貝肯斯坦上限

在物理學中，貝肯斯坦上限（）是在一有限能量之有限空間內熵或資訊的上限。反過來說，該上限是要精確描述一物理系統至量子層級的最大需要資訊量Jacob D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems" , Physical Review D, Vol. 23, No. 2, (January 15, 1981), pp. 287-298, , . （）.。這表示若要精確描述一個佔有有限空間之有限能量物理系統，只需要有限的資訊量。

緣起
貝肯斯坦從涉及黑洞的啟發式觀點導出此上限式。如果存在系統違反此不等式，也就是有太多的熵，則貝肯斯坦認為這將違反熱力學第二定律。1995年，證明了爱因斯坦场方程可以藉由假設貝肯斯坦上限和熱力學定律的真實性而導出Ted Jacobson, "Thermodynamics of Spacetime: The Einstein Equation of State", Physical Review Letters, Vol. 75, Issue 7 (August 14, 1995), pp. 1260-1263, , . Also at , April 4, 1995. Also available here  and here . Additionally available as an entry in the Gravity Research Foundation's 1995 essay competition..Lee Smolin, Three Roads to Quantum Gravity (New York, N.Y.: Basic Books, 2002), pp. 173 and 175, ISBN 0-465-07836-2, .。然而，雖然一些理論已經表明某種形式的上限必須存在，以使熱力學和廣義相對論相互一致，但該上限的確切表述一直是人們爭論的一個問題Jacob D. Bekenstein, "How Does the Entropy/Information Bound Work?", Foundations of Physics, Vol. 35, No. 11 (November 2005), pp. 1805-1823, , . Also at , April 7, 2004.Jacob D. Bekenstein, "Bekenstein bound" , Scholarpedia, Vol. 3, No. 10 (October 31, 2008), p. 7374, .Raphael Bousso, "Holography in general space-times", Journal of High Energy Physics, Vol. 1999, Issue 6 (June 1999), Art. No. 28, 24 pages, , . Mirror link. Also at , June 3, 1999.Raphael Bousso, "A covariant entropy conjecture", Journal of High Energy Physics, Vol. 1999, Issue 7 (July 1999), Art. No. 4, 34 pages, , . Mirror link. Also at , May 24, 1999.Raphael Bousso, "The holographic principle for general backgrounds", Classical and Quantum Gravity, Vol. 17, No. 5 (March 7, 2000), pp. 997-1005, , . Also at , November 2, 1999.Jacob D. Bekenstein, "Holographic bound from second law of thermodynamics", Physics Letters B, Vol. 481, Issues 2-4 (May 25, 2000), pp. 339-345, , . Also at , March 8, 2000.Jacob D. Bekenstein, "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..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, , . Also at , May 19, 2003.Jacob D. Bekenstein, "Black holes and information theory", Contemporary Physics, Vol. 45, Issue 1 (January 2004), pp. 31-43, , . Also at , November 9, 2003. Also at , November 9, 2003.Frank J. Tipler, "The structure of the world from pure numbers" , Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, , .. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything" , , 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.。

表示式
150px|thumb|雅各布·貝肯斯坦
此上限的普遍形式由雅各布·貝肯斯坦首次提出，以不等式表示之。以熵表示之該不等式為：

其中是熵、是波茲曼常數、是包圍整個系統的球殼半徑、是包含任何靜止-{質量}-的總質能、是約化普朗克常數、則是真空中的光速。然而，雖然重力在此效應中扮演著很重要的角色，但該不等式中並未出現万有引力常数。

若以二进制信息表示之，則該不等式為：

其中是資訊含量，以位元數表示球殼中所含有的量子態。而式中ln2項則來自定義資訊量為量子狀態數目的自然對數值Frank J. Tipler, "The structure of the world from pure numbers" , Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, , , p. 902. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything" , , April 24, 2007, p. 8.。若使用質能等價定理，該資訊上限式可表示為：

其中是系統-{質量}-，以公斤表示，而半徑則以公尺作為其單位。

 貝肯斯坦-霍金方程 

1972年，史蒂芬·霍金证明了黑洞视界的表面积永不会减少，两个黑洞合并后的黑洞面积不会小于原先两个黑洞面积之和。与此同时，雅各布·貝肯斯坦運用此理论提出了黑洞熵的概念。為了符合熱力學第二定律，黑洞必須擁有熵。如果黑洞沒有熵，則可以藉由將物質丟入黑洞中來違反熱力學第二定律。黑洞熵的增加必須超過被吞入物質所減少的熵。贝肯斯坦認為，黑洞的表面积与它的熵含量成正比，从而使其不违反热力学第二定律。贝肯斯坦在他的论文中指出：

貝肯斯坦認為，黑洞表面积與其熵含量的正比係數接近。1974年，霍金提出了霍金輻射A Brief History of Time, Stephen Hawking, Bantam Books, 1988.，並運用能量、溫度與熵之間的熱力學關係證實了贝肯斯坦的猜想，同時修正其正比係數為：

其中是黑洞视界的表面積，利用求得。是波茲曼常數，則是普朗克長度。此公式經常被稱為「貝肯斯坦-霍金方程」（），其中下標BH可指黑洞（）或貝肯斯坦-霍金（）的首字母縮寫。使用貝肯斯坦上限求得之最大熵含量正好等於由此方程求得之黑洞熵，此結果促成了全像原理的發展。

參見

 黑洞資訊悖論
 黑洞熵
 玻尔兹曼大脑
 數碼物理學


註釋

參考資料

外部連結
 Jacob D. Bekenstein, "Bekenstein bound"  , Scholarpedia, Vol. 3, No. 10 (2008), p. 7374, .
 Jacob D. Bekenstein, "Bekenstein-Hawking entropy"  , Scholarpedia, Vol. 3, No. 10 (2008), p. 7375, .
 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.

Category:熱力學
Category:量子信息科学