# Information Equation of State

## Abstract

**:**

^{87}bits is sufficient for information energy to account for all dark energy. A time varying equation of state with a direct link between dark energy and matter, and linked to star formation in particular, is clearly relevant to the cosmic coincidence problem. In answering the ‘Why now?’ question we wonder ‘What next?’ as we expect the information equation of state to tend towards w = 0 in the future.

## 1. Introduction

_{B}T ln2 of heat energy has to be dissipated into the surrounding environment to increase the environment’s thermodynamic entropy in compensation, and in accord with the second law of thermodynamics. The total amount of information is conserved as the surrounding environment effectively contains the erased information, although clearly no longer in a form that the computer can use.

_{2}N

_{L}, is just the number of bits needed to distinguish between the N

_{L}information bearing degrees of freedom used for computer logic states [10]. In comparison, thermodynamic entropy, k

_{B}lnN accounts for the N discrete Boltzmann microscopic states of all constituent atoms, electrons, etc. In equivalent units, the thermodynamic entropy of such a computer memory device is therefore many orders of magnitude larger than the information entropy. The energy dissipated by present day computer memory through the Landauer’s principle is thus miniscule in comparison to normal thermal dissipation. However information entropy is equivalent to thermodynamic entropy when the same degrees of freedom are considered. The information entropy of the physical world is thus the number of bits needed to account for all possible microscopic states. Then each bit of information is equivalent to ΔS = k

_{B}ln2 of thermodynamic entropy leading to Landauer’s principle that ΔS T = k

_{B}T ln2 of heat is dissipated when a computer logic bit is erased.

## 2. Applying Landauer’s principle to the universe

_{B}T ln2 of heat energy with an associated increase in the states of the environment surrounding the memory device and overall information is conserved. More generally, Landauer’s principle applies to all systems in nature so that any system, temperature T, in which information is ‘erased’ by some physical process will output k

_{B}T ln2 of heat energy per bit ‘erased’ with a corresponding increase in the information of the environment surrounding that system.

^{90}bits and that around 10

^{120}bits have been processed to present [18,19,20]. Note that 10

^{90}bits is significantly less than the theoretical maximum universe information content of 10

^{123}bits provided by applying the holographic principle [18,21] to the universe’s surface area. In the maximum information scenario, corresponding to the universe being one single black hole, 10

^{123}elemental squares of Planck length size are needed to cover the surface of the present known universe.

^{90}bits of information intrinsic to the physical world. Fortunately, for the purpose of deriving this information energy contribution to the universe energy balance it is not necessary to quantify what fraction of the information processed is being, or has been, ‘erased’ in these processes, or even to identify the possible physical processes responsible for ‘erasure’. It is sufficient to be able to say information may be ‘erased’ and thus allow us to equate each bit to k

_{B}T log2 of energy that can be dissipated as heat. Such equivalence is standard in cosmology and not fundamentally different from the way we use mc

^{2}to represent the contribution of all mass to the total universe energy budget when to date only a small fraction of baryon mass has been converted into fusion energy within stars. Most importantly note the equivalent energy of a bit depends solely on the temperature of the system of microstates described.

## 3. Information equation of state: just prior to star formation

_{tot}(including dark matter):

_{tot}c

^{2}= σ

_{v}T

^{4}

_{v}, is defined in terms of the Stefan-Boltzmann (area) constant, σ

_{a}, by σ

_{v}= 4σ

_{a}/c. Replacing σ

_{a}by its definition in terms of fundamental constants and applying Landauer’s principle we obtain equation (2), the energy, E, associated with each bit:

_{B}T ln2 = (15 ρ

_{tot}

^{3}c

^{5}/ π

^{2})

^{¼}ln2

^{¼}, and thus to a

^{-¾}, where a is the universe scale size. We expect from the second law that total information did not decrease and so we can assume that the evolution of the universe total information bit content lay between two limits. At one limit the total bit content remained fixed with the information bit number density falling in parallel with the density of matter as a

^{-3}. At the other limit the number of microstates increased with increasing volume to provide a constant bit number density. These simply defined limits have been chosen to be broad enough to include previously considered cases [18,19,20,22,23] and at the same time still provide a clear comparison against the period of star formation discussed next. Thus information energy density varied with universe scale size as a

^{-3.75}if total information bit content remained constant, and as a

^{-0.75}if information bit density was constant. These variations (as a

^{-3(1+w)}where w is the equation of state) correspond to an information equation of state, w

_{i}, just prior to star formation bounded by the range: -0.75< w

_{i}< +0.25.

## 4. Information equation of state: during star formation

^{7}K. Note that the relative difference between the interior temperatures of stars of different types and at different nuclear burning stages, or ages, is small compared with the many orders of magnitude these stellar interior temperatures exceed the temperature of non-stellar matter, primarily in the form of gas and dust. Accordingly, during the early rapid stellar evolution the average temperature and hence average characteristic information bit energy changed primarily in proportion to the fraction of all bits accounting for information in stars. As this information accounts for the Boltzmann microstates of matter, we expect that fraction equals the fraction of all baryons that are in stars.

_{b}, that are in stars has been replotted as a function of universe scale size, a, (replotted from figure 1 of [24]). To simplify modelling the data has been split into two regions by the vertical line: the main region of rapid change with redshifts, z>0.8; and a more constant region z<0.8. Data points for z>0.8 are reasonably well described by a power law variation with the best fit for f

_{b}(solid line) corresponding to a

^{+3.6}and the best fit for the specific variation as a

^{+3.0}is also drawn (dashed line) for comparison. A detailed fit to the data might entail at least three separate periods of fit with the steepest period in the middle, but, for data z>0.8, it is sufficient for our purposes to show that f

_{b}varies close to a

^{3}.

^{-0.75}reaching a temperature today only one order of magnitude higher than the 2.7K temperature of the cosmic microwave background radiation, CMB, consistent with the present matter energy density four orders of magnitude higher than the CMB energy density (see equation 1). However, the proportion of baryons in stars increased as a

^{+3.0}reaching a maximum of around 10% at z=0.8, causing the average bit temperature to also increase as a

^{+3.0}reaching a temperature of a few times 10

^{6}K (10% of typical stellar temperatures) at z=0.8.

^{+3.0}is the specific characteristic needed to counteract the a

^{-3.0}bit density dilution caused by the increasing separation between formed stars as the universe expands. Thus, for the period 10>z>0.8, there was a near constant overall information energy density (a

^{0}), or an information equation of state, w

_{i}~ -1.0.

^{-1}, necessarily limited by definition to f

_{b}<1. For reference, the black square in Figure 1 corresponds to the present day estimated fraction of all baryons that are in galaxies [24,25].

**Figure 1.**Upper plot. The cumulative fraction of all baryons, f

_{b}, that have formed stars plotted as a function of universe scale size, a, from Hu & Cowie [24]. The black line is the best power law fit for data z >0.8 (vertical line limit) and the dashed line the best a

^{+3.0}fit. The reader is referred to [24] for details of all experimental techniques and corrections applied in this survey of measurements. Lower plot. The increasing fraction of baryons found in stars during the period 10> z>0.8 causes the overall average baryon (and bit) temperature to also increase as a

^{+3.0}.

## 5. Overall time history of w_{i}

_{i}<+0.25. During stellar formation the proportion of high temperature bits increased close to a

^{+3}counteracting the a

^{-3}dilution effects from increasing star separations as the universe expanded. This resulted in a nearly constant overall information energy density, or w

_{i}~ -1.0, for the major part of cosmic time, 10>z>0.8. During this rapid growth in stars, the subsequent changes in stellar temperature with star type and age after birth would have had little effect on the average bit temperature compared with the large temperature increase at star formation.

_{b}slowed to reach a limit with the fraction of all baryons that are in stars ~10

^{-1}today as star birth and death rates become similar. Now the distribution of star types and temperature evolution with age becomes important. A detailed account of these effects to estimate the variation in average bit temperature in this recent period is beyond the scope of this work. We can assume that w

_{i}has remained significantly negative, at least initially, after z=0.8 as the average temperature continued to increase, primarily determined by the stars formed around z~0.8 as they move through hydrogen burning to hotter helium and later burning stages. In the future the overall distribution of stellar temperatures is expected to change less, leading to a more constant average temperature and a tendency for w

_{i}➔ 0.

## 6. Discussion

_{DE}~ -1.0 during the rapid star formation period 10>z>0.8, or for over one half of cosmic time. Any negative equation of state, and specifically the value w

_{i}~ -1.0, implies that information energy must make a contribution to dark energy.

^{6}K (~10% of typical stellar temperatures) and thus the average bit energy around 120eV, from k

_{B}T ln2. Then, in order for the information energy to account for all dark energy, the total universe information content would need to be ~10

^{87}bits. This value is not unreasonable and is similar to the previous estimates of 10

^{90}bits [18,20]. However, care must be taken to avoid a circular argument and not to rely solely on this similarity to support our argument since the 10

^{90}bits value of the previous work [18,19,20] was effectively estimated by dividing the universe total mass energy by the characteristic energy of our equation (2).

_{i}➔ 0 in the future since the extent of stellar formation has reached a maximum. Whilst we expect dark energy to remain the dominant energy component of the universe, dark energy density will no longer remain constant but will eventually fall along with mass energy density as a

^{-3}, or w

_{i}=0. The universe will continue to expand but no longer with an accelerating expansion.

^{7}K. CMB can be considered a ‘snapshot’ of the early universe, effectively possessing an information content equal to that of the early universe, and therefore similar in value to today’s universe, assuming CMB and the universe have both conserved information content in the intervening period. Then the very low 2.7K CMB temperature means the CMB information energy density is also insignificant compared with the hot stellar baryon component.

_{i}during stellar evolution we have ignored the loss of entropy, or reduction of information, resulting from the reduction in the number of microstates assumed to take place on star formation as initially highly disordered matter collapses to form the more ordered structure of a star [26]. However, this reduction in bit number is small compared with the major increase in temperature at star formation and will have no effect on our calculated equation of state parameter. We can illustrate this by assuming the information contained by any physical system must be similar to the information required to fully simulate that system on a computer. For example, a full simulation of baryons during star formation would require a resolution at least of the order of the Planck length, 1.6 x 10

^{-35}m. Consider changing from describing a baryon location within the universe (~10

^{26}m) to a location within a typical star, for example the sun (~10

^{9}m). This corresponds to a change in accuracy from one part in 6x10

^{60}(~2

^{202}) to one part in 6 x 10

^{43}(~2

^{145}). In order to maintain accuracy sufficient for a full simulation the minimum number of bits required to represent that location parameter on the computer only changes from 202 bits to 145 bits. Such a small 28% reduction in information is miniscule compared with a typical five orders of magnitude increase in temperature at star formation.

^{-3}eV bit energy or characteristic energy that has been associated [28] with a cosmological constant. Thus our information based argument naturally explains this low value as k

_{B}T ln2 – a value previously thought ‘difficult to explain as it is too small to relate to any interesting particle physics’ [28].

^{3}rise in the fraction of baryons in stars ceased around z=0.8. Although those stars formed at that time will continue to evolve through higher temperature nuclear cycles the average bit temperature clearly no longer rises as steeply as a

^{3}. We therefore expect to see some observable change to the dark energy equation of state towards a value w>-1 in the most recent times. The contribution information energy makes to dark energy might therefore be verified experimentally by one of the future experiments planned to measure the effects of dark energy with greater precision.

## 7. Summary

- The information equation of state was clearly negative for at least one half of cosmic time, 10>z>0.8, with a value close to the dark energy w ~ -1.0.
- Information energy can easily account for all of the dark energy with an information bit content of ~10
^{87}bits, similar in magnitude to previous estimates for the universe bit content. - The equation for the characteristic bit energy of non-stellar information is identical in form to the equation for the characteristic energy associated with a cosmological constant.
- The low characteristic energy associated with dark energy was previously thought difficult to explain from particle considerations but is a natural result of this information approach.
- Information energy is directly related to the degree of stellar evolution, and thus can provide an answer to the fundamental cosmic coincidence question ‘Why now?’
- Occam’s razor argues strongly for such a simple explanation to the main (70%) energy component of the universe.
- Whether information energy is the source of dark energy can be tested experimentally by searching for evidence of a dark energy value w>-1.0 in the most recent period (z<<0.8).

## Acknowledgements

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Gough, M.P.
Information Equation of State. *Entropy* **2008**, *10*, 150-159.
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Information Equation of State. *Entropy*. 2008; 10(3):150-159.
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2008. "Information Equation of State" *Entropy* 10, no. 3: 150-159.
https://doi.org/10.3390/entropy-e10030150