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贝肯斯坦上限

2018-08-30 10:06:58     所属分类:量子信息
大麦哲伦云面前的黑洞(中心)的模拟视图

在物理学中, 贝肯斯坦上限英语:Bekenstein bound)是在一具有有限能量之有限空间内熵或信息的上限。反过来说,该上限是要精确描述一物理系统至量子层级的最大需要信息量[1]。这表示若要精确描述一个占有有限空间之有限能量物理系统,只需要有限的信息量。

目录

  • 1 缘起
  • 2 表示式
  • 3 贝肯斯坦-霍金方程
  • 4 参见
  • 5 注释
  • 6 参考资料
  • 7 外部链接

缘起

贝肯斯坦从涉及黑洞的启发式观点导出此上限式。如果存在系统违反此不等式,也就是有太多的熵,则贝肯斯坦认为这将违反热力学第二定律。1995年,泰德·雅各布森英语Ted Jacobson证明了爱因斯坦场方程[a]可以借由假设贝肯斯坦上限和热力学定律的真实性而导出[2][3]。然而,虽然一些理论已经表明某种形式的上限必须存在,以使热力学和广义相对论相互一致,但该上限的确切表述一直是人们争论的一个问题[4][5][6][7][8][9][10][11][12][13][14]

表示式

雅各布·贝肯斯坦

此上限的普遍形式由雅各布·贝肯斯坦首次提出,以不等式表示之[1][4][5]。以熵表示之该不等式为:

其中是熵、是玻尔兹曼常数、是包围整个系统的球壳半径、是包含任何静止质量的总质能、是约化普朗克常量、则是真空中的光速。然而,虽然重力在此效应中扮演着很重要的角色,但该不等式中并未出现万有引力常数

若以二进制信息表示之,则该不等式为:

其中是信息含量,以比特数表示球壳中所含有的量子态。而式中ln2项则来自定义信息量为量子状态数目的自然对数值[15]。若使用质能等价定理,该信息上限式可表示为:

其中是系统质量,以公斤表示,而半径则以米作为其单位。

贝肯斯坦-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·贝肯斯坦运用此理论提出了黑洞熵的概念。为了匹配热力学第二定律,黑洞必须拥有熵。如果黑洞没有熵,则可以借由将物质丢入黑洞中来违反热力学第二定律。黑洞熵的增加必须超过被吞入物质所减少的熵。贝肯斯坦认为,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:

贝肯斯坦认为,黑洞表面积与其熵含量的正比系数接近。1974年,霍金提出了霍金辐射[17][18],并运用能量、温度与熵之间的热力学关系证实了贝肯斯坦的猜想,同时修正其正比系数为[19][10]

其中是黑洞视界的表面积,利用求得。是玻尔兹曼常数,则是普朗克长度。此公式经常被称为“贝肯斯坦-霍金方程”(Bekenstein–Hawking formula),其中下标BH可指黑洞(black hole)或贝肯斯坦-霍金(Bekenstein-Hawking)的首字母缩写。使用贝肯斯坦上限求得之最大熵含量正好等于由此方程求得之黑洞熵,此结果促成了全息原理的发展[10]

参见

  • 黑洞信息佯谬
  • 黑洞熵
  • 玻尔兹曼大脑
  • 数字物理学
  • 计算上的极限英语Limits to computation

注释

  1. ^ 即广义相对论。

参考资料

  1. ^ 1.0 1.1 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, doi:10.1103/PhysRevD.23.287, Bibcode: 1981PhRvD..23..287B. (页面存档).
  2. ^ 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 here and here. Additionally available as an entry in the Gravity Research Foundation's 1995 essay competition. Mirror link.
  3. ^ 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-{{{3}}}.
  4. ^ 4.0 4.1 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.
  5. ^ 5.0 5.1 Jacob D. Bekenstein, "Bekenstein bound", Scholarpedia, Vol. 3, No. 10 (October 31, 2008), p. 7374, doi:10.4249/scholarpedia.7374.
  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. Mirror link. Also at arXiv:hep-th/9906022, June 3, 1999.
  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. Mirror link. Also at arXiv:hep-th/9905177, May 24, 1999.
  8. ^ 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.
  9. ^ Jacob D. Bekenstein, "Holographic bound from second law of thermodynamics", 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.
  10. ^ 10.0 10.1 10.2 Bousso, Raphael. The Holographic Principle. Reviews of Modern Physics. 2002, 74 (3): 825–874. Bibcode:2002RvMP...74..825B. arXiv:hep-th/0203101. doi:10.1103/RevModPhys.74.825 (英语). 
  11. ^ 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. Mirror link.
  12. ^ 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.
  13. ^ 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.
  14. ^ 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, doi:10.1088/0034-4885/68/4/R04, Bibcode: 2005RPPh...68..897T. Mirror link. Also released as "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.
  15. ^ 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, doi:10.1088/0034-4885/68/4/R04, Bibcode: 2005RPPh...68..897T, p. 902. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything", arXiv:0704.3276, April 24, 2007, p. 8.
  16. ^ Jacob D. Bekenstein. Black Holes and Entropy. Phys. Rev. D. 1973-04-15, 7 (8): 2333–2346. doi:10.1103/PhysRevD.7.2333 (英语). 
  17. ^ Matson, John. Artificial event horizon emits laboratory analogue to theoretical black hole radiation. Sci. Am. Oct 1, 2010 (英语). 
  18. ^ A Brief History of Time, Stephen Hawking, Bantam Books, 1988.
  19. ^ Majumdar, Parthasarathi. Black Hole Entropy and Quantum Gravity 73: 14. 1999. Bibcode:1999InJPB..73..147M. arXiv:gr-qc/9807045 (英语). 

外部链接

  • Jacob D. Bekenstein, "Bekenstein bound", Scholarpedia, Vol. 3, No. 10 (2008), p. 7374, doi:10.4249/scholarpedia.7374.
  • Jacob D. Bekenstein, "Bekenstein-Hawking entropy", Scholarpedia, Vol. 3, No. 10 (2008), p. 7375, doi:10.4249/scholarpedia.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.
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