On Boundedness of Entropy of Photon Gas in Noncommutative Spacetime

Entropy bound for the photon gas in a noncommutative (NC) spacetime where phase space is with compact spatial momentum space, previously studied by Nozari et al., has been reexamined with the correct distribution function. While Nozari et al. have employed Maxwell-Boltzmann distribution function to investigate thermodynamic properties of photon gas, we have employed the correct distribution function, that is, Bose-Einstein distribution function. No such entropy bound is observed if Bose-Einstein distribution is employed to solve the partition function. As a result, the reported analogy between thermodynamics of photon gas in such NC spacetime and Bekenstein-Hawking entropy of black holes should be disregarded.


Introduction
According to the theories of quantum gravity, there exists a minimum length scale, below which no other length could be observed [1][2][3][4][5][6][7][8][9][10][11].Although the standard relativistic quantum mechanics does not take into account any minimal length scale, it is believed that, due to the correspondence principle in the continuum limit, the flat limit of quantum gravity will be reduced to this standard theory.Also, it should be noted that a noncommutative (NC) structure is very common in the theories with minimal length [12][13][14][15][16]. Furthermore, through the Lorentz-Fitzgerald contraction in special relativity a minimal length in one inertial frame may be different in another observer's frame.In order to take into account an observer-independent minimum length (or maximal energy) in special relativity framework the doubly special relativity (DSR) theories are investigated, which also introduces NC structure in the theory [17][18][19][20].
Phenomenological models based on noncommutative framework are necessary in order to fully explore the consequences of these results.So, it would be highly desirable to detect a definitive signature of new physics at a scale close to the Planck scale which could take place in a noncommutative geometry.It is expected that noncommutative signature will appear in experiments involving cosmic microwave background (CMB), ultrahigh-energy cosmic rays, or other high energy sources such as those of neutrinos.It is well established that the CMB radiation maintains a blackbody spectrum with high accuracy, at least for low to medium frequencies.But data are still not so accurate, and as the measurements become more accurate, deviations from the blackbody spectrum could be found, particularly in high frequency regions [16].
Recently a growing attraction [21][22][23] is noticed towards the effect of minimal length on statistical thermodynamics.Thermodynamics of photon gas has been investigated explicitly within the Magueijo-Smolin Lorentz invariant DSR model [24].Photon thermodynamics has also been studied by Nozari et al. in a Lorentz violating quantum gravity model [25].An interesting result regarding entropy bound is obtained in that study which is not present in the SR theory [26].According to that study, the entropy  bounded in NC spacetime is similar to the case of Bekenstein Hawking entropy of black holes.But there is a serious error in their calculation, as they have used Maxwell-Boltzmann statistics while calculating the partition function of photons, but it is well established that photons are integer spin quantum particles [26,27], obeying Bose-Einstein [26] distribution.So, one must use Bose-Einstein statistics to calculate the thermodynamics of the photon gas.Also, a successful study based on minimal length must coincide with the well-established results in the limit of minimal length approaching zero [24,28].Because of this severe error, the results obtained by Nozari et al. [25] do not coincide with known results of the thermodynamic quantities of photon gas in the SR theory in the limit minimal length tending to zero.For instance, the pressure calculated in that theory depends on temperature,  linearly [25], but in SR theory pressure is proportional to  4 (also known as Stefan-Boltzmann law) [26].In this paper, we will explore the photon thermodynamics in a Lorentz violating model previously visited by Nozari et al. [25] but of course with the correct distribution, that is, Bose-Einstein distribution.It would be very intriguing to check if the entropy bound is still present in the photon thermodynamics in such noncommutative spacetime, while using the Bose-Einstein distribution to solve the partition function.Another study on Lorentz violating photon thermodynamics has also been done by Camacho and Macías [28].In their work, they introduced a deformed dispersion relation as a fundamental fact and analyzed the influence of this upon the thermodynamics of photon gas.In that model, the breakdown of Lorentz symmetry entails an increase in the number of microstates, and as a consequence a growth of the entropy and other thermodynamic quantities, with respect to the case of SR theory, is observed.Also, it must be pointed out that Camacho and Macías's model have no finite upper bound on the energy of the photons [28] whether we have an upper bound in energy just like the DSR models [24].They showed that the breakdown of Lorentz symmetry entails an increase in the number of microstates, and as a consequence a growth of the entropy and other thermodynamic quantities, with respect to the case of SR theory, is observed.It would be very interesting to see if any such growth in thermodynamic quantities is observed in current study.The relation between thermodynamic quantities of photon (for example, Pressure-Energy relation (pressure  is related to internal energy  as  = 3;  is the volume)) established in SR is still intact in Lorentz invariant DSR [24] but not maintained in Lorentz violating model [28] of Camacho and Macías.We have also checked the status of such relations in the model Nozari et al. used.But most important aspiration of the current manuscript is to check if the entropy is still bounded if one takes the correct distribution function.

Photon Gas Thermodynamics in Noncommutative Spacetime
We are using the same noncommutative spacetime model considered by Nozari et al. [25] motivated by the doubly special relativity theories and noncommutative spacetime structures.The only difference is that we are going to employ Bose-Einstein distribution function to solve the partition function for photon gas.As photons are spin-one massless quantum particles, it is mandatory to use Bose-Einstein distribution.Maxwell-Boltzmann distribution is for classical particles and cannot be used to describe massless particles [24,26].

Noncommutative Spacetime Algebra.
Concept of stability plays a key role in mathematical deformation theory.A mathematical structure is considered as a stable structure for a class of deformations if any deformation from that class leads to an equivalent structure.If a given structure is not stable we can deform it until it falls into a stable one.This idea can explain the transitions of our well-known physical theories.For example, the rise of relativistic theory from nonrelativistic theory or quantum theory from classical theory can be thought of as transformations of theories from unstable to stable ones.The deformation parameters for relativistic and quantum theories are 1/ and ℎ.Vanishing values of them recover the old structure.However, putting Lorentz and Heisenberg algebra together should have given us a stable, relativistic quantum theory but unfortunately it did not.We find a candidate of stable algebra for relativistic quantum mechanics [29], where is the Minkowski metric,  = ±1,  is the deformation parameter, and I is the nontrivial operator that replaces the trivial center of the standard Heisenberg algebra.To obtain a simpler and approximate form for density of states, it is convenient to use the representation of a subalgebra {  ,   , I} with ( = 1, 2, 3).We fix  = −1 as done in [25] from the consideration of compactification of the spatial momenta space.Using the position basis representation of this subalgebra we get the deformed density of state (5) which we have introduced in the next section and used throughout the whole calculation.Deformed density of states is not rare in this kind of theories.We find similar deformed density of states in DSR theories, Snyder noncommutative space, and polymerized phase space also [30][31][32].Interested readers should go through [29,33,34] for the detailed manipulations of algebra associated with this model.

Density of States and Partition Function.
Partition function in the grand canonical ensemble for massless Bose gas is defined as Here,  is the energy and  = 1/  , with   being Boltzmann constant which we set to be 1 in natural unit system.Changing the sum to integral and introducing density of states (in SR theory), it reads But we need to make a modification in previous equation in order to incorporate the minimal length.Making the modification in the above expression as below following the spirit of [24,25], Here,  is the deformation parameter with dimension of inverse of length naturally arising in this setup (Section 2.1) which could be identified with a minimal length scale (Planck scale) [24].The density of states associated with the model, ( In the limit,  tends to infinity and both the modified expression for partition function and density of states take the usual form.Now as the partition function is fixed we can solve the thermodynamic quantities of photon gas for this model with Hamiltonian (setting  = 1),  = .As it turns out, the thermodynamic quantities cannot be obtained analytically for this model unless one makes small 1/ expansion in the density of states (as Planck scale  is a very big quantity, the inverse 1/ ∼ 10 −35 m is a very small quantity).So, we calculate the thermodynamic quantities analytically using 1/ expansion and solve it numerically for any .

Internal Energy.
One of the most important thermodynamic quantities, internal energy  is as follows: )) ) .
We expanded the expression in power series of 1/ and labeled the terms as  0 ,  2 .Here, with Li  () being the polylogarithm (also known as Jonquière's function) defined as In the limit  → ∞,  = (1/15) 2  4  which is compatible with usual result from special relativity (SR).It should be pointed out that the free energy as well as other thermodynamic quantities is obtained by Nozari et al. [25] which is unable to reproduce the known results of photon gas in special relativity.In SR, internal energy of photons in three dimensions has temperature dependency as  ∝  4 [26], but they obtained  ∝  [25], which is incorrect and it represents the result of ideal classical gas.But this is expected as they have used the Maxwell-Boltzmann distribution, which is only valid for classical particles [26].
We have plotted the internal energy in Figure 1 obtained numerically using (6).And something very interesting has been observed in current model which has not been observed in other Lorentz violating study [28] of thermodynamics of photon gas.In [35], the authors have reported that the internal energy obtained from their deformed theory grows at a much faster rate (for the whole temperature range) than in the SR theory as temperature increases.But our result does not quite agree with it.From Figure 1(b), it is clear that when  < 0.26  the internal energy obtained from our NC theory also increases at a much higher rate than the case of the SR theory.At  = 0.26, the two graphs intersect and for  > 0.26  the situation reverses.This happens due to the presence of cutoff in our theory (see (4)), unlike [28].We must point out that this behaviour has also been noticed in the case of other thermodynamic quantities.As the temperature increases and approached the Planck temperature, the existence of cutoff in our considered model restricts the thermodynamic quantities to change in a slower rate than SR or any other model which does not have a cutoff.Figure 1: Plot (a) of internal energy  of photon gas against temperature  for both the SR theory and the NC model.The dashed line corresponds to the NC model, and the solid line represents the corresponding quantity SR theory.The NC model results have been obtained numerically using Mathematica [35].We have used the Planck units, and the corresponding parameters take the following values:  = 10000,   = 1,  = 1, ℎ = 1, and  = 1.In this scale,  = 10000 K is the Planck temperature,   (the plotting style has been adopted from [24]).Plot (a) is done for the entire temperature range, that is, from zero temperature to Planck temperature.But in plot (b) we have plotted the internal energy in a range from  = 0 to  = 3000 K so that it becomes clear where two graphs intersect each other.

Free Energy.
From the definition of free energy, ) .
In Figure 2 the free energy of photon is plotted for NC and SR theory.The intriguing phenomenon of an intersection between SR and NC results has also been observed in Figure 2. Our model under consideration is unique in this way as this phenomenon has not been observed in other quantum gravity inspired study of photon thermodynamics [24,28].In special relativity, the internal energy  is related to free energy  by And this result is also maintained in Lorentz invariant quantum gravity study [24].But, it has been reported that this relation is not preserved in Lorentz violating deformed dispersion study of Camacho and Macías [28], although they did not have upper bound in partition function.Though we have an upper bound in our Lorentz violating model, we have Figure 3: Entropy in SR and NC model.The units and the parameters are the same as in Figure 1.The NC model results have been obtained numerically using Mathematica [35].
found that our theory also does not obey such relation (11).It is clear from (7) and (10).But both of these expressions are analytic; both of them depend upon small 1/ expansion.We have checked the status of relation (11) in our model numerically without any expansion and found it to be violated.
The standard Pressure-Energy relation [26] in SR is much used in cosmology [36], But this relation is not preserved in our model.But the standard relation is recovered in the limit  → ∞.We have also plotted the results of entropy in Figure 3.But we must point out that the entropy of photon gas is not bounded in the presence of minimal length as claimed by Nozari et al. [25].The obtained result in [25] is merely a consequence of Maxwell-Boltzmann distribution, which they used to calculate the thermodynamics of photon gas.But if one employs the correct distribution (Bose-Einstein distribution) the entropy of the photon gas is not bounded (Figure 3), just like the SR.

Conclusion
In this paper, we successfully calculated the thermodynamics of photon gas in a DSR motivated Lorentz violating theory.Because of the deformed density of states and an upper bound, the task of solving the partition function analytically becomes highly nontrivial.But, our results match with the known results [26] of SR theory in the limit  → ∞ unlike [25], as expected.It was found that it is not possible to compute the thermodynamic quantities analytically in our model where an invariant energy scale is present.But analytic computation is possible if one makes small 1/ expansion (( 7) and ( 10)).Nevertheless, we have also calculated all the thermodynamic quantities numerically using Mathematica without any expansion.quantities of photon gas break down in our theory.Such relations were maintained in previously reported Lorentz invariant study [24] but invalid in Lorentz violating model [28].This is the key differences in the study of photon thermodynamics in Lorentz symmetry violating and Lorentz symmetry obeying models.The current technique could be applied to relativistic massive fluid to check how the equation of state is modified.(c) The most important finding of the current study is that entropy is not bounded in the present model just like SR. Nozari et al. obtained this result as they employed wrong distribution function.So, the analogy with the Bekenstein-Hawking entropy for black holes should be disregarded.
The fascinating new results obtained in our study inspire us to look at the status of Bose-Einstein Condensation and Fermi degeneracy within this model.It would be very interesting if these key differences are also maintained in the case of massive quantum gases.The condensation temperature for Bose gas as well as the Fermi temperature for Fermi gas should be affected with such NC spacetime.The former case is interesting in the scalar field dark matter model [37], where the dark matter particle is a spin-0 boson, while the latter is in the case of Chandrashekhar limit [38].Generalisation of standard cosmology result in this current framework is the next program that we wish to take up.

Figure 2 :
Figure 2: Free energy in SR and NC model.The units and the parameters are the same as in Figure 1.The NC model results have been obtained numerically using Mathematica [35].
The obtained numerical results are plotted in Figures 1, 2, and 3.The reason Nozari et al. was able to calculate the thermodynamic quantities analytically is because they used Maxwell-Boltzmann distribution.But it certainly gave the wrong result in  → ∞ limit.The central features of the obtained results in this manuscript are (a) Up to a certain temperature, the thermodynamic quantities obtained from our model change with temperature in a greater rate than SR.But after that temperature the predicted results from our model change in a slower rate than SR.In the Lorentz violating model of Camacho and Macías [28] the results obtained from modified dispersion relation changed in a greater rate than SR throughout the whole temperature range.But we do not notice this in our model, due to the presence of an upper bound.(b) The established relations among the thermodynamic