Bekenstein bound

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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.

Upon exceeding the Bekenstein bound a storage medium would collapse into a black hole.^[2] This finds parallels with the concept of a kugelblitz, a concentration of light or radiation so intense that its energy forms an event horizon and becomes self-trapped: according to general relativity and the equivalence of mass and energy.

Equations

The universal form of the bound was originally found by Jacob Bekenstein as the inequality^[1][3][4]

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

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 the gravitational constant G.
In informational terms, the bound is given by

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

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.^[5] Using mass energy equivalence, the informational limit may be reformulated as

${\displaystyle I\leq {\frac {2\pi cRm}{\hbar \ln 2}}\approx 2.5769082\times 10^{43}mR}$

where ${\displaystyle m}$ is the mass of the system in kilograms, and the radius ${\displaystyle R}$ 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.^[6][7] 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.^{[3][4][8][9][10][11][12][13][14][15][16]}

Examples

Black holes

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

${\displaystyle S={\frac {kA}{4}}}$

where A is the two-dimensional area of the black hole's event horizon in units of the Planck area, ${\displaystyle \hbar G/c^{3}}$.
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 ${\displaystyle \approx 2.6\times 10^{42}}$ bits and represents the maximum information needed to perfectly recreate an average human brain down to the quantum level. This means that the number ${\displaystyle O=2^{I}}$ of states of the human brain must be less than ${\displaystyle \approx 10^{7.8\times 10^{41}}}$.