Zobrazenie zdroja stránky Bekensteinova hranica

'''Bekensteinova hranica''', vo [[fyzika|fyzike]], predstavuje horný limit [[entropia|entropie]] ''S'' alebo [[informácia|informácie]] ''I'', ktorá môže byt obsiahnutá v určitej danej konečnej oblasti vesmíru, ktorý má určité konečné množstvo energie - alebo obrátene, maximálne množstvo informácií, potrebných na úplné popísanie daného fyzikálneho systému na kvantovej úrovni.<ref name="Bekenstein1972">J. D. Bekenstein, [http://www.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Beken-Entropy.pdf "Black Holes and the Second Law"], ''[[Nuovo Cimento|Lettere al Nuovo Cimento]]'', Vol. 4, No 15 (August 12, 1972), pp. 737-740, {{doi|10.1007/BF02757029}}, {{bibcode|1972NCimL...4..737B}}..</ref><ref name="Bekenstein1973">Jacob D. Bekenstein, [http://www.physics.princeton.edu/~mcdonald/examples/QM/bekenstein_prd_7_2333_73.pdf "Black Holes and Entropy"], ''[[Physical Review|Physical Review D]]'', Vol. 7, No. 8 (April 15, 1973), pp. 2333-2346, {{doi|10.1103/PhysRevD.7.2333}}, {{bibcode|1973PhRvD...7.2333B}}. [http://www.webcitation.org/5pvpyakNu Mirror link].</ref><ref name="Bekenstein1974">Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/PRD9-3292-1974.pdf "Generalized second law of thermodynamics in black-hole physics"], ''[[Physical Review|Physical Review D]]'', Vol. 9, No. 12 (June 15, 1974), pp. 3292-3300, {{doi|10.1103/PhysRevD.9.3292}}, {{bibcode|1974PhRvD...9.3292B}}..</ref><ref name="Bekenstein1975">Jacob D. Bekenstein, [http://www.physics.princeton.edu/~mcdonald/examples/QM/bekenstein_prd_12_3077_75.pdf "Statistical black-hole thermodynamics"], ''[[Physical Review|Physical Review D]]'', Vol. 12, No. 10 (November 15, 1975), pp. 3077-3085, {{doi|10.1103/PhysRevD.12.3077}}, {{bibcode|1975PhRvD..12.3077B}}. [http://www.webcitation.org/5pvqVyq9K Mirror link].</ref><ref name="Bekenstein1980">Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/PT,33,24(1980).pdf "Black-hole thermodynamics"], ''[[Physics Today]]'', Vol. 33, Issue 1 (January 1980), pp. 24-31, {{doi|10.1063/1.2913906}}, {{bibcode|1980PhT....33a..24B}}..</ref><ref name="Bekenstein1981-1">Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdf "Universal upper bound on the entropy-to-energy ratio for bounded systems"], ''[[Physical Review|Physical Review D]]'', Vol. 23, No. 2, (January 15, 1981), pp. 287-298, {{doi|10.1103/PhysRevD.23.287}}, {{bibcode|1981PhRvD..23..287B}}..</ref><ref name="Bekenstein1981-3">Jacob D. Bekenstein, [http://www.physics.princeton.edu/~mcdonald/examples/QM/bekenstein_prl_46_623_81.pdf "Energy Cost of Information Transfer"], ''[[Physical Review Letters]]'', Vol. 46, No. 10 (March 9, 1981), pp. 623-626, {{doi|10.1103/PhysRevLett.46.623}}, {{bibcode|1981PhRvL..46..623B}}. [http://www.webcitation.org/5pvuFpfRA Mirror link].</ref><ref name="Bekenstein1982">Jacob D. Bekenstein, "Specific entropy and the sign of the energy", ''[[Physical Review|Physical Review D]]'', Vol. 26, No. 4 (August 15, 1982), pp. 950-953, {{doi|10.1103/PhysRevD.26.950}}, {{bibcode|1982PhRvD..26..950B}}.</ref><ref name="Bekenstein1984">Jacob D. Bekenstein, [http://128.112.100.2/~kirkmcd/examples/QM/bekenstein_prd_30_1669_84.pdf "Entropy content and information flow in systems with limited energy"], ''[[Physical Review|Physical Review D]]'', Vol. 30, No. 8, (October 15, 1984), pp. 1669-1679, {{doi|10.1103/PhysRevD.30.1669}}, {{bibcode|1984PhRvD..30.1669B}}. [http://www.webcitation.org/5pvud0uay Mirror link].</ref><ref name="Bekenstein1988">Jacob D. Bekenstein, [http://pm1.bu.edu/~tt/qcl/pdf/bekenstj19867766031b.pdf "Communication and energy"] {{Webarchive|url=https://web.archive.org/web/20100813204732/http://pm1.bu.edu/~tt/qcl/pdf/bekenstj19867766031b.pdf |date=2010-08-13 }}, ''[[Physical Review|Physical Review A]]'', Vol. 37, Issue 9 (May 1988), pp. 3437-3449, {{doi|10.1103/PhysRevA.37.3437}}, {{bibcode|1988PhRvA..37.3437B}}..</ref><ref name="SchifferBekenstein1989">Marcelo Schiffer and Jacob D. Bekenstein, "Proof of the quantum bound on specific entropy for free fields", ''[[Physical Review|Physical Review D]]'', Vol. 39, Issue 4 (February 15, 1989), pp. 1109-1115, {{doi|10.1103/PhysRevD.39.1109}}, {{bibcode|1989PhRvD..39.1109S}}.</ref><ref name="Bekenstein1989">Jacob D. Bekenstein, "Is the Cosmological Singularity Thermodynamically Possible?", ''[[International Journal of Theoretical Physics]]'', Vol. 28, Issue 9 (September 1989), pp. 967-981, {{doi|10.1007/BF00670342}}, {{bibcode|1989IJTP...28..967B}}.</ref><ref name="Bekenstein1994">Jacob D. Bekenstein, "Entropy bounds and black hole remnants", ''[[Physical Review|Physical Review D]]'', Vol. 49, Issue 4 (February 15, 1994), pp. 1912-1921, {{doi|10.1103/PhysRevD.49.1912}}, {{bibcode|1994PhRvD..49.1912B}}. Also at {{arxiv|gr-qc|9307035}}, July 25, 1993.</ref><ref name="Zaslavskii1996">Oleg B. Zaslavskii, "Generalized second law and the Bekenstein entropy bound in ''Gedankenexperiments'' with black holes", ''[[Classical and Quantum Gravity]]'', Vol. 13, No. 1 (January 1996), pp. L7-L11, {{doi|10.1088/0264-9381/13/1/002}}, {{bibcode|1996CQGra..13L...7Z}}. See also O. B. Zaslavskii, "Corrigendum to 'Generalized second law and the Bekenstein entropy bound in ''Gedankenexperiments'' with black holes'", ''[[Classical and Quantum Gravity]]'', Vol. 13, No. 9 (September 1996), p. 2607, {{doi|10.1088/0264-9381/13/9/024}}, {{bibcode|1996CQGra..13.2607Z}}.</ref><ref name="Bekenstein1999">Jacob D. Bekenstein, "Non-Archimedean character of quantum buoyancy and the generalized second law of thermodynamics", ''[[Physical Review|Physical Review D]]'', Vol. 60, Issue 12 (December 15, 1999), Art. No. 124010, 9 pages, {{doi|10.1103/PhysRevD.60.124010}}, {{bibcode|1999PhRvD..60l4010B}}. Also at {{arxiv|gr-qc|9906058}}, June 16, 1999.</ref> To znamená, že ak oblasť vesmíru a jej energia je určitá a konečná, tak informácia o fyzikálnom systéme alebo informácie potrebné na úplné popísanie systému musia byť konečné. V prípade [[Veda o počítačoch|IT]] to znamená, že existuje maximálna rýchlosť spracovania informácií pre fyzický systém, ktorý má určitú veľkosť a energiu, a že [[Turingov stroj]], s jeho nekonečnou pamäťou, nie je fyzikálne možný, pokiaľ nemá nekonečnú energiu. Všeobecný tvar bol objavený [[Jacobom Bekenstein]]om pôvodne ako nezrovnalosť.<ref name="Bekenstein1981-1"/><ref name="Bekenstein2005"/><ref name="Bekenstein2008"/> 
 
:<math>S \leq \frac{2 \pi k R E}{\hbar c}</math>

kde ''S'' je [[entropia]], ''k'' je [[Boltzmannova konštanta]], ''R'' je polomer gule, ktorá obklopuje daný systém, ''E'' je maximálna [[Einsteinov vzťah|hmotnosť-energia]] vrátane všetkých zvyškových hmotností, ''ħ'' je [[redukovaná Planckova konštanta]] a ''c'' je [[rýchlosť svetla]]. Zaujímavé je, že zatiaľ čo gravitácia hrá dôležitú úlohu jej posilnení, tak hranica je nezávislá od Newtonovej konštanty G.

V informačnom vzťahu je hranica daná

:<math>I \leq \frac{2 \pi R E}{\hbar c \ln 2}</math>
 
kde ''I'' je [[informácia]] vyjadrená počtom [[bit]]ov obsiahnutých v kvantovom stave gule. [[logaritmus|Ln]]2 faktor pochádza z definície informácie ako logaritmus počtu kvantových stavov so základom 2.<ref name="Tipler2005b">[[Frank J. Tipler]], [http://math.tulane.edu/~tipler/theoryofeverything.pdf "The structure of the world from pure numbers"], ''[[Reports on Progress in Physics]]'', Vol. 68, No. 4 (April 2005), pp. 897-964, {{doi|10.1088/0034-4885/68/4/R04}}, {{bibcode|2005RPPh...68..897T}}, p. 902.. Also released as [http://arxiv.org/abs/0704.3276 "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything"], {{arxiv|0704.3276}}, April 24, 2007, p. 8.</ref> Pravá strana predchádzajúceho vzťahu je približne rovná 2.5769087×10<sup>43</sup>×(hmotnosť v kg)×(polomer v metroch).

Bekenstein odvodil  hranicu z metodických dôkazov zahŕňajúcich [[čierna diera|čierne diery]]. Ak existuje systém porušujúci túto hranicu, tak by bolo možné porušiť [[Druhá termodynamická veta|2. termodynamický zákon]] jeho znížením na úroveň čiernej diery, argumentoval Bekenstein. V roku 1995 [[Ted Jacobson]] demonštroval, že Einstein field equations môžu byť odvodené za predpokladu,že Bekensteinova hranica a zákony termodynamiky platia.<ref name="Jacobson1995">[[Theodore Jacobson|Ted Jacobson]], "Thermodynamics of Spacetime: The Einstein Equation of State", ''[[Physical Review Letters]]'', Vol. 75, Issue 7 (August 14, 1995), pp. 1260-1263, {{doi|10.1103/PhysRevLett.75.1260}}, {{bibcode|1995PhRvL..75.1260J}}. Also at {{arxiv|gr-qc|9504004}}, April 4, 1995. Also available [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.6675 here] and [http://www.math.ohio-state.edu/~gerlach/jacobson/jacobson.ps here]. Additionally available as [http://www.gravityresearchfoundation.org/pdf/awarded/1995/jacobson.pdf an entry] {{Webarchive|url=https://web.archive.org/web/20111001230201/http://www.gravityresearchfoundation.org/pdf/awarded/1995/jacobson.pdf |date=2011-10-01 }}in the [[Gravity Research Foundation]]'s 1995 essay competition..</ref><ref name="Smolin2002">[[Lee Smolin]], ''[[Three Roads to Quantum Gravity]]'' (New York, N.Y.: [[Basic Books]], 2002), pp. 173 and 175, ISBN 0465078362, {{lccn|2007||310371}}.</ref> Aj keď doteraz bolo predstavených viacero dôkazov, že nejaká hranica musí existovať, aby zákony termodynamiky boli navzájom konzistentné, tak presná formulácia hranice je stále predmetom diskusií.<ref name="Bekenstein2005">Jacob D. Bekenstein, "How Does the Entropy/Information Bound Work?", ''[[Foundations of Physics]]'', Vol. 35, No. 11 (November 2005), pp. 1805-1823, {{doi|10.1007/s10701-005-7350-7}}, {{bibcode|2005FoPh...35.1805B}}. Also at {{arxiv|quant-ph|0404042}}, April 7, 2004.</ref><ref name="Bekenstein2008">Jacob D. Bekenstein, [http://www.scholarpedia.org/wiki/index.php?title=Bekenstein_bound&oldid=50791 "Bekenstein bound"], ''[[Scholarpedia]]'', Vol. 3, No. 10 (October 31, 2008), p. 7374, {{doi|10.4249/scholarpedia.7374}}.</ref><ref name="Bousso1999-6">Raphael Bousso, "Holography in general space-times", ''[[Journal of High Energy Physics]]'', Vol. 1999, Issue 6 (June 1999), Art. No. 28, 24 pages, {{doi|10.1088/1126-6708/1999/06/028}}, {{bibcode|1999JHEP...06..028B}}. [http://www.webcitation.org/5pyuEsv6m Mirror link]. Also at {{arxiv|hep-th|9906022}}, June 3, 1999.</ref><ref name="Bousso1999-7">Raphael Bousso, "A covariant entropy conjecture", ''[[Journal of High Energy Physics]]'', Vol. 1999, Issue 7 (July 1999), Art. No. 4, 34 pages, {{doi|10.1088/1126-6708/1999/07/004}}, {{bibcode|1999JHEP...07..004B}}. [http://www.webcitation.org/5pyuO4nVn Mirror link]. Also at {{arxiv|hep-th|9905177}}, May 24, 1999.</ref><ref name="Bousso2000">Raphael Bousso, "The holographic principle for general backgrounds", ''[[Classical and Quantum Gravity]]'', Vol. 17, No. 5 (March 7, 2000), pp. 997-1005, {{doi|10.1088/0264-9381/17/5/309}}, {{bibcode|2000CQGra..17..997B}}. Also at {{arxiv|hep-th|9911002}}, November 2, 1999.</ref><ref name="Bekenstein2000">Jacob D. Bekenstein, "Holographic bound from second law of thermodynamics", ''[[Physics Letters|Physics Letters B]]'', Vol. 481, Issues 2-4 (May 25, 2000), pp. 339-345, {{doi|10.1016/S0370-2693(00)00450-0}}, {{bibcode|2000PhLB..481..339B}}. Also at {{arxiv|hep-th|0003058}}, March 8, 2000.</ref><ref name="Bousso2002">Raphael Bousso, [https://web.archive.org/web/20110812102152/http://bib.tiera.ru/DVD-005/Bousso_R._The_holographic_principle_(2002)(en)(50s).pdf "The holographic principle"], ''[[Reviews of Modern Physics]]'', Vol. 74, No. 3 (July 2002), pp. 825-874, {{doi|10.1103/RevModPhys.74.825}}, {{bibcode|2002RvMP...74..825B}}. . Also at {{arxiv|hep-th|0203101}}, March 12, 2002.</ref><ref name="Bekenstein2003">Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/Holographic_Univ.pdf "Information in the Holographic Universe: Theoretical results about black holes suggest that the universe could be like a gigantic hologram"], ''[[Scientific American]]'', Vol. 289, No. 2 (August 2003), pp. 58-65..</ref><ref name="BoussoEtAl2003">Raphael Bousso, Éanna É. Flanagan and [[Donald Marolf]], "Simple sufficient conditions for the generalized covariant entropy bound", Physical Review D, Vol. 68, Issue 6 (September 15, 2003), Art. No. 064001, 7 pages, {{doi|10.1103/PhysRevD.68.064001}}, {{bibcode|2003PhRvD..68f4001B}}. Also at {{arxiv|hep-th|0305149}}, May 19, 2003.</ref><ref name="Bekenstein2004">Jacob D. Bekenstein, "Black holes and information theory", ''[[Contemporary Physics]]'', Vol. 45, Issue 1 (January 2004), pp. 31-43, {{doi|10.1080/00107510310001632523}}, {{bibcode|2003ConPh..45...31B}}. Also at {{arxiv|quant-ph|0311049}}, November 9, 2003. Also at {{arxiv|quant-ph/0311049}}, November 9, 2003.</ref><ref name="Tipler2005">[[Frank J. Tipler]], [http://math.tulane.edu/~tipler/theoryofeverything.pdf "The structure of the world from pure numbers"], ''[[Reports on Progress in Physics]]'', Vol. 68, No. 4 (April 2005), pp. 897-964, {{doi|10.1088/0034-4885/68/4/R04}}, {{bibcode|2005RPPh...68..897T}}. [http://www.webcitation.org/5nx3CxKm0 Mirror link]. Also released as [http://arxiv.org/abs/0704.3276 "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything"], {{arxiv|0704.3276}}, April 24, 2007. Tipler gives a number of arguments for maintaining that Bekenstein's original formulation of the bound is the correct form. See in particular the paragraph beginning with "A few points ..." on p. 903 of the ''Rep. Prog. Phys.'' paper (or p. 9 of the ''arXiv'' version), and the discussions on the Bekenstein bound that follow throughout the paper.</ref>

Stáva sa, že Bekenstein-hawkingova entropia  trojrozmernej čiernej diery presne napĺňa hranicu 

:<math>S =\frac{kA}{4}</math>

kde ''A'' je  dvojrozmerná oblasť  horizontu udalostí čiernej diery (v jednotkách [[Planckova dĺžka|Planckovej plochy]]) <math>\hbar G/c^3</math>.

Bekensteinova hranica je úzko prepojená s termodynamikou čiernych dier, [[Holografický princíp|holografickým princípom]] a Buosseho holografickou hranicou kvantovej gravitácie.

==Príklady==
===Ľudské telo===

Napríklad priemerný američan váži {{kg|88.3}} a je vysoký {{m|1.763|m}}. Takáto osoba môže byť vložená do guľe s polomerom {{m|0.8815|m}} (polomer = polovica výšky).. Maximálnu energiu máme danú vzťahom mc2 = E  , v SI jednotkách  hmotnosť 88.3 kilograms × 299792458<sup>2</sup>(rýchlosť svetla v m/s na druhú)= 7936008228246099761.2 Joulov energie. Doporučená hodnota redukovanej Planckovej konštanty je 1.054571628×10<sup>−34</sup> joul-sekúnd. 
Vložením týchto čísel do Bekensteinovej hranice dostaneme 

:<math>I_{\text{human}} \leq \frac{2 \pi (0.8815) (7936008228246099761.2)}{(1.054571628\times 10^{-34}) (299792458) \ln 2} \approxeq 2.0057742\times 10^{45},</math>
  
ako výsledok , že maximálny počet bitov  potrebný na dokonalé vytvorenie priemerného američana na kvantovej úrovni v počítači je 2.0057742×10<sup>45</sup>  bitov informácií alebo 2.5072178×10<sup>38</sup>  megabajtov. V prípade ľudí je skutočný počet bitov, ktoré môžu byť kódované ich telom  omnoho menšie (ako aj pri ostatných známych formách hmoty), pretože  toto je horná hranica, dosiahnutá iba čiernou dierou.

===Ľudský mozog===

Priemerný ľudský mozog váži {{kg|1.5}} a má objem {{cm3|1260}} . Energia(E&nbsp;=&nbsp;m·c<sup>2</sup>)  bude  1.34813·10<sup>17</sup> J  a keĎže mozog je približne guľa, potom  polomer (V&nbsp;=&nbsp;4·π·r<sup>3</sup>/3) bude  6.70030·10<sup>−2</sup> m. 
Bekensteinova hranica bude 210<sup>7.79640·10<sup>41</sup></sup>  bitov  a predstavuje maximum informácií potrebných na dokonalé vytvorenie  priemerného ľudského mozgu na kvantovej úrovni.

== Pozri aj ==
* [[Entropia]]

== Referencie ==
{{reflist}}

== Zdroj ==
{{preklad|en|Bekenstein bound|418882723}}

== Externé odkazy ==
* Jacob D. Bekenstein, [http://www.scholarpedia.org/article/Bekenstein_bound "Bekenstein bound"], ''[[Scholarpedia]]'', Vol. 3, No. 10 (2008), p.&nbsp;7374 {{doi|10.4249/scholarpedia.7374}}
* Jacob D. Bekenstein, [http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy "Bekenstein-Hawking entropy"], ''[[Scholarpedia]]'', Vol. 3, No. 10 (2008), p.&nbsp;7375, {{doi|10.4249/scholarpedia.7375}}
* [http://www.phys.huji.ac.il/~bekenste/ Jacob D. Bekenstein's website] at the Racah Institute of Physics, [[Hebrew University of Jerusalem]], which contains a number of articles on the Bekenstein bound

[[Kategória:Kvantová informatika]]