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Journal articles on the topic 'Entropy map'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

KOSSAKOWSKI, A., M. OHYA, and N. WATANABE. "QUANTUM DYNAMICAL ENTROPY FOR COMPLETELY POSITIVE MAP." Infinite Dimensional Analysis, Quantum Probability and Related Topics 02, no. 02 (June 1999): 267–82. http://dx.doi.org/10.1142/s021902579900014x.

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A dynamical entropy for not only shift but also completely positive (CP) map is defined by generalizing the AOW entropy1 defined through quantum Markov chain and AF entropy defined by a finite operational partition. Our dynamical entropy is numerically computed for several models.

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2

Ma, Dongkui, and Bin Cai. "TOPOLOGICAL ENTROPY OF PROPER MAP." Taiwanese Journal of Mathematics 18, no. 4 (August 2014): 1219–41. http://dx.doi.org/10.11650/tjm.18.2014.3339.

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3

Dolgopyat, D. "Entropy of coupled map lattices." Journal of Statistical Physics 86, no. 1-2 (January 1997): 377–89. http://dx.doi.org/10.1007/bf02180211.

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4

Fullwood, James, and Arthur J. Parzygnat. "The Information Loss of a Stochastic Map." Entropy 23, no. 8 (August 8, 2021): 1021. http://dx.doi.org/10.3390/e23081021.

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We provide a stochastic extension of the Baez–Fritz–Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure that we call conditional information loss. Although not functorial, these information measures are semi-functorial, a concept we introduce that is definable in any Markov category. We also introduce the notion of an entropic Bayes’ rule for information measures, and we provide a characterization of conditional entropy in terms of this rule.

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5

Bjørke, Jan T. "Framework for Entropy-based Map Evaluation." Cartography and Geographic Information Systems 23, no. 2 (January 1996): 78–95. http://dx.doi.org/10.1559/152304096782562136.

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6

BYSZEWSKI, JAKUB, FRYDERYK FALNIOWSKI, and DOMINIK KWIETNIAK. "Transitive dendrite map with zero entropy." Ergodic Theory and Dynamical Systems 37, no. 7 (March 8, 2016): 2077–83. http://dx.doi.org/10.1017/etds.2015.136.

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Hoehn and Mouron [Hierarchies of chaotic maps on continua. Ergod. Th. & Dynam. Sys.34 (2014), 1897–1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn–Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Entropy estimates for transitive maps on trees. Topology40(3) (2001), 551–569].

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7

Xiong, Jincheng. "A CHAOTIC MAP WITH TOPOLOGICAL ENTROPY." Acta Mathematica Scientia 6, no. 4 (October 1986): 439–43. http://dx.doi.org/10.1016/s0252-9602(18)30503-4.

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8

Boling, Jess, Casey Kelleher, and Jeffrey Streets. "Entropy, stability and harmonic map flow." Transactions of the American Mathematical Society 369, no. 8 (April 24, 2017): 5769–808. http://dx.doi.org/10.1090/tran/6949.

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9

Fox, Ronald F. "Entropy evolution for the Baker map." Chaos: An Interdisciplinary Journal of Nonlinear Science 8, no. 2 (June 1998): 462–65. http://dx.doi.org/10.1063/1.166327.

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10

Sun, Tai Xiang. "Topological entropy of a graph map." Acta Mathematica Sinica, English Series 34, no. 2 (November 28, 2017): 194–208. http://dx.doi.org/10.1007/s10114-017-7236-6.

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11

Perić, Milan. "Polynomial Entropy of the Logistic Map." Studia Scientiarum Mathematicarum Hungarica 58, no. 2 (June 29, 2021): 206–15. http://dx.doi.org/10.1556/012.2021.58.2.1494.

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We study the polynomial entropy of the logistic map depending on a parameter, and we calculate it for almost all values of the parameter. We show that polynomial entropy distinguishes systems with a low complexity (i.e. for which the topological entropy vanishes).

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12

Hellmund, Meis. "Entanglement and output entropy of the diagonal map." Quantum Information and Computation 13, no. 5&6 (May 2013): 379–92. http://dx.doi.org/10.26421/qic13.5-6-2.

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We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real $N=3$ states $\omega(z)$ and solve the case $z<0$ where $z$ is the non-diagonal entry in the density matrix. We also report a surprising result about the behavior of the output entropy of the diagonal map for arbitrary dimensions $N$; showing a bifurcation at $N=6$.

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13

Galias, Zbigniew. "On Topological Entropy of Finite Representations of the Hénon Map." International Journal of Bifurcation and Chaos 29, no. 13 (December 10, 2019): 1950175. http://dx.doi.org/10.1142/s021812741950175x.

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The topological entropy of finite representations of the Hénon map is studied. Efficient methods to compute the topological entropy of finite representations of maps are presented. Accurate finite representations of the Hénon map and its iterates are constructed and the topological entropy of these representations is calculated. The relation between the topological entropy of the Hénon map and the topological entropy of its finite representations is discussed.

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14

ROGA, WOJCIECH, KAROL ŻYCZKOWSKI, and MARK FANNES. "ENTROPIC CHARACTERIZATION OF QUANTUM OPERATIONS." International Journal of Quantum Information 09, no. 04 (June 2011): 1031–45. http://dx.doi.org/10.1142/s0219749911007794.

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We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system, there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 ⊗ Ψ2 holds.

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15

Acosta, D., P. Fernández de Cordóba, J. M. Isidro, and J. L. G. Santander. "A holographic map of action onto entropy." Journal of Physics: Conference Series 361 (May 10, 2012): 012027. http://dx.doi.org/10.1088/1742-6596/361/1/012027.

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16

Schack, Rüdiger, and Carlton M. Caves. "Information and entropy in the baker’s map." Physical Review Letters 69, no. 23 (December 7, 1992): 3413–16. http://dx.doi.org/10.1103/physrevlett.69.3413.

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17

Baldovin, F., C. Tsallis, and B. Schulze. "Nonstandard entropy production in the standard map." Physica A: Statistical Mechanics and its Applications 320 (March 2003): 184–92. http://dx.doi.org/10.1016/s0378-4371(02)01584-4.

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18

Ren, YunLi, LianFa He, JinFeng Lü, and GuoPing Zheng. "Topological r-entropy and measure-theoretic r-entropy of a continuous map." Science China Mathematics 54, no. 6 (February 23, 2011): 1197–205. http://dx.doi.org/10.1007/s11425-011-4181-1.

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19

BLOCK, LOUIS, ALEXANDER M. BLOKH, and ETHAN M. COVEN. "ZERO ENTROPY PERMUTATIONS." International Journal of Bifurcation and Chaos 05, no. 05 (October 1995): 1331–37. http://dx.doi.org/10.1142/s0218127495001009.

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The entropy of a permutation is the (topological) entropy of the "connect-the-dots" map determined by it. We give matrix- and graph-theoretic, geometric, and dynamical characterizations of zero entropy permutations, as well as a procedure for constructing all of them. We also include some information about the number of zero entropy permutations.

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20

MISIUREWICZ, MICHAŁ, and PIOTR ZGLICZYŃSKI. "TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS." International Journal of Bifurcation and Chaos 11, no. 05 (May 2001): 1443–46. http://dx.doi.org/10.1142/s021812740100281x.

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21

Huynh, Van, Adel Ouannas, Xiong Wang, Viet-Thanh Pham, Xuan Nguyen, and Fawaz Alsaadi. "Chaotic Map with No Fixed Points: Entropy, Implementation and Control." Entropy 21, no. 3 (March 14, 2019): 279. http://dx.doi.org/10.3390/e21030279.

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A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map.

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22

CLIMENHAGA, VAUGHN, and DANIEL J. THOMPSON. "Intrinsic ergodicity via obstruction entropies." Ergodic Theory and Dynamical Systems 34, no. 6 (April 3, 2013): 1816–31. http://dx.doi.org/10.1017/etds.2013.16.

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AbstractBowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then there is a unique measure of maximal entropy.

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23

Kaiser, Susanna, Maria Garcia Puyol, and Patrick Robertson. "Measuring the Uncertainty of Probabilistic Maps Representing Human Motion for Indoor Navigation." Mobile Information Systems 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/9595306.

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Indoor navigation and mapping have recently become an important field of interest for researchers because global navigation satellite systems (GNSS) are very often unavailable inside buildings. FootSLAM, a SLAM (Simultaneous Localization and Mapping) algorithm for pedestrians based on step measurements, addresses the indoor mapping and positioning problem and can provide accurate positioning in many structured indoor environments. In this paper, we investigate how to compare FootSLAM maps via two entropy metrics. Since collaborative FootSLAM requires the alignment and combination of several individual FootSLAM maps, we also investigate measures that help to align maps that partially overlap. We distinguish between the map entropy conditioned on the sequence of pedestrian’s poses, which is a measure of the uncertainty of the estimated map, and the entropy rate of the pedestrian’s steps conditioned on the history of poses and conditioned on the estimated map. Because FootSLAM maps are built on a hexagon grid, the entropy and relative entropy metrics are derived for the special case of hexagonal transition maps. The entropy gives us a new insight on the performance of FootSLAM’s map estimation process.

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24

WANG, YI, and FANG-YUE CHEN. "THE ENTROPY OF STATIONARY SOLUTIONS' MAP OF CELLULAR NEURAL NETWORKS." International Journal of Bifurcation and Chaos 14, no. 12 (December 2004): 4317–23. http://dx.doi.org/10.1142/s0218127404011831.

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In this paper, the entropy of the stationary solutions' map of one-dimensional Cellular Neural Networks with threshold is restudied. Under certain parameters, the map is topological conjugate to a Beruonulli shift of certain symbolic space. Further, the topological entropy of the map can be obtained explicitly as a spacial devil-staircase function.

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25

Zhu, Zhan Long, Gong Liu Yang, Yan Yong Wang, and Yuan Yuan Liu. "Geomagnetic Reference Map Denoising Based on Singular Entropy." Applied Mechanics and Materials 543-547 (March 2014): 912–16. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.912.

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To weaken the noise disturbance of GRM and improve the matching precision and matching probability of inertial/geomagnetic system, this paper proposed a method for denoising based on SVD. Firstly, from the perspective of information entropy, the singular entropy is introduced and the inner link between singular entropy and signal-to-noise ratio (SNR) is analyzed. Secondly, the method based on the asymptotic characteristic of the probabilities associated with the different singular values order (SVO) is proposed. Lastly, by utilizing practical GRM, the denoising analysis about the proposed method is demonstrated and later simulation experiments of GMN are accomplished. Simulation results show that the method is feasible and reliable.

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26

Wolf, Christian. "A shift map with a discontinuous entropy function." Discrete & Continuous Dynamical Systems - A 40, no. 1 (2020): 319–29. http://dx.doi.org/10.3934/dcds.2020012.

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27

Csirmaz, László. "Using multiobjective optimization to map the entropy region." Computational Optimization and Applications 63, no. 1 (June 16, 2015): 45–67. http://dx.doi.org/10.1007/s10589-015-9760-6.

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28

Wu–ming, Liu, Ying Yang–jun, Chen Shi–gang, and He Xian–tu. "Symbolic Dynamics and Topological Entropy of Henon Map." Communications in Theoretical Physics 15, no. 1 (January 1991): 1–8. http://dx.doi.org/10.1088/0253-6102/15/1/1.

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29

Cánovas, J. S. "A chaotic interval map with zero sequence entropy." aequationes mathematicae 64, no. 1-2 (August 2002): 53–61. http://dx.doi.org/10.1007/s00010-002-8030-8.

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30

HASSELBLATT, BORIS, and JAMES PROPP. "Degree-growth of monomial maps." Ergodic Theory and Dynamical Systems 27, no. 5 (October 2007): 1375–97. http://dx.doi.org/10.1017/s0143385707000168.

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AbstractFor projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

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31

Luo, Lvlin. "The Topological Entropy Conjecture." Mathematics 9, no. 4 (February 3, 2021): 296. http://dx.doi.org/10.3390/math9040296.

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For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇp(X;Z), where 0≤p≤n=nJ. For a continuous self-map f on X, let α∈J be an open cover of X and Lf(α)={Lf(U)|U∈α}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);U⊂X}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages f−n(U) for n∈N. Then, we prove the conjecture logρ≤entL(f) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f*, which is the linear transformation associated with f on the Čech homology group Hˇ*(X;Z)=⨁i=0nHˇi(X;Z).

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32

EBELING, WERNER, JAN FREUND, and KATJA RATEITSCHAK. "ENTROPY AND EXTENDED MEMORY IN DISCRETE CHAOTIC DYNAMICS." International Journal of Bifurcation and Chaos 06, no. 04 (April 1996): 611–25. http://dx.doi.org/10.1142/s0218127496000308.

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We investigate simple one-dimensional maps which allow for exact solutions of their related statistical properties. In addition to the originally refined dynamical description a coarsegrained level of description based on certain partitions of the phase space is selected. The deterministic micropscopic dynamics is shifted to a stochastic symbolic dynamics. The higher order entropies are studied for the logistic map, the tent map, and the shark fin map. Markov sources of any prescribed order are constructed explicitly. In a special case, long memory tails are observed. Systems of this type may be of interest for modelling naturally ocurring phenomena.

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33

Berta, Mario, Marius Lemm, and Mark M. Wilde. "Monotonicity of quantum relative entropy and recoverability." Quantum Information and Computation 15, no. 15&16 (November 2015): 1333–54. http://dx.doi.org/10.26421/qic15.15-16-5.

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The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, 2014] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, 2003]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.

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34

Steeb, Willi-Hans. "Maximum Entropy Formalism and Genetic Algorithms." Zeitschrift für Naturforschung A 61, no. 10-11 (November 1, 2006): 556–58. http://dx.doi.org/10.1515/zna-2006-10-1106.

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35

Glykos, Nicholas M., and Michael Kokkinidis. "GraphEnt: a maximum-entropy program with graphics capabilities." Journal of Applied Crystallography 33, no. 3 (June 1, 2000): 982–85. http://dx.doi.org/10.1107/s0021889800004246.

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A maximum-entropy formalism aimed at the production of a `maximally noncommittal' map is a standard method in fields of science like radioastronomy, but a rare exception in both X-ray crystallography and electron microscopy (or crystallography). This is rather unfortunate, given the wealth of information that a maximum-entropy map can reveal, especially when the map itself is the end product (for example, low-resolution electron or potential density maps, Patterson functions, deformation maps). The programGraphEntattempts to automate the procedure of calculating maximum-entropy maps, with emphasis on the calculation of difference Patterson functions for macromolecular crystallographic problems, while providing a useful graphical output of the current stage of the calculation.

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36

Deng, He, Chao Pan, Tongxin Wen, and Jianguo Liu. "Entropy Flow-Aided Navigation." Journal of Navigation 64, no. 1 (November 26, 2010): 109–25. http://dx.doi.org/10.1017/s0373463310000354.

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Integrating the visual navigation mechanism of flying insects with a nonlinear Kalman filter, this paper proposes a novel navigation algorithm. New concepts of entropic map and entropy flow are presented, which can characterize topographic features and measure changes of the image respectively. Meanwhile, an auto-selecting algorithm of assessment threshold is proposed to improve computational accuracy and efficiency of global motion estimation. The simulation results suggest that the navigation algorithm can perform real-time rectification of the missile's trajectory well, and can reduce the cost of the missile's hardware.

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37

YANG, XIAO-SONG, and QINGDU LI. "ON ENTROPY OF CHUA'S CIRCUITS." International Journal of Bifurcation and Chaos 15, no. 05 (May 2005): 1823–28. http://dx.doi.org/10.1142/s0218127405012818.

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In this paper we revisit the well-known Chua's circuit and give a discussion on entropy of this circuit. We present a formula for the topological entropy of a Chua's circuit in terms of the Poincaré map derived from the ordinary differential equations of this Chua's circuit by computer simulation arguments.

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38

Sbitnev, Valery I. "Chaos Structure and Spiral Wave Self-Organization in 2D Coupled Map Lattice." International Journal of Bifurcation and Chaos 08, no. 12 (December 1998): 2341–52. http://dx.doi.org/10.1142/s0218127498001881.

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Methods borrowed from nonequilibrium thermodynamics and statistical physics have been employed in the quantitative analysis of spatiotemporal chaos in a 2D coupled map lattice (CML). Emphasis is made on entropy, entropy variation and entropy production. These quantities manifest peculiar changes in a region where spiral waves emerge. The spiral waves observed in the 2D CML are found to be dissipative objects with an elevated entropy production.

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39

Zhao, Xi Qing, Wen Ying Zhang, Peng Sun, and Li Jun Wang. "The Adjustment of Texture Mapping Based on Image-Entropy." Advanced Materials Research 765-767 (September 2013): 2866–69. http://dx.doi.org/10.4028/www.scientific.net/amr.765-767.2866.

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Mass terrain in four fork tree index structure Xia, draws Shi needs for effective of texture map, viewpoint of moved and texture retrieved Shi scene dynamic update directly effect to displayed efficiency, used macro points block match micro-texture map of method, effective to reduced has data of frequency refresh; introduced image entropy reflect image color distribution of space features, to improve original terrain texture, to out a more meet actual of terrain draws method; experimental indicates that in mass terrain texture map process in the, Increases the level of authenticity and real-time terrain rendering

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40

Coven, Ethan M., and Melissa C. Hidalgo. "On the topological entropy of transitive maps of the interval." Bulletin of the Australian Mathematical Society 44, no. 2 (October 1991): 207–13. http://dx.doi.org/10.1017/s0004972700029634.

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The topological entropy of a continuous map of the interval is the supremum of the topological entropies of the piecewise linear maps associated to its finite invariant sets. We show that for transitive maps, this supremum is attained at some finite invariant set if and only if the map is piecewise monotone and the set contains the endpoints of the interval and the turning points of the map.

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41

Kruglikov, Boris, and Martin Rypdal. "A piece-wise affine contracting map with positive entropy." Discrete & Continuous Dynamical Systems - A 16, no. 2 (2006): 393–94. http://dx.doi.org/10.3934/dcds.2006.16.393.

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42

Shlyakhtenko, Dimitri. "Free entropy with respect to a completely positive map." American Journal of Mathematics 122, no. 1 (2000): 45–82. http://dx.doi.org/10.1353/ajm.2000.0005.

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43

D'Or, Dimitri, and Patrick Bogaert. "Continuous-valued map reconstruction with the Bayesian Maximum Entropy." Geoderma 112, no. 3-4 (March 2003): 169–78. http://dx.doi.org/10.1016/s0016-7061(02)00304-x.

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44

Torres, H. M., J. A. Gurlekian, H. L. Rufiner, and M. E. Torres. "Self-organizing map clustering based on continuous multiresolution entropy." Physica A: Statistical Mechanics and its Applications 361, no. 1 (February 2006): 337–54. http://dx.doi.org/10.1016/j.physa.2005.05.073.

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45

Olbrich, Eckehard, Rainer Hegger, and Holger Kantz. "Local Estimates for Entropy Densities in Coupled Map Lattices." Physical Review Letters 84, no. 10 (March 6, 2000): 2132–35. http://dx.doi.org/10.1103/physrevlett.84.2132.

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46

VanHulle, M. M. "Entropy-Based Kernel Mixture Modeling for Topographic Map Formation." IEEE Transactions on Neural Networks 15, no. 4 (July 2004): 850–58. http://dx.doi.org/10.1109/tnn.2004.828763.

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47

Andries, J., M. Fannes, P. Tuyls, and R. Alicki. "The dynamical entropy of the quantum Arnold cat map." Letters in Mathematical Physics 35, no. 4 (December 1995): 375–83. http://dx.doi.org/10.1007/bf00750844.

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48

ŠPITALSKÝ, VLADIMÍR. "Topological entropy of transitive dendrite maps." Ergodic Theory and Dynamical Systems 35, no. 4 (November 14, 2013): 1289–314. http://dx.doi.org/10.1017/etds.2013.97.

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AbstractWe show that every dendrite$X$satisfying the condition that no subtree of$X$contains all free arcs admits a transitive, even exactly Devaney chaotic map with arbitrarily small entropy. This gives a partial answer to a question of Baldwin from 2001.

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49

MISIUREWICZ, MICHAŁ. "ENTROPY OF MAPS WITH HORIZONTAL GAPS." International Journal of Bifurcation and Chaos 14, no. 04 (April 2004): 1489–92. http://dx.doi.org/10.1142/s0218127404010047.

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We study the behavior of topological entropy in one-parameter families of interval maps obtained from a continuous map f by truncating it at the level depending on the parameter. When f is piecewise monotone, the entropy function has the devil's staircase structure.

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DIÓSI, LAJOS, TOVA FELDMANN, and RONNIE KOSLOFF. "ON THE EXACT IDENTITY BETWEEN THERMODYNAMIC AND INFORMATIC ENTROPIES IN A UNITARY MODEL OF FRICTION." International Journal of Quantum Information 04, no. 01 (February 2006): 99–104. http://dx.doi.org/10.1142/s0219749906001645.

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Abstract:

We consider an elementary collision model of a molecular reservoir upon which an external field is applied and the work is dissipated into heat. To realize macroscopic irreversibility at the microscopic level, we introduce a "graceful" irreversible map which randomly mixes the identities of the molecules. This map is expected to generate informatic entropy exactly equal to the independently calculable irreversible thermodynamic entropy.

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