in part by the National Science Foundation.     

Statistical Mechanical and Thermodynamic Entropies of the Einstein-Maxwell Dilaton-Axion Black Hole

We discuss the connection between the statistical mechanical andthermodynamic entropies due to the nonminimally coupled scalar fields on theEinstein-Maxwell dilaton-axion black hole spacetime. It is demonstrated thatalthough the statistical mechanical entropy and one-loop correction to thethermodynamic entropy are equivalent for coupling 0, the presence ofthe bare pure geometrical contribution excludes the possibility to identify thestatistical mechanical entropy with the thermodynamic entropy if we use thestandard renormalization scheme.     

Observer-Dependence of Chaos Under Lorentz and Rindler Transformations

The behavior of Lyapunov exponents and dynamical entropies h, whose positivity characterizes chaotic motion, under Lorentz and Rindler transformations is studied. Under Lorentz transformations, and h are changed, but their positivity is preserved for chaotic systems. Under Rindler transformations, and h are changed in such a way that systems, which are chaotic for an accelerated Rindler observer, can be nonchaotic for an inertial Minkowski observer. Therefore, the concept of chaos is observer-dependent.