Menelaus' theorem

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. 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

$S\leq {\frac {2\pi kRE}{\hbar c}}$

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

$I\leq {\frac {2\pi RE}{\hbar c\ln 2}}$

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. Using mass energy equivalence, the informational limit may be reformulated as

$I\leq {\frac {2\pi cRm}{\hbar \ln 2}}\approx 2.577\times 10^{43}mR$

where $m$ is the mass of the system.

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. 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.

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 $\approx 10^{7.8\times 10^{41}}$.

The existence of the Bekenstein bound implies that the storage capacity of human brain is finite, although potentially very large, if constrained only by ultimate physical limits. This makes mind uploading possible from the point of view of quantum mechanics, provided that physicalism is true.

See also

• Limits to computation
• Black hole entropy
• Digital physics
• Entropy

Further reading

• J. D. Bekenstein, "Black Holes and the Second Law", Lettere al Nuovo Cimento, Vol. 4, No 15 (August 12, 1972), pp. 737-740, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link.
• Jacob D. Bekenstein, "Black Holes and Entropy", Physical Review D, Vol. 7, No. 8 (April 15, 1973), pp. 2333-2346, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link.
• Jacob D. Bekenstein, "Statistical black-hole thermodynamics", Physical Review D, Vol. 12, No. 10 (November 15, 1975), pp. 3077-3085, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link.
• Jacob D. Bekenstein, "Black-hole thermodynamics", Physics Today, Vol. 33, Issue 1 (January 1980), pp. 24-31, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link.
• Jacob D. Bekenstein, "Energy Cost of Information Transfer", Physical Review Letters, Vol. 46, No. 10 (March 9, 1981), pp. 623-626, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode.
• 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link.
• Jacob D. Bekenstein, "Communication and energy", Physical Review A, Vol. 37, Issue 9 (May 1988), pp. 3437-3449, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park. PMID 9959747, Template:Bibcode.
• Jacob D. Bekenstein, "Is the Cosmological Singularity Thermodynamically Possible?", International Journal of Theoretical Physics, Vol. 28, Issue 9 (September 1989), pp. 967-981, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode.
• Jacob D. Bekenstein, "Entropy bounds and black hole remnants", Physical Review D, Vol. 49, Issue 4 (February 15, 1994), pp. 1912-1921, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode.
• 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, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, June 16, 1999.

References

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External links

• Jacob D. Bekenstein, "Bekenstein bound", Scholarpedia, Vol. 3, No. 10 (2008), p. 7374, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park..
• Jacob D. Bekenstein, "Bekenstein-Hawking entropy", Scholarpedia, Vol. 3, No. 10 (2008), p. 7375, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park..
• 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.