Advances in High Energy Physics

Volume 2018 (2018), Article ID 8968732, 7 pages

https://doi.org/10.1155/2018/8968732

## Quantum Gravity Effects in Statistical Mechanics with Modified Dispersion Relation

Correspondence should be addressed to Mir Mehedi Faruk; moc.liamg@6141.3azrutum

Received 25 September 2017; Revised 14 December 2017; Accepted 15 February 2018; Published 18 March 2018

Academic Editor: Elias C. Vagenas

Copyright © 2018 Shovon Biswas and Mir Mehedi Faruk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Planck scale inspired theories which are also often accompanied with maximum energy and/or momentum scale predict deformed dispersion relations compared to ordinary special relativity and quantum mechanics. In this paper, we resort to the methods of statistical mechanics in order to determine the effects of a deformed dispersion relation along with an upper bound in the partition function that maximum energy and/or momentum scale can have on the thermodynamics of photon gas. We also analyzed two distinct quantum gravity models in this paper.

#### 1. Introduction

Different quantum gravity approaches such as string theory, loop quantum gravity, noncommutative geometry, doubly special relativity (DSR), and generalized uncertainty principle (GUP) suggest the existence of a minimal length scale of the order of Planck length, . This length and its inverse, the Planck energy , mark thresholds beyond which the old description of space-time breaks down and qualitatively new phenomena are expected to appear [1]. DSR is a very well-known approach to deal with minimal length situations. DSR type theories predict a modified dispersion relation compared to usual dispersion relation of special relativity. But one should notice that DSR can be formulated in different approaches. One of the approaches [2] which maintain a varying speed of light (VSL) scenario can be entitled with maximum momenta or maximum energy depending on the value of the parameter which describes the minimal length scale of the theory (we will refer to this as model A in this manuscript). Another possible DSR scenario [3, 4] in which the speed of light is constant is also heavily studied with the help of nonlinear realizations of the Lorentz group (this is referred to as model 1 in the rest of the paper). A point to note, these different models predict different dispersion relations and distinct physical phenomena. The implication of nonzero minimal length has been considered recently in different contexts that include harmonic oscillators [5–10], hydrogen atoms [11–14], gravitational quantum wells [15], the Casimir effect [16, 17], particles scattering [18, 19], relativistic thermodynamics of ideal fluids [20], blackbody radiation [21], and Unruh effect [22, 23]. It has been reported [24] and well established that a consistent marriage of ideas of quantum mechanics and gravitation needs a noncommutative description of space-time to avoid the paradoxical condition of creating a black hole for an event that is sufficiently localized in space-time. As DSR admits noncommutative space-time, DSR theory fits the criterion for being a quantum gravity framework.

Invariant energy scales, that is, Planck scale modified dispersion relations, are usually identified with maximum energy or maximum momenta or both [1, 2, 25]. It is interesting to note that dispersion modifications can lead to many further predictions which can be tested in a series of experiments such as gamma ray burst observation [26] or inflationary cosmology [27]. In this spirit, lots of studies are done on statistical thermodynamics of quantum gases [3, 4, 28, 29]. One has to be very careful while calculating the partition function of the quantum gases within these models as different models change the structure of the partition function. For instance, in the case of a massless boson, model A changes both the integration measure and the upper limit of the partition function [2], while in model 1 only the upper limit is changed [3, 4, 30]. Different errors have been noticed in several manuscripts while evaluating the partition function for quantum gases. For example, in [3, 28], Maxwell-Boltzmann distribution functions are used to calculate the thermodynamics of photon gas. Reference [3] is based on model 1. One of us (MMF) has recently made an attempt [4] to evaluate the partition function of photon gas with a correct distribution function, that is, Bose-Einstein distribution function just as in [2]. But while calculating the partition function in a previous manuscript [4], an error has been noticed, which will be corrected in this paper (we have adopted ). The deformed dispersion relation predicted by model 1 can be written as [1, 3] (we have adopted )Here, and are the energy and the magnitude of the three momenta of the particle, respectively, while is the mass of the particle and is the Planck energy. Now, for massless particles, the dispersion reduces to . That is why an upper energy bound in energy is equivalent to an upper energy bound in momentum in this case. However, in model A, the dispersion relation for massless particles [2] takes the formwhere , so can be both positive and negative. Now, when , (2) implies a maximum momentum , which remains invariant under deformed transformation laws. In contrast, corresponds to an energy upper bound for photons. So, in case of model A, we find an upper bound either in momentum or in energy, whereas in model 1 we see an upper bound both in energy and in momentum. In this manuscript, we will solve the partition function for photon gas with model 1.

Now, another well studied modified dispersion relation based on a phenomenological study is [30, 31]Here, is the Planck energy and is a coefficient of order 1, whose precise value depends upon the considered quantum gravity model, while , the lowest power in Planck’s length leading to a nonvanishing contribution, is also a model-dependent quantity. This type of dispersion relation is observed in loop quantum gravity [31]. In [31–34], it has been argued that can be chosen as or . See [31] to understand the different physical scenarios with and . Now, choosing and following the spirit of Zhang et al. [2], we find out from (3) that indicates an upper bound in energy . A bound in momentum can be obtained with , which is unphysical, so we discard the scenario. We must note that Camacho and Macías [30] have already visited the thermodynamics of photon gas within this model. But they did not take into account the upper bound of maximum energy while calculating the partition function. Any model aimed at describing the behavior of thermodynamic quantities near Planck scale must take into account the effect of the upper bound. In this paper, we will also visit the thermodynamics of photon gas with this deformed dispersion relation, but with maximum energy bound as argued above. We will refer to this model as model 2 in this paper. We have found that, due to this maximum energy bound, the final results of this model are drastically changed compared with what was reported before. A remarkable similarity is noticed between the results of this Lorentz violating model and another recently reported Lorentz violating model [35].

#### 2. Thermodynamics of Photon Gas with Deformed Dispersion Relation in Model 1

##### 2.1. Partition Function

The expression of the partition function in usual statistical mechanics for massless bosons in grand canonical ensemble can be obtained [36]:Here, . Now, changing the sum to integral, we find outIn [1], Magueijo and Smolin proposed a DSR model where the dispersion relation for a massive particle takes the formFor massless photon gas, (6) takes the usual form . Take a look at [3] to see how the phase space volume remains unaltered in this model. One might also wonder whether phase space is invariant under canonical transformation. Canonical transformation are those which keep the form of Hamiltonian’s equation invariant. In case of usual special relativity (SR), the Hamiltonian equation for massless particles is . Now, of course, any canonical transformation must keep the form of this equation invariant. And it is known that these canonical transformations also keep the usual phase space volume invariant. In case of model 1, the dispersion relation changes due to the existence of the Planck scale [see (1)]. Now, if we only consider the massless case of model 1, one can see then that the dispersion relation is nothing but (1) (just like the usual SR scenario). Therefore, in the massless scenario, the dispersion relation of model 1 reduces to the usual special relativity. As the dispersion relation (Hamiltonian’s equation) is of the usual form (same as special relativity), the canonical transformations associated with the massless case of model 1 are the same as the usual special relativity scenarios. And we already know that these canonical transformations not only keep the Hamiltonian equation invariant, but also keep the phase space volume invariant. Therefore, in this case, it is enough to take the usual phase space volume as phase space volume for the massless case of model 1. But a point to note is that it is only true for the massless case. In case of massive particles, the dispersion relation (Hamiltonian equation) changes [see (1)] and therefore the canonical transformation changes compared to the usual SR. As a result, the phase space volume also changes. But as we are dealing only with the massless scenarios, the phase space volume mentioned in here is justified. Now, due to the presence of an upper bound in energy, in this model, the partition function becomesHere, is the total volume of the system. Equation (7) can be written aswhere the function is defined in the Appendix. (In order to calculate (7), one can choose . As a result, the upper limit of integration should also be changed accordingly. Although a correct distribution function, i.e., Bose-Einstein distribution, was taken instead of Maxwell-Boltzmann distribution in our study [4], the change of the upper limit of integration with respect to this variable shift was not mistakenly taken into account. Once this variable shift is also taken into account in the upper bound of the integration, one finds out that the upper bound of the integration contains a temperature-dependent term, which was completely absent in [4].)

##### 2.2. Thermodynamic Quantities

Thermodynamic quantities can be derived easily from the partition function following the simple and effective rules of statistical mechanics [36]. For example, the internal energy is related to the partition function byThe pressure can be found from the relationWe see that the well-known relationship between pressure and energy is not valid anymore as obtained in the usual statistical mechanics. Another important thermodynamic quantity is entropy :where . As , we notice indicating that Nernst’s postulate is valid. Finally, specific heat can be obtained from the relationAll the thermodynamic quantities reduce to the usual results of statistical mechanics in the limit [36].

#### 3. Thermodynamics of Photon Gas with Deformed Dispersion Relation in Model 2

##### 3.1. The Partition Function

Now, let us focus on the model based on the phenomenological study, first discussed in detail by Amelino-Camelia [26, 31]. Later, thermodynamics of photon gas were studied within this model by Camacho and Marcias [30]. As discussed in the Introduction, Camacho and Marcias did not take into account the upper energy bound that exists in this model, while evaluating the partition function. In this section, we evaluate the partition function in the presence of the upper bound.

The modified dispersion relation of massless bosons in this model isNow, it is clear from the above equation with that tends to , that is, the maximum value of , considering . As [26, 31], we find out .

The partition function can be equivalently written aswhere is the density of states [30]:Based on the above observation, we can write the partition function:Equation (16) can be rewritten as

##### 3.2. Thermodynamic Quantities

Just like in Section 2.2, the thermodynamic quantities can be derived from the partition function.

*Internal Energy**Pressure**Entropy**Specific Heat*The thermodynamic quantities of this model also coincide with the results of the usual statistical mechanics in the limit . The presence of an upper bound in the partition function drastically changes the final outcome of this model. Several other observations will be noted in the next section.

#### 4. Discussion and Conclusion

We are now in a position to present the results, obtained in the previous two sections. The results have been obtained using .

In Figure 1, we have plotted the pressure obtained from model 1 and model 2, with and , and performed a comparison with the usual results of special relativity (SR). In the study of model 1 (DSR), we have noticed that the values of pressure exactly coincide with SR results in low temperature regions but show deviation from the usual result at a high temperature limit. At any given temperature, the thermodynamic quantities in the DSR model pick smaller values compared to the usual result in that temperature. This is expected in the DSR model, as the dispersion relation is unchanged compared to SR, and due to the presence of an upper energy scale, the total number of microstates becomes less than SR. Although this result has been reported before in [4], we should note that, due to the correction (we have adopted ) we have presented in the current paper in evaluating the partition function for the DSR model, the resulting thermodynamic quantities take lesser values compared to the previous study [4].