Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 98 tok/s
Gemini 2.5 Pro 58 tok/s Pro
GPT-5 Medium 25 tok/s Pro
GPT-5 High 23 tok/s Pro
GPT-4o 112 tok/s Pro
Kimi K2 165 tok/s Pro
GPT OSS 120B 460 tok/s Pro
Claude Sonnet 4 29 tok/s Pro
2000 character limit reached

Dynamics of quantum discommensurations in the Frenkel-Kontorova chain (2401.12614v1)

Published 23 Jan 2024 in cond-mat.stat-mech, nlin.PS, and quant-ph

Abstract: The ability for real-time control of topological defects can open up prospects for dynamical manipulation of macroscopic properties of solids. A sub-category of these defects, formed by particle dislocations, can be effectively described using the Frenkel-Kontorova chain, which characterizes the dynamics of these particles in a periodic lattice potential. This model is known to host solitons, which are the topological defects of the system and are linked to structural transitions in the chain. This work addresses three key questions: Firstly, we investigate how imperfections present in concrete implementations of the model affect the properties of topological defects. Secondly, we explore how solitons can be injected after the rapid change in lattice potential or nucleated due to quantum fluctuations. Finally, we analyze the propagation and scattering of solitons, examining the role of quantum fluctuations and imperfections in influencing these processes. Furthermore, we address the experimental implementation of the Frenkel-Kontorova model. Focusing on the trapped ion quantum simulator, we set the stage for controllable dynamics of topological excitations and their observation in this platform.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (107)
  1. Pietro Maria Bonetti, Andrea Rucci, Maria Luisa Chiofalo,  and Vladan Vuletić, “Quantum effects in the aubry transition,” Phys. Rev. Research 3, 013031 (2021).
  2. Lukas Hormann, Johannes J. Cartus,  and Oliver T. Hofmann, “Does incommensurability mean superlubricity?”  (2023), arXiv:2304.12427 [cond-mat.mtrl-sci] .
  3. Yi-Wei Li, Peng-Fei Xu,  and Yong-Ge Yang, “Nano-friction phenomenon of frenkel–kontorova model under gaussian colored noise,” Chinese Physics B 31, 050501 (2022).
  4. L Timm, LA Rüffert, H Weimer, L Santos,  and TE Mehlstäubler, “Quantum nanofriction in trapped ion chains with a topological defect,” Physical Review Research 3, 043141 (2021a).
  5. Yun Dong, Fangming Lian, Weibin Hui, Yusong Ding, Zhiyuan Rui, Yi Tao,  and Rong Fu, “Velocity-dependent phononic friction in commensurate and incommensurate states,” Tribology International , 108224 (2023).
  6. Weichao Zheng, “Structural lubricity and fractures in liquid crystals,”  (2023), arXiv:2305.06191 [cond-mat.soft] .
  7. Gourab Kumar Sar, Dibakar Ghosh,  and Kevin O’Keeffe, “Solvable model of driven matter with pinning,”  (2023), arXiv:2306.09589 [nlin.AO] .
  8. O.M. Braun and A.G. Naumovets, “Nanotribology: Microscopic mechanisms of friction,” Surface Science Reports 60, 79–158 (2006).
  9. L. Timm, L. A. Rüffert, H. Weimer, L. Santos,  and T. E. Mehlstäubler, “Quantum nanofriction in trapped ion chains with a topological defect,” Phys. Rev. Res. 3, 043141 (2021b).
  10. A. G. Naumovets and A. G. Fedorus, “Disordering of submonolayer films of electropositive elements adsorbed on metals,” Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 73, 1085–1092 (1977).
  11. D. N. Aristov and A. Luther, “Correlations in the sine-gordon model with finite soliton density,” Phys. Rev. B 65, 165412 (2002).
  12. VL Pokrovsky and AL Talapov, “Theory of incommensurate crystals, harwood acad,” Publ., New York  (1984).
  13. Thomas Bohlein, Jules Mikhael,  and Clemens Bechinger, “Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces,” Nature materials 11, 126–130 (2012).
  14. T. Goto, T. Kimura, G. Lawes, A. P. Ramirez,  and Y. Tokura, “Ferroelectricity and giant magnetocapacitance in perovskite rare-earth manganites,” Phys. Rev. Lett. 92, 257201 (2004).
  15. P. Coullet, “Commensurate-incommensurate transition in nonequilibrium systems,” Phys. Rev. Lett. 56, 724–727 (1986).
  16. R. Schuster, I. K. Robinson, K. Kuhnke, S. Ferrer, J. Alvarez,  and K. Kern, “Pokrovsky-talapov commensurate-incommensurate transition in the co/pd(100) system,” Phys. Rev. B 54, 17097–17101 (1996).
  17. Andrea Vanossi, Nicola Manini, Michael Urbakh, Stefano Zapperi,  and Erio Tosatti, “Colloquium: Modeling friction: From nanoscale to mesoscale,” Reviews of Modern Physics 85, 529 (2013).
  18. CR Woods, Liam Britnell, Axel Eckmann, RS Ma, JC Lu, HM Guo, X Lin, GL Yu, Y Cao, Roman V Gorbachev, et al., “Commensurate–incommensurate transition in graphene on hexagonal boron nitride,” Nature physics 10, 451–456 (2014).
  19. Md Tusher Ahmed, Chenhaoyue Wang, Amartya S Banerjee,  and Nikhil Chandra Admal, “Bicrystallography-informed frenkel-kontorova model for interlayer dislocations in strained 2d heterostructures,” arXiv preprint arXiv:2309.07325  (2023).
  20. P Bak, “Commensurate phases, incommensurate phases and the devil's staircase,” Reports on Progress in Physics 45, 587–629 (1982).
  21. D. N. Aristov, Vadim V. Cheianov,  and A. Luther, “Optical conductivity of one-dimensional doped hubbard-mott insulator,” Phys. Rev. B 66, 073105 (2002).
  22. J Zhang, BT Chow, S Ejtemaee,  and PC Haljan, “Spectroscopic characterization of the quantum linear-zigzag transition in trapped ions,” npj Quantum Information 9, 68 (2023).
  23. Raphaël Menu, Jorge Yago Malo, Vladan Vuletić, Maria Luisa Chiofalo,  and Giovanna Morigi, “Commensurate-incommensurate transition in frustrated wigner crystals,”  (2023), arXiv:2311.14396 [cond-mat.quant-gas] .
  24. H. P. Büchler, G. Blatter,  and W. Zwerger, “Commensurate-incommensurate transition of cold atoms in an optical lattice,” Phys. Rev. Lett. 90, 130401 (2003).
  25. V. Kasper, J. Marino, S. Ji, V. Gritsev, J. Schmiedmayer,  and E. Demler, “Simulating a quantum commensurate-incommensurate phase transition using two raman-coupled one-dimensional condensates,” Phys. Rev. B 101, 224102 (2020).
  26. Elisabeth Wybo, Alvise Bastianello, Monika Aidelsburger, Immanuel Bloch,  and Michael Knap, “Preparing and analyzing solitons in the sine-gordon model with quantum gas microscopes,”  (2023), arXiv:2303.16221 [cond-mat.quant-gas] .
  27. Farokh Mivehvar, Francesco Piazza, Tobias Donner,  and Helmut Ritsch, “Cavity qed with quantum gases: new paradigms in many-body physics,” Advances in Physics 70, 1–153 (2021).
  28. Clemens Neuenhahn, Anatoli Polkovnikov,  and Florian Marquardt, “Localized phase structures growing out of quantum fluctuations in a quench of tunnel-coupled atomic condensates,” Phys. Rev. Lett. 109, 085304 (2012).
  29. Jorge Mellado Muñoz, Rahul Sawant, Anna Maffei, Xi Wang,  and Giovanni Barontini, “Realizing the frenkel-kontorova model with rydberg-dressed atoms,” Phys. Rev. A 102, 043308 (2020).
  30. Ivo A. Maceira, Natalia Chepiga,  and Frédéric Mila, “Conformal and chiral phase transitions in rydberg chains,” Phys. Rev. Res. 4, 043102 (2022).
  31. Natalia Chepiga and Frédéric Mila, “Kibble-zurek exponent and chiral transition of the period-4 phase of rydberg chains,” Nature Communications 12, 414 (2021).
  32. Jin Zhang, Sergio H. Cantú, Fangli Liu, Alexei Bylinskii, Boris Braverman, Florian Huber, Jesse Amato-Grill, Alexander Lukin, Nathan Gemelke, Alexander Keesling, Sheng-Tao Wang, Y. Meurice,  and S. W. Tsai, “Probing quantum floating phases in rydberg atom arrays,”  (2024), arXiv:2401.08087 [quant-ph] .
  33. Haggai Landa, Cecilia Cormick,  and Giovanna Morigi, “Static kinks in chains of interacting atoms,” Condensed Matter 5 (2020), 10.3390/condmat5020035.
  34. Dorian A. Gangloff, Alexei Bylinskii,  and Vladan Vuletić, “Kinks and nanofriction: Structural phases in few-atom chains,” Phys. Rev. Research 2, 013380 (2020).
  35. Andrea Vanossi, Clemens Bechinger,  and Michael Urbakh, “Structural lubricity in soft and hard matter systems,” Nature Communications 11, 1–11 (2020).
  36. K Pyka, J Keller, HL Partner, R Nigmatullin, T Burgermeister, DM Meier, K Kuhlmann, A Retzker, Martin B Plenio, WH Zurek, et al., “Topological defect formation and spontaneous symmetry breaking in ion coulomb crystals,” Nature communications 4, 2291 (2013).
  37. S Ejtemaee and PC Haljan, “Spontaneous nucleation and dynamics of kink defects in zigzag arrays of trapped ions,” Physical Review A 87, 051401 (2013).
  38. Gabriele De Chiara, Adolfo Del Campo, Giovanna Morigi, Martin B Plenio,  and Alex Retzker, “Spontaneous nucleation of structural defects in inhomogeneous ion chains,” New Journal of Physics 12, 115003 (2010).
  39. S Ulm, J Roßnagel, G Jacob, C Degünther, ST Dawkins, UG Poschinger, R Nigmatullin, A Retzker, MB Plenio, F Schmidt-Kaler, et al., “Observation of the kibble–zurek scaling law for defect formation in ion crystals,” Nature communications 4, 2290 (2013).
  40. Oksana Chelpanova, Shane P. Kelly, Giovanna Morigi, Ferdinand Schmidt-Kaler,  and Jamir Marino, “Injection and nucleation of topological defects in the quench dynamics of the frenkel-kontorova model,” Europhysics Letters 143, 25002 (2023).
  41. Ya I Frenkel and TA Kontorova, “see phys. z. sowietunion 13 1,” Zh. Eksp. Teor. Fiz 8, 89 (1938).
  42. F. C. Frank, J. H. van der Merwe,  and Nevill Francis Mott, “One-dimensional dislocations. i. static theory,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 198, 205–216 (1949).
  43. Thaned Pruttivarasin, Michael Ramm, Ishan Talukdar, Axel Kreuter,  and Hartmut Häffner, “Trapped ions in optical lattices for probing oscillator chain models,” New Journal of Physics 13, 075012 (2011).
  44. Ignacio García-Mata, OV Zhirov,  and DL Shepelyansky, “Frenkel-kontorova model with cold trapped ions,” The European Physical Journal D 41, 325–330 (2007).
  45. Mariana Malard, “Sine-gordon model: renormalization group solution and applications,” Brazilian Journal of Physics 43, 182–198 (2013).
  46. David Ruelle, “A review of linear response theory for general differentiable dynamical systems,” Nonlinearity 22, 855 (2009).
  47. Giuseppe Del Vecchio Del Vecchio, Márton Kormos, Benjamin Doyon,  and Alvise Bastianello, “Exact large-scale fluctuations of the phase field in the sine-gordon model,” arXiv preprint arXiv:2305.10495  (2023).
  48. Kerson Huang, Introduction to statistical physics (Chapman and Hall/CRC, 2009).
  49. A. Lazarides, O. Tieleman,  and C. Morais Smith, “Pokrovsky-Talapov model at finite temperature: A renormalization-group analysis,” Phys. Rev. B 80, 245418 (2009).
  50. Milton Abramowitz, Irene A Stegun,  and Robert H Romer, “Handbook of mathematical functions with formulas, graphs, and mathematical tables,”  (1988).
  51. Neil W Ashcroft and N David Mermin, “Festkörperphysik,” in Festkörperphysik (Oldenbourg Wissenschaftsverlag, 2001).
  52. B. Kutnjak-Urbanc and B. Žekš, “Phase-excitation spectrum of ferroelectric liquid crystals in an external static electric field,” Phys. Rev. E 52, 3892–3903 (1995).
  53. Michel Peyrard and Martin D Kruskal, “Kink dynamics in the highly discrete sine-gordon system,” Physica D: Nonlinear Phenomena 14, 88–102 (1984).
  54. Alidad Askari, Aliakbar Moradi Marjaneh, Zhanna G Rakhmatullina, Mahdy Ebrahimi-Loushab, Danial Saadatmand, Vakhid A Gani, Panayotis G Kevrekidis,  and Sergey V Dmitriev, “Collision of ϕitalic-ϕ\phiitalic_ϕ4 kinks free of the peierls–nabarro barrier in the regime of strong discreteness,” Chaos, Solitons & Fractals 138, 109854 (2020).
  55. C. Willis, M. El-Batanouny,  and P. Stancioff, “Sine-gordon kinks on a discrete lattice. i. hamiltonian formalism,” Phys. Rev. B 33, 1904–1911 (1986).
  56. Mark J Ablowitz, Justin T Cole, Pipi Hu,  and Peter Rosenthal, “Peierls-nabarro barrier effect in nonlinear floquet topological insulators,” Physical Review E 103, 042214 (2021).
  57. Heather L Partner, Ramil Nigmatullin, Tobias Burgermeister, Karsten Pyka, Jonas Keller, Alex Retzker, Martin B Plenio,  and Tanja E Mehlstäubler, “Dynamics of topological defects in ion coulomb crystals,” New Journal of Physics 15, 103013 (2013).
  58. Subir Sachdev, “Quantum phase transitions,” Physics world 12, 33 (1999).
  59. Rafael M. Fernandes, Peter P. Orth,  and Jörg Schmalian, “Intertwined vestigial order in quantum materials: Nematicity and beyond,” Annual Review of Condensed Matter Physics 10, 133–154 (2019).
  60. Efrat Shimshoni, Giovanna Morigi,  and Shmuel Fishman, “Quantum zigzag transition in ion chains,” Phys. Rev. Lett. 106, 010401 (2011a).
  61. Yu. N. Gornostyrev, M. I. Katsnelson, A. V. Kravtsov,  and A. V. Trefilov, “Fluctuation-induced nucleation and dynamics of kinks on dislocation: Soliton and oscillation regimes in the two-dimensional frenkel-kontorova model,” Phys. Rev. B 60, 1013–1018 (1999).
  62. Efrat Shimshoni, Giovanna Morigi,  and Shmuel Fishman, “Quantum structural phase transition in chains of interacting atoms,” Phys. Rev. A 83, 032308 (2011b).
  63. Alessio Lerose, Jamir Marino, Bojan Žunkovič, Andrea Gambassi,  and Alessandro Silva, “Chaotic dynamical ferromagnetic phase induced by nonequilibrium quantum fluctuations,” Phys. Rev. Lett. 120, 130603 (2018).
  64. Alessio Lerose, Bojan Žunkovič, Jamir Marino, Andrea Gambassi,  and Alessandro Silva, “Impact of nonequilibrium fluctuations on prethermal dynamical phase transitions in long-range interacting spin chains,” Phys. Rev. B 99, 045128 (2019).
  65. Jamir Marino, Martin Eckstein, Matthew Foster,  and Ana-Maria Rey, “Dynamical phase transitions in the collisionless pre-thermal statesof isolated quantum systems: theory and experiments,” Reports on Progress in Physics  (2022).
  66. William Berdanier, Jamir Marino,  and Ehud Altman, “Universal dynamics of stochastically driven quantum impurities,” Phys. Rev. Lett. 123, 230604 (2019).
  67. John W Negele and Henri Orland, Quantum many–particle systems (CRC Press, 2018).
  68. Anatoli Polkovnikov, “Phase space representation of quantum dynamics,” Annals of Physics 325, 1790–1852 (2010).
  69. Jarrett Lancaster, Emanuel Gull,  and Aditi Mitra, “Quenched dynamics in interacting one-dimensional systems: Appearance of current-carrying steady states from initial domain wall density profiles,” Phys. Rev. B 82, 235124 (2010).
  70. R. Bistritzer and E. Altman, “Intrinsic dephasing in one-dimensional ultracold atom interferometers,” Proceedings of the National Academy of Sciences 104, 9955–9959 (2007).
  71. A. I. Yakimenko, Y. M. Bidasyuk, M. Weyrauch, Y. I. Kuriatnikov,  and S. I. Vilchinskii, “Vortices in a toroidal bose-einstein condensate with a rotating weak link,” Phys. Rev. A 91, 033607 (2015).
  72. K. C. Wright, R. B. Blakestad, C. J. Lobb, W. D. Phillips,  and G. K. Campbell, “Driving phase slips in a superfluid atom circuit with a rotating weak link,” Phys. Rev. Lett. 110, 025302 (2013).
  73. B. A. Kochetov, O. G. Chelpanova, V. R. Tuz,  and A. I. Yakimenko, “Spontaneous and engineered transformations of topological structures in nonlinear media with gain and loss,” Phys. Rev. E 100, 062202 (2019).
  74. Xuekai Ma, Bernd Berger, Marc Aßmann, Rodislav Driben, Torsten Meier, Christian Schneider, Sven Höfling,  and Stefan Schumacher, “Realization of all-optical vortex switching in exciton-polariton condensates,” Nature communications 11, 1–7 (2020).
  75. R Rajaraman, “Solitons and instantons. an introduction to solitons and instantons in quantum field theory. 1. reprint as a paperback,”  (1987).
  76. T. Dauxois and M. Peyrard, Physics of Solitons (Cambridge University Press, 2006).
  77. J. Brox, P. Kiefer, M. Bujak, T. Schaetz,  and H. Landa, “Spectroscopy and directed transport of topological solitons in crystals of trapped ions,” Phys. Rev. Lett. 119, 153602 (2017).
  78. R. Carretero-González, L.A. Cisneros-Ake, R. Decker, G.N. Koutsokostas, D.J. Frantzeskakis, P.G. Kevrekidis,  and D.J. Ratliff, “Kink–antikink stripe interactions in the two-dimensional sine–gordon equation,” Communications in Nonlinear Science and Numerical Simulation 109, 106123 (2022).
  79. Patrick B. Greene, Lev Kofman,  and Alexei A. Starobinsky, “Sine-gordon parametric resonance,” Nuclear Physics B 543, 423–443 (1999).
  80. Michael V Berry and KE Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315 (1972).
  81. Yuri S Kivshar and Boris A Malomed, “Dynamics of solitons in nearly integrable systems,” Reviews of Modern Physics 61, 763 (1989).
  82. Adalto R Gomes, FC Simas, KZ Nobrega,  and PP1402 Avelino, “False vacuum decay in kink scattering,” Journal of High Energy Physics 2018, 1–17 (2018).
  83. Vakhid A Gani, Aliakbar Moradi Marjaneh,  and Danial Saadatmand, “Multi-kink scattering in the double sine-gordon model,” The European Physical Journal C 79, 620 (2019).
  84. Oleg M Braun and Yuri S Kivshar, “Nonlinear dynamics of the frenkel–kontorova model,” Physics Reports 306, 1–108 (1998).
  85. Cyrille Marquet, Ferdinand Schmidt-Kaler,  and Daniel FV James, “Phonon–phonon interactions due to non-linear effects in a linear ion trap,” Applied Physics B 76, 199–208 (2003).
  86. D. von Lindenfels, O. Gräb, C. T. Schmiegelow, V. Kaushal, J. Schulz, Mark T. Mitchison, John Goold, F. Schmidt-Kaler,  and U. G. Poschinger, “Spin heat engine coupled to a harmonic-oscillator flywheel,” Phys. Rev. Lett. 123, 080602 (2019).
  87. Rasmus B. Linnet, Ian D. Leroux, Mathieu Marciante, Aurélien Dantan,  and Michael Drewsen, “Pinning an ion with an intracavity optical lattice,” Phys. Rev. Lett. 109, 233005 (2012).
  88. Marko Cetina, Alexei Bylinskii, Leon Karpa, Dorian Gangloff, Kristin M Beck, Yufei Ge, Matthias Scholz, Andrew T Grier, Isaac Chuang,  and Vladan Vuletić, “One-dimensional array of ion chains coupled to an optical cavity,” New Journal of Physics 15, 053001 (2013).
  89. Bambi Hu and Baowen Li, “Quantum frenkel–kontorova model,” Physica A: Statistical Mechanics and its Applications 288, 81–97 (2000), dynamics Days Asia-Pacific: First International Conference on NonLinear Science.
  90. Giovanna Morigi and Shmuel Fishman, “Dynamics of an ion chain in a harmonic potential,” Phys. Rev. E 70, 066141 (2004).
  91. Daniel H. E. Dubin, “Minimum energy state of the one-dimensional coulomb chain,” Phys. Rev. E 55, 4017–4028 (1997).
  92. M. R. Kamsap, C. Champenois, J. Pedregosa-Gutierrez, S. Mahler, M. Houssin,  and M. Knoop, “Experimental demonstration of an efficient number diagnostic for long ion chains,” Phys. Rev. A 95, 013413 (2017).
  93. M. Mazzanti, R. X. Schüssler, J. D. Arias Espinoza, Z. Wu, R. Gerritsma,  and A. Safavi-Naini, “Trapped ion quantum computing using optical tweezers and electric fields,” Phys. Rev. Lett. 127, 260502 (2021).
  94. Liam Bond, Lisa Lenstra, Rene Gerritsma,  and Arghavan Safavi-Naini, “Effect of micromotion and local stress in quantum simulations with trapped ions in optical tweezers,” Phys. Rev. A 106, 042612 (2022).
  95. L. Timm, H. Weimer, L. Santos,  and T. E. Mehlstäubler, “Heat transport in an ion coulomb crystal with a topological defect,” Phys. Rev. B 108, 134302 (2023).
  96. Elisabeth Wybo, Michael Knap,  and Alvise Bastianello, “Quantum sine-gordon dynamics in coupled spin chains,” Phys. Rev. B 106, 075102 (2022).
  97. H. Landa, S. Marcovitch, A. Retzker, M. B. Plenio,  and B. Reznik, “Quantum coherence of discrete kink solitons in ion traps,” Phys. Rev. Lett. 104, 043004 (2010).
  98. Manuel Mielenz, J Brox, Steffen Kahra, Günther Leschhorn, Magnus Albert, Tobias Schätz, Haggai Landa,  and Benni Reznik, “Trapping of topological-structural defects in coulomb crystals,” Physical review letters 110, 133004 (2013).
  99. Sidney Coleman, “Fate of the false vacuum: Semiclassical theory,” Phys. Rev. D 15, 2929–2936 (1977).
  100. Gianluca Lagnese, Federica Maria Surace, Márton Kormos,  and Pasquale Calabrese, “False vacuum decay in quantum spin chains,” Phys. Rev. B 104, L201106 (2021).
  101. Gianluca Lagnese, Federica Maria Surace, Sid Morampudi,  and Frank Wilczek, “Detecting a long lived false vacuum with quantum quenches,” arXiv preprint arXiv:2308.08340  (2023).
  102. Jens D. Baltrusch, Cecilia Cormick,  and Giovanna Morigi, “Quantum quenches of ion coulomb crystals across structural instabilities,” Phys. Rev. A 86, 032104 (2012).
  103. Jens D. Baltrusch et al., “Quantum superpositions of crystalline structures,” Phys. Rev. A 84, 063821 (2011).
  104. Alexander Streltsov, Gerardo Adesso,  and Martin B. Plenio, “Colloquium: Quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
  105. Shane P. Kelly, Ulrich Poschinger, Ferdinand Schmidt-Kaler, Matthew P. A. Fisher,  and Jamir Marino, “Coherence requirements for quantum communication from hybrid circuit dynamics,”  (2022).
  106. PA Bromiley, NA Thacker,  and E Bouhova-Thacker, “Shannon entropy, renyi entropy, and information,” Statistics and Inf. Series (2004-004) 9, 2–8 (2004).
  107. T.M. Cover and J.A. Thomas, Elements of Information Theory (Wiley, 2012).
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 post and received 0 likes.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up to View