Landauer’s Principle of Minimum Energy might place limits on the detectability of gravitons of certain mass

NOVA, Department of Mathematics, 8333 Little River Turnpike Annandale, VA 22003 USA Abstract According to Landauer’s principle, the energy of a particle may be used to record or erase N number of information bits within the thermal bath. The maximum number of information N recorded by the particle in the heat bath is found to be inversely proportional to its temperature T. If at least one bit of information is transferred from the particle to the medium, then the particle might exchange information with the medium. Therefore for at least one bit of information, the limiting mass that can carry or transform information assuming a temperature T= 2.73 K is equal to m = 4.71810 kg which is many orders of magnitude smaller that the masse of most of today’s elementary particles. Next, using the corresponding temperature of a graviton relic and assuming at least one bit of information the corresponding graviton mass is calculated and from that, a relation for the number of information N carried by a graviton as a function of the graviton mass mgr is derived. Furthermore, the range of information number contained in a graviton is also calculated for the given range of graviton mass as given by Nieto and Goldhaber, from which we find that the range of the graviton is inversely proportional to the information number N. Finally, treating the gravitons as harmonic oscillators in an enclosure of size R we derive the range of a graviton as a function of the cosmological parameters in the present era.


Introduction
The existence of any physical systems presupposes the registration of information just because it exists. Dynamically evolved systems not only process but they also transform information. In this case is the laws of physics determine the precise amount of information that a system can register or process or transfer as well as the exact number of logical operation that the system can perform. In Landauer's original paper (1988), the author went on making the statement that information is physical but at the same time all this information is registered and processed by physical systems. Furthermore, the laws of physics that describe a certain system can involve information and information processing. On the other hand, Landauer's principle is a physical principle that defines the lower possible theoretical limit of energy consumption during a computation. Landauer postulated that any "logically" irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase of non-information bearing degrees of freedom in the information processing apparatus or its environment" (Bennett 2003). In other words, we can say that Landauer (1961) simply postulated that information being physical has an energy equivalent, it must also obey the laws of physics and in particular the laws of the thermodynamics. At this point, we must also say that even though non-equilibrium extensions as well as quantum extensions to the Landauer principle have not been considered yet. Furthermore, Landauer's principle is a physical principle pertaining to the lower theoretical limit of energy consumption of a computation. He also postulated that any "logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information bearing degrees of freedom of the information processing apparatus or its environment" (Bennett 2003). Landauer's principle asserts that there is a minimum possible amount of energy required to change one bit of information, known as the Landauer's limit, and it is equal to: where kB =1.38×10 −23 J/K is Boltzmann's constant, and T is the temperature of the circuit. Therefore, Landauer's energy formula is written in terms of the information number N  1 in the following way (2) In a recent paper by Vopson (2019) the author postulates the idea of a new principle of mass-energyinformation equivalence. He simply proposes that a bit of information it's not physical as it has already been proposed but there is actually a certain mass associated with it where the information its stored. For example, at room temperature T = 300 K the mass corresponding to one bit of information is equal to

The theory
In a private communication paper by Bormashenko (2019), the author postulates and considers a particle with energy E in contact but not necessarily in thermal equilibrium with the thermal bath at temperature T. According to Landauer's principle, the energy of a particle may be used to record or erase N number of information bits within the thermal bath. The maximum number of information recorded by the particle in the heat bath is equal to: Equation 3 can be considered as the maximum number of information contents in bits of a particle at rest (Bormanshenko, 2019). If at least one bit of information is transferred from the particle to the medium, then the particle might exchange information with the medium. In this case we can write that Nmax ≥ 1should be true. Following Bormanshenko, ibid (2019) using Eq. 3 we obtain the following relation for the mass of the particle we have that: To get an idea with of the limiting mass that can carry or transform information to the medium following (Bormanshenko (2019) let us assume the temperature to be that of the background radiation of the universe namely T = 2.725 K we obtain that: In reference to the numerical result of equation (5) we must say that in today's standard model the mass values of the elementary particles including that of neutrino with estimated mass are much heavier than the mass predicted above. In table 1, we remind the reader of the order of magnitude of the some of the elementary particle masses. Therefore, particles with masses smaller than m0 ≤ 10 -40 kg will not transform/transmit the information they carry to the universe and therefore as a result they will not be detectable.
Specifically, the dispersive term is ascribed to the quantum of gravitation having non-zero rest mass mgr or equivalently a non infinite Compton wavelength / gr mc  = and where  is the potential function of the gravitational field. Theories like M-theory, loop quantum gravity, string theory and superstring theory and quantum field theory all predict the existence of graviton particles. In relation to quantum field theory gravitons are the elementary particles that mediate the force of gravity, expected to be massless which results to an infinite range gravitational force. Furthermore, gravitons are bosons of spin-2 which is related to the stress energy momentum tensor, and can give rise to a force that is indistinguishable from gravitation simply because the massless spin-2 field is coupled to the stress-energy tensor in the same way that gravity field does (Lightman et al. 1975). Therefore, if a particle of spin 2 is discovered it must be graviton. Its detection will be an important step in the validation of the gravitational theories above (ibid, 1973) and unify quantum mechanics with general relativity. At the same time, we should say that the extreme weakness of the gravitational force makes the graviton detection a very hard issue. In a recent book by Dyson (2004), the author suggests "the detection of a single graviton may in fact ruled out in the real universe". If this proves to be true, then issues will be raised immediately for the quantization of gravity. On the other hand, attempts to extend the standard model by including gravitons have failed at high energies because of the infinities arising at quantum effects, which means that gravitation is nonrenormalizable (Birrel and Davies, 1994). This also means that quantum mechanics and gravity are incompatible at these energies and thus the situation becomes tenable.
Concluding in relation to the mass of the graviton, we say that in a recent paper by Novello and Neves (2003) the authors indicate a link between the cosmological constant and the graviton mass mgr.
Theory predicts that particles that travel with the speed of light have practically zero mass m ≥ 10 -68 kg.
This result is in excellent agreement with the current experimental mass bound of photon and graviton, something that suggests that entropic gravity might result of a softly broken recent local symmetry (Mureika and Mann, 2011). Moreover, cosmological holography postulates all the information content in our universe is encoded at the cosmological horizon, a proposal forwarded by Smoot (2010).

Discussion and Numerical Results
At this point, we will use Landauer's principle and calculate the corresponding mass of a graviton if the graviton exchanges information with the medium at least one bit of information, using the result in for the temperature of graviton. Thus, according to Evans and Lineweaver (2010) a graviton thermal background is expected to exist which decoupled from the photon bath around the Planck time, and has been cooling since as To be more specific Evans and Lineweaver claim that photons cooled less quickly because they have been heated by the annihilation of heavy species, and therefore the current graviton temperature to the current photon temperature and can be written as: Where: ( ) .... 8 In previous estimates of the background graviton entropy the authors have assumed Using the above temperature instead as the temperature under which the graviton now can exchange information and if it exchanges at least one bit of information with the surrounding medium we find that the mass of the graviton to be equal to: The above-calculated graviton mass exchanging one information bit with the medium is an order of magnitude less than the mass limit predicted by Bormanshenko (2019) where the author has used for temperature of the cosmic microwave background radiation instead. This is the lower bound mass limit required for the graviton to transform information to the medium at its own temperature as calculated by Evans and Lineweaver (2010). Therefore, we conclude that at one bit of information gravitons in the range kg which is four orders of magnitude heavier that the mass predicted above.
Next using the temperature of the graviton (ibid, 2010) we can now write the number of information bits as a function of the graviton mass in the following way (Haranas et al. 2013 This is the number of information bits, which various graviton masses will be able to transfer to the medium because of their masses. This has been calculated is under the assumption that all gravitons have a temperature equal with that of the graviton relic T = 0.61 K. We can easily see that the number of information bits is a linear function of the mass of the graviton particle in the following way: From equation (12) we see that it takes a quantifiable graviton of mass mgr = 6.48510 -41 kg to contain and transfer one bit of information to the medium, which is impossible since the calculated graviton mass range is many orders of magnitudes smaller that the limiting mass. In our calculation, we have used the corresponding graviton temperature as calculated by Evans and Lineweaver (2010), but the mas is still many orders of magnitude when compared with that mass of most elementary particles as given in table 1 above. In reference to Eq. (13) in Haranas and Gkigkitzis (2013), certain cosmological scenarios involve fractional number of information bits. According to information theory, fractional information bits imply uncompressed data. This will also imply information that is not possible to be decompressed (or sometimes it has not been decompressed yet) through matter and energy. Thus, we may perceive matter in our universe as a system of specific information in which any interaction and law takes place with the exchange of specific amount of information under the assumption that matter is an entity analogous to a computer.
Next, is known that the Compton wavelength or range of a boson  in this case the graviton it is given by the following relation (Haranas, 2017): Using Eq. (3) for or the mass of graviton, as function of the maximum number of information N that may be recorded by the particle with the bath, we can write the range of graviton in the following way: In other words, the range of the graviton scales inversely proportional to the information number N. Thus, if a graviton contains a very small number of information bits is expected to have a very large range or Compton wavelength. In that aspect we postulate that the less information a graviton carries the less will probably interact with the rest of the universe and it will propagate in large cosmic distances. Therefore, we propose that the information bit might be an alternative way via which graviton, particles and matter could interact with the rest of the universe. Thus, we can picture information as a new way via which matter interacts with the rest of the universe. It could be that information also tells matter how to interact. Using the values predicted for the mass of graviton according to  we find that  falls in the range: 3.513 10 m 3.513 10 m Talking into account that the radius of the universe is 26 4.4 10 uni R = m ( Bars and Teming, 2009) and (Davis and Lineweaver 2004) we say that such a graviton of different masses can have a ranges approximately equal to the size of the visible universe and the solar system respectively. In Finn and Sutton (2002) applied this idea in a investigation of the data from the Hulse-Taylor binary pulsar and from the pulsar PSR B1534+12. From their analysis of the data, they found the graviton range to be 12 2.60 10 m

 
which is almost identical to our lower bound in equation (16). That was under the assumption that, some linearized theories, allow a massive graviton which would propagate freely via the Klein-Gordon equation of a particle with mass mgr. If the graviton had a rest mass, the decay rate of an orbiting binary would be affected (Taylor et al., 1992). As the decay rates of binary pulsars agree very well with GR, the errors in their agreements provide a limit on a graviton mass.
Next, in a more general way following Viaggiu (2017) where: where again R is the dimension of the "box" and is In the present era x << 1 has an extremely small value and therefore expanding (17) Using the value of x and after some algebra we can write eq. (20) in the following form: for which we can say that the entropy for a total number of gravitons gr i N treating gravitons as harmonic oscillators in an enclosure of size R and temperature T and it's equal to an effective kB Boltzmann constant, by twice the value of the square bracket in the above equation. It is the proportionality factor that relates the average relative kinetic energy of the gravitons in the graviton gas with the thermodynamic temperature of the Taking R to be the Hubble radius RH = c/H0 and assuming one graviton i N = 1. equation (21) where we have used that qua min EH = is the so-called minimum quantum energy of the graviton (Gershtein et al. 2003). Next, let us now consider the total number of gravitons in the universe as its given in Haranas and Gkigkitzis (2014): where  is the cosmological and pl is the Planck length. Substituting (27) in (15) we obtain the total number of information for the total number of gravitons in the universe is equal to: Next let us assume that the graviton can have an "infinite" range, would imply that the denominator of eq.
(28) should be equal to zero for two different possibilities for the thermal bath temperature, i.e. T = 0 and also T equal to equation (25) Similarly, and infinite range will require a Hubble constant that is equal to: In flat universe where the Hubble constant depends on information in the following way (Haranas et al. 2013a) Equating (30) and (31) we can write this temperature of the bath as a function of information in the following way: (34) The above entropy is approximately 0.271 times smaller than the one calculated by Egan and Lineweaver (2010) at the visible horizon of the universe, namely  (40) in Egan and Lineweaver (2017) paper namely: the radius of the horizon and solving for the radius of the horizon we obtain: where today's radius of the universe is calculated using the black body background radiation temperature T= 2.725 K and also the individual graviton temperature as given by Egan and Lineweaver (2010) i.e. T = 0.61 K in the above equation we find that: where the radius of the universe in Egan and Lineweaver is calculated to be 26 1.550 10 u R = m (ibid, 2010).
At his point, we estimate the temperature required for which the graviton has an infinite range. In the present era we find that the corresponding temperature is: This is an extremely low temperature. To obtain an idea of such a low temperature let us consider the mass universe of Similarly using equation (15) and solving for the number of information N we obtain that: Assuming that the range of the graviton is at least equal to the visible horizon of the universe i.e.
Using the temperature of microwave background radiation i.e. T = 2.725 K as well as the temperature of the graviton relic as given in equation (9) This number is the same order of magnitude as the total number of gravitons in the universe being equal to: where N is the number of the total information in the universe. Looking at equation (45) it might be worth trying to understand this numerical result. We postulate that the result might be related to signal processing, data compression, and source coding, or bit-rate reduction which is the process of encoding information by making use of fewer bits that the original representation. This is known as lossless compression and reduces the number of bits by identifying and eliminating the statistical redundancy but without any information lost during the process, which is an algorithmic one. This is taking place by removing unnecessary or unimportant information. In our case we postulate that the universe performs data compression, playing the role of an encoder, and the graviton which performs the decompression plays the role of a decoder.
Such a compression perhaps uses the spacetime physics and geometry as resources required to store and transmit data. Computational resources such as history of orbits, interactions and configurations are consumed in the compression and decompression processes. Data compression is subject to a space-time complexity trade-off. The design of data compression schemes involves trade-offs among various factors, including the degree of compression, the amount of distortion introduced (when using lossy data compression, perhaps due to an increasing entropy in the universe), and the computational resources required to compress and decompress the data.
For example, highly repetitive input such as the orbit of a planet or particle can effectively be compressed, for instance, such as in biology a biological data collection of the same or closely related species can be compressed. The basic task can be achieved through a context free physics. However, just as the human eye is more sensitive to subtle variations in luminance than it is to the variations in color, it is very likely that our perception is restricted to the use of several compression formats that lead to paradoxes, such as a graviton having information context greater than the universe.
Inspired by the close connection between machine learning and compression where a system can predict the posterior probabilities of a sequence given its entire history for optimal data compression, we may expect that a particle such as the graviton, can read the history of the universe and has more bits of information as predicted posterior probabilities of the system (the concept of "general intelligence. (Mahoney, 2013), (Shmilovici et al., 2009) and(Ben Gal 2008). When we study the universe, our rational perception resembles the psychoacoustics of our ears where not all data in an audio stream can be perceived by the human auditory system. But as a lossy compression, it reduces redundancy by first identifying perceptually irrelevant sounds, that is, sounds that are very hard to hear.
Another hypothesis we can make is that perhaps the universe manifests only certain aspects of its information context that can be measured or identified by our scientific methods. Something like it happens to speech encoding, an important category of audio data compression. The range of frequencies needed to convey the sounds of a human voice are normally far narrower than that needed for music, and the sound is normally less complex. As a result, speech can be encoded at high quality using a relatively low bit rate.
Perhaps the information context of the universe has an intrinsic mechanism that "hides" away more of the information bits-keeping just enough to reconstruct an "intelligible" universe to the observer rather than the full realization range of events, that can be modeled even beyond quantum probabilities.

Conclusions
In this paper the idea that the max number of information N recorded by one particle emerged in a heat bath is inversely proportional to the particle's temperature T. Furthermore if at least one bit of information it is transferred from the particle to the medium, then the corresponding particle might exchange information with the medium. If the particle exists in our universe and the temperature to be the same of that of black body radiation T= 2.73 K, the corresponding mass of the particle carrying at least one bit of information is equal m = 4.71810 -40 kg which is many orders of magnitude smaller that the masse of most of today's elementary particles. Next, using the corresponding temperature of a graviton i.e. and assuming at least for one bit of information the corresponding graviton mass is calculated and from that, a relation for the number of information N carried by a graviton as a function of the graviton mass mgr is derived. Furthermore, the range of information number contained in a graviton is also calculated for the given range of graviton mass as given by Nieto and Goldhaber and, we find that the range of the graviton is inversely proportional to the information number N. Finally, treating gravitons as harmonic oscillators in an enclosure of size R we derive the graviton range as a function of cosmological constant, graviton temperature and the Hubble parameter.