ベッケンシュタイン境界のソースを表示

{{要改訳}}
物理学では、'''ベッケンシュタイン境界'''（ベッケンシュタインきょうかい、Bekenstein bound）は、[[エントロピー]] ''S''、あるいは、[[情報量]] ''I'' の上界であり、与えられた有限な領域の空間内には有限なエネルギーしか持たない、また逆に、与えられた量子レベルへ落とした物理系を完全に記述する情報の最大量があることを意味する。<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}}. [https://webcitation.org/5pvt5c96N Mirror link].</ref> このことは、物理系の情報量、あるいは系を完全に記述するのに必要な情報量は、空間の大きさやエネルギーが有限であれば、有限でなければいけないことを意味する。[[計算機科学]]では、このことは有限の大きさとエネルギーを持つ物理系に対して最大の情報量プロセス率（{{仮リンク|ブレマーマンの境界|en|Bremermann's limit}}(Bremermann's limit)）が存在し、有限の物理的次元で無限のメモリを持つ[[チューリングマシン]]は、物理的に不可能であることを意味する。{{Citation needed|date=January 2014}}
<!--__NOTOC__
In [[physics]], the '''Bekenstein bound''' is an upper limit on the [[entropy]] ''S'', or [[information]] ''I'', that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level.<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}}. [https://webcitation.org/5pvt5c96N Mirror link].</ref> It implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy is finite. In [[computer science]], this implies that there is a maximum information-processing rate ([[Bremermann's limit]]) for a physical system that has a finite size and energy, and that a [[Turing machine]] with finite physical dimensions and unbounded memory is not physically possible.-->

==式==
T境界の普遍的な形式は、元々は、[[ヤコブ・ベッケンシュタイン]](Jacob Bekenstein)により<ref name="Bekenstein1981-1"/><ref name="Bekenstein2005"/><ref name="Bekenstein2008"/>、[[不等式]]
:<math>S \leq \frac{2 \pi k R E}{\hbar c}</math>
として発見された。ここに ''S'' は[[エントロピー]]、''k'' は[[ボルツマン定数]]、''R'' は与えられた系を囲むことの可能な[[球]]の[[半径]]、''E'' はすべての[[不変質量]]を含む全[[E=mc2|質量エネルギー]]、''ħ'' は[[ディラック定数]]、''c'' は[[光速度]]である。重力は力として重要な役割を果たすが、それに対し、境界の表現は[[重力定数|ニュートン定数]]&nbsp;''G'' を含まないことに注意する。
<!--==Equations==
The universal form of the bound was originally found by [[Jacob Bekenstein]] as the [[inequality (mathematics)|inequality]]<ref name="Bekenstein1981-1"/><ref name="Bekenstein2005"/><ref name="Bekenstein2008"/>

:<math>S \leq \frac{2 \pi k R E}{\hbar c}</math>

where ''S'' is the [[entropy]], ''k'' is [[Boltzmann's constant]], ''R'' is the [[radius]] of a [[sphere]] that can enclose the given system, ''E'' is the total [[mass–energy equivalence|mass–energy]] including any [[invariant mass|rest masses]], ''ħ'' is the [[Planck constant#Reduced Planck constant|reduced Planck constant]], and ''c'' is the [[speed of light]]. Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain [[Gravitational constant|Newton's Constant]]&nbsp;''G''.-->

情報量の項として境界は、
:<math>I \leq \frac{2 \pi R E}{\hbar c \ln 2}</math>
として与えられる。ここに ''I'' は球の中の量子状態を意味する[[ビット]]の数であらわされる[[情報量]]である。[[自然対数|ln]] 2 の要素は、情報量を量子状態の数の[[二進法|2進数]]の[[対数]]として定義することから来る。<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. [https://webcitation.org/5nx3CxKm0 Mirror link]. Also released as [https://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>[[E=mc2|質量とエネルギーの等価性]]を使うと、
:<math>I \leq \frac{2 \pi c R m}{\hbar \ln 2} \approx 2.577\times 10^{43} (m / \mathrm{kg}) (R / \mathrm{m})</math>

<!--In informational terms, the bound is given by

:<math>I \leq \frac{2 \pi R E}{\hbar c \ln 2}</math>

where ''I'' is the [[information]] expressed in number of [[bit]]s contained in the quantum states in the sphere. The [[Natural logarithm|ln]] 2 factor comes from defining the information as the [[logarithm]] to the [[radix|base]] [[Binary numeral system|2]] of the number of quantum states.<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. [https://webcitation.org/5nx3CxKm0 Mirror link]. Also released as [https://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> Using [[mass energy equivalence]], the informational limit may be reformulated as

:<math>I \leq \frac{2 \pi c R m}{\hbar \ln 2} \approx 2.577\times 10^{43} m R</math>

where <math>m</math> is the mass of the system in kilograms, and the radius <math>R</math> is expressed in meters.-->
==起源==
ベッケンシュタインは[[ブラックホール]]を意味する発見的方法から境界を導出した。境界を破る、つまり、大きすぎるエントロピーを持つような系は、ブラックホールの中でエントロピーを下げることにより[[熱力学第二法則]]を破ることは可能かもしれないとベッケンシュタインは論じた。1995年に{{仮リンク|テオドール・ジェイコブソン|en|Theodore Jacobson}}(Theodore Jacobson)は、[[アインシュタイン方程式|アインシュタイン場の方程式]]（つまり[[一般相対論]]）がベッケンシュタイン境界と[[熱力学#熱力学の法則|熱力学の法則]]が正しいことを前提とすると導出できることを示した。<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] in the [[重力研究財団|Gravity Research Foundation]]'s 1995 essay competition. [https://webcitation.org/5pw2xrwBb Mirror link].</ref><ref name="Smolin2002">[[Lee Smolin]], ''[[Three Roads to Quantum Gravity]]'' (New York, N.Y.: [[Basic Books]], 2002), pp. 173 and 175, ISBN 0-465-07836-2, {{LCCN|2007310371}}.</ref> しかしながら、熱力学の法則と一般相対性が互いに整合性を持つために、ある境界が存在する必要があることを占めることには数々の議論があるが、一方、境界の正確な定式化は、論争となっている。<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}}. [https://webcitation.org/5pyuEsv6m?url=http://iopscience.iop.org/1126-6708/1999/06/028/pdf/1126-6708_1999_06_028.pdf 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}}. [https://webcitation.org/5pyuO4nVn?url=http://iopscience.iop.org/1126-6708/1999/07/004/pdf/1126-6708_1999_07_004.pdf 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, [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}}. [https://webcitation.org/5pw1VZbGO Mirror link]. 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. [https://webcitation.org/5pvxM7hws Mirror link].</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}}. [https://webcitation.org/5nx3CxKm0 Mirror link]. Also released as [https://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>
<!--==Origins==
Bekenstein derived the bound from heuristic arguments involving [[black hole]]s. If a system exists that violates the bound, i.e. by having too much entropy, Bekenstein argued that it would be possible to violate the [[second law of thermodynamics]] by lowering it into a black hole. In 1995, [[Theodore Jacobson|Ted Jacobson]] demonstrated that the [[Einstein field equations]] (i.e., [[general relativity]]) can be derived by assuming that the Bekenstein bound and the [[laws of thermodynamics]] are true.<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] in the [[Gravity Research Foundation]]'s 1995 essay competition. [https://webcitation.org/5pw2xrwBb Mirror link].</ref><ref name="Smolin2002">[[Lee Smolin]], ''[[Three Roads to Quantum Gravity]]'' (New York, N.Y.: [[Basic Books]], 2002), pp. 173 and 175, ISBN 0-465-07836-2, {{LCCN|2007310371}}.</ref> However, while a number of arguments have been devised which show that some form of the bound must exist in order for the laws of thermodynamics and general relativity to be mutually consistent, the precise formulation of the bound has been a matter of debate.<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}}. [https://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}}. [https://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, [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}}. [https://webcitation.org/5pw1VZbGO Mirror link]. 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. [https://webcitation.org/5pvxM7hws Mirror link].</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}}. [https://webcitation.org/5nx3CxKm0 Mirror link]. Also released as [https://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>-->

==例==
===ブラックホール===
3次元[[ブラックホール]]の[[ブラックホールの熱力学#ブラックホール|ホーキング・ベッケンシュタインのエントロピー]]は、正確に境界
:<math>S =\frac{kA}{4}</math>
で飽和することが起きる。ここに ''A'' は[[プランク長|プランク面積]] <math>\hbar G/c^3</math> の単位でブラックホールの事象の地平線の 2次元面積である。

境界は密接に、[[ブラックホールの熱力学]]や[[ホログラフィック原理]]や量子重力の[[:en:Bousso's holographic bound|共変エントロピー境界]](covariant entropy bound)<ref>'''共変エントロピー境界'''とはブゾー(Bousso)のホログラフィック境界のこのことを言っている。</ref>と関連していて、後者の予想されている強い形から導出することができる。
<!--==Examples==
===Black holes===
It happens that the [[Black_hole_thermodynamics#Black_hole_entropy|Bekenstein-Hawking Entropy]] of three-dimensional [[black hole]]s exactly saturates the bound

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

where ''A'' is the two-dimensional area of the black hole's event horizon in units of the [[Planck area]], <math>\hbar G/c^3</math>.

The bound is closely associated with [[black hole thermodynamics]], the [[holographic principle]] and the [[covariant entropy bound]] of quantum gravity, and can be derived from a conjectured strong form of the latter.-->

===人間の脳===
平均的な[[脳#ヒトの脳について|人間の脳]]は、1.5&nbsp;kg の重さと 1260&nbsp;cm³ の体積を持っている。脳が球に近似しているとすると、[[球面]]の半径は 6.7 cm となる。

情報量的なベッケンシュタイン境界は <math>\approx 2.6 \times 10^{42}</math> ビットとなり、量子レベルに落とした平均的な人間の脳を完全に再現するのに必要な最大の情報量を表している。このことは、人間の脳の[[量子状態|状態]]の数 <math>O=2^I</math> が、<math>\approx 10^{7.8 \times 10^{41}}</math> よりも小さいはずであることを意味している。
<!--===Human brain===
An average [[human brain]] has a mass of 1.5&nbsp;kg and a volume of 1260&nbsp;cm³. If the brain is approximated by a sphere then [[Sphere#Volume_of_a_sphere|the radius will be]] 6.7 cm.

The informational Bekenstein bound will be <math>\approx 2.6 \times 10^{42}</math> bit and represents the maximum information needed to perfectly recreate an average human brain down to the quantum level. This means that the number <math>O=2^I</math> of [[quantum state|states]] of the human brain must be less than <math>\approx 10^{7.8 \times 10^{41}}</math>.-->

==さらに先の文献==
*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}}. [https://webcitation.org/5pvpODjvK Mirror link].
*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}}. [https://webcitation.org/5pvpyakNu Mirror link].
*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}}. [https://webcitation.org/5pvqDR9Rs Mirror link].
*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}}. [https://webcitation.org/5pvqVyq9K Mirror link].
*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}}. [https://webcitation.org/5pvqqPGuq Mirror link].
*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}}. [https://webcitation.org/5pvt5c96N Mirror link].
*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}}. [https://webcitation.org/5pvuFpfRA Mirror link].
*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}}.
*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}}. [https://webcitation.org/5pvud0uay Mirror link].
*Jacob D. Bekenstein, [http://pm1.bu.edu/~tt/qcl/pdf/bekenstj19867766031b.pdf "Communication and energy"], ''[[Physical Review|Physical Review A]]'', Vol. 37, Issue 9 (May 1988), pp. 3437-3449, {{doi|10.1103/PhysRevA.37.3437}}, {{bibcode|1988PhRvA..37.3437B}}. [https://webcitation.org/5pvutswfg Mirror link].
*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}} {{PMID|9959747}}, {{bibcode|1989PhRvD..39.1109S}}.
*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}}.
*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.
*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}}.
*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.

==脚注==
{{reflist}}

==関連項目==
* {{仮リンク|計算の限界|en|Limits to computation}}(Limits to computation)
* [[ブラックホールの熱力学|ブラックホールエントロピー]]
* [[デジタル物理学]](Digital physics)
* [[エントロピー]]

==外部リンク==
* 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.

{{DEFAULTSORT:へつけんしゆたいんきようかい}}
[[Category:エントロピー]]
[[Category:量子情報科学]]
[[Category:エポニム]]