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Bekenstein bound

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Bekenstein bound

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.[1] 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.

Equations

The universal form of the bound was originally found by Jacob Bekenstein as the inequality[1][2][3]

S \leq \frac{2 \pi k R E}{\hbar c}

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 including any rest masses, ħ is the 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 Newton's Constant G.

In informational terms, the bound is given by

I \leq \frac{2 \pi R E}{\hbar c \ln 2}

where I is the information expressed in number of bits contained in the quantum states in the sphere. The ln 2 factor comes from defining the information as the logarithm to the base 2 of the number of quantum states.[4] Using mass energy equivalence, the informational limit may be reformulated as

I \leq \frac{2 \pi c R m}{\hbar \ln 2} \approx 2.577\times 10^{43} m R

where m is the mass of the system in kilograms, and the radius R is expressed in meters.

Origins

Bekenstein derived the bound from heuristic arguments involving black holes. 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, 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.[5][6] 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.[2][3][7][8][9][10][11][12][13][14][15]

Examples

Black holes

It happens that the Bekenstein-Hawking Entropy of three-dimensional black holes exactly saturates the bound

S =\frac{kA}{4}

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

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.

Human brain

An average human brain has a mass of 1.5 kg and a volume of 1260 cm³. If the brain is approximated by a sphere then the radius will be 6.7 cm.

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

• J. D. Bekenstein, "Black Holes and the Second Law", Lettere al Nuovo Cimento, Vol. 4, No 15 (August 12, 1972), pp. 737-740, doi:10.1007/BF02757029, Bibcode: 1972NCimL...4..737B. Mirror link.
• Jacob D. Bekenstein, "Black Holes and Entropy", Physical Review D, Vol. 7, No. 8 (April 15, 1973), pp. 2333-2346, doi:10.1103/PhysRevD.7.2333, Bibcode: 1973PhRvD...7.2333B. Mirror link.
• Jacob D. Bekenstein, "Generalized second law of thermodynamics in black-hole physics", Physical Review D, Vol. 9, No. 12 (June 15, 1974), pp. 3292-3300, doi:10.1103/PhysRevD.9.3292, Bibcode: 1974PhRvD...9.3292B. Mirror link.
• Jacob D. Bekenstein, "Statistical black-hole thermodynamics", Physical Review D, Vol. 12, No. 10 (November 15, 1975), pp. 3077-3085, doi:10.1103/PhysRevD.12.3077, Bibcode: 1975PhRvD..12.3077B. Mirror link.
• Jacob D. Bekenstein, "Black-hole thermodynamics", Physics Today, Vol. 33, Issue 1 (January 1980), pp. 24-31, doi:10.1063/1.2913906, Bibcode: 1980PhT....33a..24B. Mirror link.
• 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. Mirror link.
• Jacob D. Bekenstein, "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. Mirror link.
• Jacob D. Bekenstein, "Specific entropy and the sign of the energy", 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, "Entropy content and information flow in systems with limited energy", Physical Review D, Vol. 30, No. 8, (October 15, 1984), pp. 1669-1679, doi:10.1103/PhysRevD.30.1669, Bibcode: 1984PhRvD..30.1669B. Mirror link.
• Jacob D. Bekenstein, "Communication and energy", Physical Review A, Vol. 37, Issue 9 (May 1988), pp. 3437-3449, doi:10.1103/PhysRevA.37.3437, Bibcode: 1988PhRvA..37.3437B. Mirror link.
• Marcelo Schiffer and Jacob D. Bekenstein, "Proof of the quantum bound on specific entropy for free fields", 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 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 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.

References

1. ^ a b 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. Mirror link.
2. ^ a b 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.
3. ^ a b Jacob D. Bekenstein, "Bekenstein bound", Scholarpedia, Vol. 3, No. 10 (October 31, 2008), p. 7374, doi:10.4249/scholarpedia.7374.
4. ^ 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.
5. ^ 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.
6. ^ Lee Smolin, Three Roads to Quantum Gravity (New York, N.Y.: Basic Books, 2002), pp. 173 and 175, ISBN 0-465-07836-2, LCCN 2007-310371.
7. ^ 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.
8. ^ 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.
9. ^ 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.
10. ^ 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.
11. ^ Raphael Bousso, "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. Mirror link. Also at arXiv:hep-th/0203101, March 12, 2002.
12. ^ 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.
13. ^ 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.
14. ^ 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.
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. 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.