I recently attended a talk by Dr. Ravi Gomatam on 'quantum reality', where the speaker suggested, that conservation of energy is not a fundamental law, and is conditional, but the conservation of information is fundamental. What exactly is the meaning of information? Can it be quantified? How is it related to energy?


4 Answers 4


If one measures lack of information by the entropy (as usual in information theory), and equates it with the entropy in thermodynamics then the laws of thermodynamics say just the opposite: In a mechanically, chemically and thermally isolated system described in the hydrodynamic approximation, energy is conserved whereas entropy can be created, i.e., information can get lost. (In an exact microscopic Hamiltonian description, the concept of information makes no sense, as it assumes a coarse-grained description in which not everything is known.)

The main stream view in physics (aside from speculations about future physics not yet checkable by experiment) is that on the most fundamental level energy is conserved (being a consequence of the translation symmetry of the universe), while entropy is a concept of statistical mechanics that is applicable only to macroscopic bodies and constitutes an approximation, though at the human scale a very good one.

  • $\begingroup$ What you have said, is just a statistical definition of information. Can physical information be defined is some other way ( a quantum mechanical definition?) $\endgroup$
    – user7757
    Commented Mar 8, 2012 at 15:41
  • $\begingroup$ Information is a statistical concept, also in telecommunication engineering, say. It captures the scientific aspect of information, though not its subjective value for human beings. Maybe you can ask morespecifically after having read en.wikipedia.org/wiki/Physical_information $\endgroup$ Commented Mar 8, 2012 at 16:00
  • $\begingroup$ Ramanujan, do you happen to know any online links to explain what Dr Gomatam means by the assertion that conservation of energy (and I'll assume this really means mass-energy) is not fundamental? That alone is a very unusual assertion, so it's not clear exactly what he intended. I looked at his home page, but nothing looked promising based on the titles. $\endgroup$ Commented Mar 8, 2012 at 23:41
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    $\begingroup$ @LegitStack: Energy is never described in terms of information but in terms of a Hamiltonian. $\endgroup$ Commented Mar 21, 2018 at 13:38
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    $\begingroup$ @JánLalinský: Imposing a probability distribution means for a classical system (implied by your reference to phase space) having coarse-grained the system, as without coarse-graining the system is deterministic. $\endgroup$ Commented Mar 23, 2018 at 14:05

In contrast to @ArnoldNeumaier, I'd argue that the information content of the World could be constant: it almost certainly can't get smaller and how it and if it gets bigger depends on the resolution of questions about the correct interpretation of what exactly happens when one makes a quantum measurement. I'll leave the latter (resolution of quantum interpretation) aside, and instead discuss situations wherein information is indeed constant. See here for definition of "information": the information in a thing is essentially the size in bits of the smallest document one can write and still uniquely define that thing. For the special case of a statistically independent string of symbols, the Shannon information is the mean of the negative logarithms of their probabilities $p_j$ of appearance in an infinite string:

$H=-\sum_j p_j \log p_j$

If the base of the logarithm is 2, H is in bits. How this relates to the smallest defining document for the string is defined in Shannon's noiseless coding theorem.

In the MaxEnt interpretation of the second law of thermodynamics, pioneered by E. T. Jaynes (also of the Jaynes-Cumming model for two level atom with one electromagnetic field mode interaction fame), the wonted "observable" or "experimental" entropy $S_{exp}$ (this is what the Boltzmann H formula yields) of a system comprises what I would call the true Shannon information, or Kolmogorov complexity, $S_{Sha}$, plus the mutual information $M$ between the unknown states of distinguishable subsystems. In a gas, $M$ measures the predictability of states of particles conditioned on knowledge about the states of other particles, i.e. is is a logarithmic measure of statistical correlation between particles:

$S_{Exp} = S_{Sha} + M$ (see this reference, as well as many other works by E. T. Jaynes on this subject)

$S_{Sha}$ is the minimum information in bits needed to describe a system, and is constant because the basic laws of physics are reversible: the World, therefore, has to "remember" how to undo any evolution of its state. $S_{Sha}$ cannot in general be measured and indeed, even given a full description of a system state, $S_{Sha}$ is not computable (i.e. one cannot compute the maximum reversible compression of that description). The Gibbs entropy formula calculates $S_{Sha}$ where the joint probability density function for the system state is known.

The experimental (Boltzmann) entropy stays constant in a reversible process, and rises in a non-reversible one. Jaynes's "proof" of the second law assumes that a system begins with all its subsystems uncorrelated, and therefore $S_{Sha} = S_{exp}$. In this assumed state, the subsystems are all perfectly statistically independent. After an irreversible change (e.g. a gas is allowed to expand into a bigger container by opening a tap, the particles are now subtly correlated, so that their mutual information $M > 0$. Therefore one can see that the observable entropy $S_{exp}$ must rise. This ends Jaynes's proof.

See also this answer, for an excellent description of entropy changes an irreversible change. The question is also relevant to you.

Energy is almost unrelated to information, however, there is a lower limit on the work must do to "forget" information in a non reversible algorithm: this is the Landauer limit and arises to uphold the second law of thermodynamics simply because the any information must be encoded in a physical system's state: there is no other "ink" to write in in the material world. Therefore, if we swipe computer memory, the Kolmogorov complexity of the former memory state must get pushed into the state of the surrounding World.

Afterword: I should declare bias by saying I subscribe to many of the ideas of the MaxEnt interpretation, but disagree with Jaynes's "proof" of the second law. There is no problem with Jayne's logic but (Author's i.e. My Opinion): after an irreversible change, the system's substates are correlated and one has to describe how they become uncorrelated again before one can apply Jaynes's argument again. So, sadly, I don't think we have a proof of the second law here.

  • $\begingroup$ I think you've misunderstood Jaynes' proof. Jaynes doesn't say the system's components become uncorrelated, he says that (some of) the correlations become irrelevant for making predictions about the system's future behaviour, so you can safely forget about them. Thus we pretend that the system's components have become uncorrelated, even though we know they haven't really, because this allows us to do calculations that would be completely intractable otherwise. $\endgroup$
    – N. Virgo
    Commented Jul 16, 2013 at 6:44
  • $\begingroup$ He makes the argument much more clearly in section 4 of this paper. $\endgroup$
    – N. Virgo
    Commented Jul 16, 2013 at 6:47
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    $\begingroup$ @Nathaniel I'm trying not to get too far off the track here. I am quite familiar with the argument you cite above, but I still think it's begging the question (by brining in further assumptions about what is and what is not "relevant" mutual information). I am not convinced there is an argument that fully resolves the Loschmidt paradox aside from saying that the second law is about boundary conditions of the universe. I understand that not everyone agrees on this point. Moreover, please understand that I would not call ANY of Jayne's assumptions unreasonable. $\endgroup$ Commented Jul 16, 2013 at 7:01
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    $\begingroup$ Jaynes argument for the second law is very explicitly based on an empirical fact. That fact is that we, as experimenters and engineers, are able to directly influence the initial conditions of an experiment, but we can't affect the final conditions except indirectly, by changing the initial conditions. Jaynes' argument says that given this asymmetry, the second law follows. However, this empirical fact in itself is then in need of an explanation, which Jaynes' argument can't help us with, and that's probably where you have to start thinking about the boundary conditions of the universe. $\endgroup$
    – N. Virgo
    Commented Jul 16, 2013 at 16:55
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    $\begingroup$ @Nathaniel I'm glad we seem to agree then. One sometimes reads people citing Jaynes's work as a "mathematical proof" for the second law, independently of experiment and that it explains the arrow of time. I don't believe that Jaynes himself ever claimed this status for his work - indeed he seems often to be "going the other way", i.e. beginning with physical reality and experiment and calling on these to shed light on the philosophical foundations of chance and randomness. This is why I baulk at calling his work a "proof" of the second law. $\endgroup$ Commented Jul 17, 2013 at 4:10

Energy is the relationship between information regimes. That is, energy is manifested, at any level, between structures, processes and systems of information in all of its forms, and all entities in this universe is composed of information. To understand information and energy, consider a hypothetical universe consisting only of nothingness. In this universe imagine the presence of the smallest most fundamental possible instance of deformation which constitutes a particle in this otherwise pristine firmament of nothingness. Imagine there is only one instance of this most fundamental particle and let us dub this a Planck-Particle PP. What caused this PP to exist is not known, but the existence of the PP constitutes the existence of one Planck-Bit (PB) of information. Resist the temptation to declare that energy is what caused our lone PP to exist. In this analogy, as in our reality, the ‘big’ bang that produced our single PP is not unlike the big bang that caused our known universe in that neither can be described in terms of any energy relationship or regime known to the laws of physics in this universe.

This PB represents the most fundamental manifestation of information possible in this universe. Hence, the only energy that exists in this conceptual universe will be described by the relationship (there’s that word again) between the lone PP and the rest of the firmament of nothingness that describes its universe. Call this energy a Planck-quantum (PQ). Note that this PQ of energy in this universe only exists by virtue of the existence of the PP alone in relation to the surrounding nothingness. With only one PP there are few descriptions of energy that can be described. There is no kinetic energy, no potential energy no gravity no atomic or nuclear energy no entropy, no thermodynamics etc.. However, there will be some very fundamental relationships pertaining to the degrees-of freedom defined by our PP compared to is bleak environment that may be describable as energy.

Should we now introduce a second PP into our sparse universe, you may now define further relationship and energy regimes within our conceptual growing universe, and formulate Nobel worthy theories and equations which describe these relationships. Kinetic energy suddenly manifest as the relationship of distance between our lonely PP’s suddenly comes into existence. Likewise, energy as we know it describes the relationships manifested between information regimes which are describable by the language of mathematics.

  • $\begingroup$ I appreciate the intuitive nature of this answer. $\endgroup$
    – MetaStack
    Commented Mar 21, 2018 at 16:02

Electron and Information.

Information is transferred through EM waves. There isn't EM wave without electron. (H. Lorentz)

Information is the new atom or electron, the fundamental building block of the universe ... We now see the world as entirely made of information: it's bits all the way down. (Bryan Appleyard)

It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. (Richard Feynman about an electron)

Electron is a quantum of information. Electron is a keeper of information. Why? An electron has six ( 6 ) formulas: $$E=h*f\qquad \text{and}\qquad e^2=ah*c ,$$ $$E=Mc^2\qquad \text{and}\qquad -E=Mc^2 ,$$ $$E=-me^4/2h^2= -13.6eV\qquad \text{and}\qquad E= \infty$$ and obeys five (5) Laws :

  • a) The Law of conservation and transformation energy/ mass
  • b) The Heisenberg Uncertainty Principle / Law
  • c) The Pauli Exclusion Principle/ Law
  • d) Dirac - Fermi statistic
  • e) Maxwell / Lorentz EM law

It means in different actions electron must know six different formulas and must observe five laws. To behave in such different conditions a single electron itself must be a keeper of information.

The laws of physics dictate that information, like energy, cannot be destroyed, which means it must go somewhere. (Michael Brooks, Book ‘ The big questions’. Page 195-196.)

It means an electron (as a little blobs of a definite amount of energy) even in different situations never loses its information.

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    $\begingroup$ Unfortunately, this is mostly incoherent. By the time you catch yourself signing your answers with your full name in bold, it's time to step back and reevaluate things. $\endgroup$
    – Nat
    Commented May 13, 2018 at 13:20