# Is there an equivalence between information, energy and matter?

Consider the physical world as made of information, with energy and matter as incidentals.

This statement must be placed in the context of the holographic principle and the works of Bekenstein ...

Indeed, according to Bekenstein

the thermodynamic entropy and Shannon entropy are conceptually equivalent.

the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information that would be needed to implement any particular arrangement ...

...of matter and energy

The only fundamental difference between the thermodynamic entropy of physics and the entropy of Shannon lies in the units of measurement; the first is expressed in units of energy divided by the temperature, the second in "bits" of information essentially dimensionless.

moreover, the Bekenstein limit asserts that there is a maximum to the amount of information that can potentially be stored in a given region of space that has a finite amount of energy is similar to the holographic principle and that this finally returns to say that the ultimate particle is an information bit (1 or 0).

Can we talk about an equivalence of information, energy and matter ...?

EDIT:

To enlarge the context ,...

Energy and matter

As Einstein showed us, E = MC^2, there is equivalence between mass and energy

Information is energy

Information is a form of energy; indeed to store or transmit information it takes energy

and for his reciproque

energy is information

indeed, Photons emitted by any source, in the form of radiation such as for example stars, these carry information on the source itself its location in space and time, its colors and temperatures, its atomic and molecular composition, its speed of rotation and its direction of movement, ..

• Welcome to SE.Physics! This looks like an interesting question, though I'm having trouble parsing parts of it. I modified the first quote at the start of your question to show how to use a quotation box; if you could do that for the other quotes in your question, it'd probably help to clear it up. – Nat May 25 '18 at 21:28
• It seems the axiom of purification may contribute plus.maths.org/content/… and physics.aps.org/articles/v4/55 – CriglCragl May 25 '18 at 23:02
• Hey, thx for the welkome,... imporved formatting,...can you tell me what parts you have trouble parsing it this in order to improve "readability" – alamata May 25 '18 at 23:08
• this is a fascinating topic indeed, which I personally do not fully understand... an important note: practitioners have different mathematical definitions of "entropy" depending on whether they are dealing with steam, quantum particles, or bits of information- bekenstein notwithstanding. volumes have been written about this, some of which you will find here on the stack exchange. Happy hunting! – niels nielsen May 26 '18 at 6:36

However, we can also take a classical thermodynamic / statistical mechanic interpretation to the question. In this case we have the quantities of entropy $S$ and internal energy $U$, which can be related (via the first law): $dU = TdS + dW$. This version of entropy is related to the Shannon entropy by a constant factor ($k_B/\ln{2}$?) as $S=k_B\ln{\Omega}$, where $\Omega$ is all possible states that the system can be in. If we want to separate matter from mass, we might talk about individual particles (eg those that make up a gas), in which case their multiplicity (amount of matter) determines the entropy of the full system $S$. $S$ is a measure of how well we know the states of each unit of matter. There's also a competition between entropy $S$ and internal energy $U$, where at low temperatures the system will be in a state where $U$ is minimized but at high temperatures $S$ is minimized.
Thermodynamics is well defined only in equilibria, but let us consider a system that moves arbitraily from state A to state B. There is a free energy difference between states A and B. The work involved to move the state from A to B can be greater than the free energy difference (eg due to friction). By rewriting the first law, we see that this work lost ("dissipated") must be accounted for by an increase in entropy. So you could equate the work $W$ of driving a process to an increase in uncertainty $S$. But in such dynamic processes, the work of defining entropy is an area of active study, so we need to be careful in our words.
But all of this depends on context. What is the energy or information of interest? In terms of your two direct questions, we see from the classical thermodynamic description that the entropy $S$ is indeed related to average number of bits needed to describe the system (Shannon entropy). The second on a restriction of information density can be implied from Bekenstein-Hawking entropy / entanglement entropy of a black hole (As the size of the black hole is the limit).