⚛️ phenomenology

Photon Gas Thermodynamics in Doubly Special Relativity at the Planck Scale

This paper investigates the thermodynamics of a photon gas within the Magueijo-Smolin formulation of doubly special relativity, deriving and numerically evaluating key thermodynamic quantities that smoothly recover standard special relativistic results at low energies while exhibiting systematic suppression near the Planck scale due to the invariant energy cutoff.

Original authors: Qi Xiong, Xinyi Yang, Guifeng Su, Yi Zhang

Published 2026-07-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Qi Xiong, Xinyi Yang, Guifeng Su, Yi Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic dance floor where particles like photons (particles of light) are the dancers. For over a century, our best map for how these dancers move has been a theory called Special Relativity. It tells us that no matter how fast you run or how fast the dancers move, the speed of light is always the same, and there are no speed limits other than that. But deep down, physicists suspect this map might have a tiny crack in it. They think that at the very smallest scales imaginable—the "Planck scale," which is so small it's hard to even imagine—space and time might behave differently, perhaps even having a "pixelated" structure like a video game screen. If this is true, there might be a maximum energy a particle can have, a cosmic speed limit for energy itself. This idea is called Doubly Special Relativity (DSR). It's a thrilling "what if" scenario: what if the rules of the dance floor change when the music gets too loud and the energy gets too high? Understanding this isn't just about abstract math; it could help us explain the most violent explosions in the universe, like gamma-ray bursts, and perhaps even unlock the secrets of how the universe began.

In this paper, a team of researchers from Shanghai Normal University and the Chinese Academy of Sciences decided to test this idea by looking at a "photon gas"—a box full of light particles—under these new, extreme rules. They used a specific version of DSR called the Magueijo-Smolin (MS) model. Think of this model as a rulebook that says: "Light always travels at the same speed, but there is a hard ceiling on how much energy a single photon can carry." This ceiling is set at the Planck energy, a value so huge it's roughly $10^{19}$ GeV. The researchers asked a simple question: If we fill a box with light and heat it up until it's almost as hot as the Big Bang (the Planck temperature), how does this energy ceiling change the way the light behaves?

To find the answer, the team treated the photon gas like a crowd of people at a concert. In the old, standard rules (Special Relativity), you could keep adding more and more energetic fans to the front row as the temperature rose, making the crowd's total energy and pressure grow without bound. But in the new MS model, there's a VIP section with a strict capacity limit. Once the energy of a photon hits the Planck ceiling, it simply can't exist in the system. The researchers calculated the "grand partition function," which is basically a master scorecard that tells you how many different ways the photons can arrange themselves given this new limit. They then used this scorecard to figure out the gas's temperature, pressure, and how much heat it can hold.

What they found is fascinating. When the temperature is low (like the temperature of a star or even a hot summer day), the energy ceiling is so far away that the photons don't even notice it. The gas behaves exactly as we expect it to, following the standard rules of physics. However, as the temperature climbs closer to the Planck scale, the ceiling starts to matter. The researchers discovered that the finite limit acts like a sieve, filtering out the most energetic photons. Because these high-energy "super-dancers" are missing from the crowd, the total energy of the gas, its pressure, and its entropy (a measure of disorder) are all significantly lower than they would be in the standard universe.

One of the most interesting findings is about the relationship between pressure and energy. In the standard world, the pressure of light is always exactly one-third of its energy density. But in this DSR world, that neat ratio breaks down. The pressure drops even more than the energy does, creating a new, modified relationship that depends on how close the temperature is to the Planck limit. Furthermore, while the heat capacity of a normal photon gas keeps growing as it gets hotter, the heat capacity in this model eventually hits a wall. It stops growing and levels off at a constant value, because the gas simply runs out of new, high-energy states to fill.

The authors also took a moment to correct a previous mistake in the field. Another study had tried to do similar calculations but used the wrong statistical rules (treating photons like classical billiard balls instead of quantum waves). The new paper shows that when you use the correct quantum rules, the results are very different and actually make sense: the energy ceiling suppresses the gas's activity rather than enhancing it.

Ultimately, this paper suggests that if the universe really does have this Planck-scale energy limit, we might see the effects in the most extreme environments in the cosmos. While we can't reach these temperatures in a lab, the universe does it all the time in events like gamma-ray bursts. The authors point out that recent observations of ultra-high-energy photons from these bursts could be the perfect place to look for these subtle deviations. If we can measure the pressure and energy of light from these cosmic explosions with enough precision, we might finally see the "pixels" of space-time and confirm whether the universe really has a hard energy limit. For now, the math suggests that if such a limit exists, it quietly tames the most energetic particles in the universe, keeping the cosmic dance floor from getting too chaotic.

Technical Summary: Photon Gas Thermodynamics in Doubly Special Relativity at the Planck Scale

Problem Statement
While a complete theory of quantum gravity remains elusive, established physics suggests novel effects near the Planck scale ( $E_P \sim 10^{19}$ GeV). A fundamental tension exists regarding the definition of the Planck scale: if it represents a fundamental, observer-independent length or energy, it conflicts with the standard Lorentz contraction of Special Relativity (SR). To resolve this, Doubly Special Relativity (DSR) theories have been proposed, introducing a second invariant scale (typically the Planck energy) alongside the speed of light. Specifically, the Magueijo-Smolin (MS) model modifies the relativistic dispersion relation to $E^2 - p^2 = m^2(1 - E/E_P)^2$ (in $c=1$ units).

Existing literature has explored the thermodynamics of photon gases under modified dispersion relations (MDR). However, previous studies contain significant inconsistencies. One study [36] assumed a variable speed of light and Lorentz symmetry breaking, leading to an increase in entropy. Another study [37] utilized the MS model but incorrectly applied Maxwell-Boltzmann (MB) statistics to the photon gas. Since photons are bosons with vanishing chemical potential ( $\mu=0$ ), the correct equilibrium description requires the Grand Canonical Ensemble with Bose-Einstein (BE) statistics. The use of MB statistics in [37] failed to recover standard SR results in the appropriate limit and yielded classical ideal gas forms rather than photon gas thermodynamics. This paper addresses these gaps by rigorously applying BE statistics within the MS framework to derive the thermodynamic properties of a photon gas subject to a Planck-scale energy cutoff.

Methodology
The authors adopt the MS formulation of DSR, where the speed of light $c$ and the Planck energy $E_P$ are both observer-independent invariants. For massless photons ( $m=0$ ), the dispersion relation remains identical to standard SR ($E=pc$); the DSR effect manifests solely as a finite upper bound on single-particle energy, $E \leq E_P$ .

Partition Function Derivation: The authors derive the logarithmic grand partition function, $\ln \Xi$ , for a photon gas in a volume $V$ . They replace the standard sum over states with an integral over energy space, explicitly imposing the upper cutoff $E_P$ .
$\ln \Xi^{(DSR)} = -\frac{V}{\pi^2 \hbar^3 c^3} \int_0^{E_P} E^2 \ln(1 - e^{-\beta E}) dE$
Using integration by parts and introducing dimensionless variables $x = \beta E$ and $x_P = \beta E_P$ , the expression is reformulated to separate the standard SR contribution from the correction induced by the finite cutoff. The derivation utilizes an incomplete Bose-Einstein integral, defined as $J(y) = \int_0^y \frac{x^3}{e^x - 1} dx$ .
Thermodynamic Quantities: Based on the derived partition function, the authors analytically compute the Helmholtz free energy ( $F$ ), internal energy ( $U$ ), entropy ( $S$ ), pressure ( $P$ ), and heat capacity at constant volume ( $C_V$ ) using standard statistical mechanical relations.
Numerical Evaluation: The derived expressions are evaluated numerically using Planck units ( $E_P = c = \hbar = k_B = 1$ ) to compare the DSR results against standard SR predictions across a range of temperatures, particularly approaching the Planck temperature $T_P$ .

Key Contributions and Results

Correction of Statistical Formalism: The paper corrects the statistical approach used in prior MS-model studies by employing the correct Bose-Einstein statistics. This ensures that the derived thermodynamic quantities correctly reduce to standard SR expressions in the limit $E_P \to \infty$ .
Systematic Suppression of Thermodynamic Functions: Unlike models involving Lorentz symmetry breaking (which predict increased entropy), the MS model preserves Lorentz symmetry but constrains the phase space via the energy cutoff. Consequently, the number of accessible microstates is reduced. The results show that for temperatures approaching the Planck scale, the Helmholtz free energy, internal energy, entropy, pressure, and heat capacity are all systematically suppressed relative to their standard SR counterparts.
Modified Equation of State: The standard photon gas relation $P = U/(3V)$ is modified. The authors derive a new equation of state:
$P^{(DSR)} = \frac{U^{(DSR)}}{3V} - \frac{k_B T E_P^3}{3\pi^2 \hbar^3 c^3} \ln(1 - e^{-E_P/k_B T})$
This introduces a positive correction term arising from the boundary at $E_P$ , altering the pressure-to-energy-density ratio ( $P/u$ ). While $P/u \to 1/3$ at low temperatures, the ratio deviates upward as $T$ approaches $T_P$ .
Heat Capacity Saturation: A distinct feature of the DSR photon gas is the behavior of the heat capacity at high temperatures ( $k_B T \gg E_P$ ). While the SR heat capacity scales as $T^3$ , the DSR heat capacity approaches a finite constant ( $C_V \to \text{const}$ ) as $T \to \infty$ . This saturation reflects the finite volume of the one-particle phase space available below the cutoff $E_P$ .
Low-Temperature Behavior: At temperatures significantly below the Planck scale, the thermal population of modes near the cutoff is exponentially suppressed. In this regime, the DSR results are indistinguishable from standard SR predictions, satisfying the third law of thermodynamics ( $S \to 0$ as $T \to 0$ ).

Significance and Claims
The paper claims to provide the first consistent derivation of photon gas thermodynamics within the MS-DSR framework using the correct Bose-Einstein statistics. The primary significance lies in demonstrating that the Planck-scale energy cutoff acts as a phase-space constraint that reduces thermodynamic quantities rather than enhancing them, a direct consequence of maintaining Lorentz invariance while imposing an energy bound.

The authors note that while these corrections are negligible at ordinary temperatures, they become significant in extreme high-temperature environments. They suggest that the modified equation of state and the saturation of heat capacity could potentially manifest in early Universe cosmology or other extreme astrophysical scenarios. The paper concludes by referencing recent observations of ultra-high-energy photons (e.g., by HAWC, Tibet AS $\gamma$ , and LHAASO) as a potential observational arena for testing Planck-scale modifications, though it does not propose specific new experimental tests or claim that current data definitively confirms the DSR model. The work serves as a theoretical foundation for future phenomenological investigations into modified dispersion relations.

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