### The Mass of Information

December 5, 2014

Einstein's most famous

equation, at least to people who aren't that interested in

general relativity, is the

mass-energy equivalence equation,

**E = mc**^{2}. As I wrote in a

previous article (Mass-Energy Equivalence, June 2, 2014), Einstein's

publication of mass-energy equivalence didn't contain the

**E = mc**^{2} equation; instead, Einstein wrote

**L = mc**^{2}, where L is the

Lagrangian.[1]

The Lagrangian (

**L = T - V**) is the difference between the

kinetic energy (

**T**) and the

potential energy (

**V**) of a

system. It's a useful concept in

classical mechanics, which was essentially the only developed form of mechanics when Einstein published his paper, just five years after the discovery of the

quantum of energy by

Max Planck.

The equation, as written in its traditional form, puts the emphasis of

energy, but it's possible to invert the equation to associate a

mass with a quantity of energy. If you have an energy, you can use Einstein's equation to calculate its equivalent mass. Such masses are small, since the

speed of light, 300 million

meters per second,

squared, is the

conversion factor between the

SI units for energy and mass, the

joule and the

kilogram; viz., c

^{2} = 9 x 10

^{16} meters

^{2}/sec

^{2}.

Since even

scientists don't have a good idea of the magnitude of some SI units as they relate to

everyday life, we note that a

kilowatt-hour is 3.6 million joules (3.6 x 10

^{6} joules). We can convert this amount of energy, enough to power a

hair dryer for an hour, to an equivalent mass, which gives us just 4 x 10

^{-8} grams (40

nanograms).

In 1961, at the time when

computers were becoming more common,

Rolf Landauer (1927-1999), a

physicist working at

IBM, investigated the

thermodynamics of computing. In most cases, computing is an

irreversible process, so computation should cause some energy to be lost to the

environment.

Landauer's calculation of the minimum energy lost per irreversible bit operation gives a result that appears obvious in retrospect; namely, E = kT ln(2), where k is the Boltzmann constant (1.38 x 10

^{-23} J/K), T is the temperature, and ln() is the natural logarithm. The factor of 2 comes, of course, from the idea that a bit has two states.

All this is related to the concept of entropy

**S**, as given by Boltzmann's equation,

**S = k ln(Ω)**, in which

**Ω** is the number of possible system states, combined with the idea that the product of temperature and entropy,

**TS**, gives the internal energy of an

adiabatic system; that is, a system that doesn't exchange heat or mass with its surroundings. The minimum energy of a

bit operation is very small. At

room temperature (about 25

°C), it's just 2.85 zeptojoules. For those who have trouble sorting their

zeptos from their

femtos and

attos, a zeptojoule is 10

^{-21} joule. This energy has actually been measured in a

model memory cell comprising a

colloidal particle moving between two

potential wells.[2]

The discussion above about the

Landauer's limit in computation relates to the energy we see as the result of bit changes, and this includes the creation of a bit of

information. We can associate a mass with such an energy, and that's an idea posed in a paper posted on

arXiv earlier this year by Luis Herrera of the

School of Physics, the

Universidad Central de Venezuela, Caracas, Venezuela.[3-4] Essentially, he applies the mass-energy equivalence to the Landauer energy. As a result, the mass of a single bit of information at room temperature is about 3 x 10

^{-35} grams, which is a hundred million times smaller than the mass of an

electron (9.11 x 10

^{-28} grams).[3]

It's also possible to plug this energy into

Heisenberg's uncertainty relation, as written in

time and energy, ΔE Δt ≈

*h*/2π, where

*h* is the

Planck constant (6.626 x 10

^{-34} joule-sec), to get a time

**t** associated with the energy change. Inverting this time gives you the maximum

frequency at which information can be changed, 10

^{5} GHz, which is comfortably higher than the several GHz of today's

desktop computers.[3]

### References:

- A. Einstein, "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Annalen der Physik, vol. 18, no. 13 (1905). A PDF file of an English translation (Does The Inertia Of A Body Depend Upon Its Energy-Content?) can be found here.
- Antoine Bérut, Artak Arakelyan, Artyom Petrosyan, Sergio Ciliberto, Raoul Dillenschneider, and Eric Lutz, "Experimental verification of Landauer’s principle linking information and thermodynamics," Nature, vol. 483, no. 7388 (March 8, 2012), pp. 187-189, doi:10.1038/nature10872.
- L. Herrera, "The mass of a bit of information and the Brillouin's principle," arXiv, March 18, 2014.
- L. Herrera, "The mass of a bit of information and the Brillouin's principle," Fluctuation and Noise Letters, vol. 13, no. 1 (March 2014), Article No. 1450002 (5 pages, DOI: 10.1142/S0219477514500023).