🔬 condensed matter

Ideal heat engine cycles at maximal efficiency -- the ideal gas and beyond

Contrary to the assumption that the ideal gas is universally optimal, this paper demonstrates that for Stirling, Otto, and Brayton cycles, the maximal efficiency is actually achieved by any thermodynamic system whose working medium is linear in temperature, a category that includes ideal gases, classical harmonic oscillators, and rubber band models.

Original authors: Gregory Behrendt, Sebastian Deffner

Published 2026-08-13

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CC BY 4.0

Original authors: Gregory Behrendt, Sebastian Deffner

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 you are an engineer trying to build the ultimate machine. Not a robot or a spaceship, but a heat engine—a device that turns heat into motion, like the pistons in a car or the turbines in a power plant. For over a century, physics students have been taught that the "perfect" fuel for these machines is the ideal gas. Think of this as a collection of tiny, invisible billiard balls bouncing around in a box, never sticking together and never getting tired. It's the standard, go-to model for how things work in thermodynamics.

But here is the twist: what if the ideal gas isn't actually the best choice? What if there's a secret ingredient that could make your engine even more efficient, squeezing out more work from the same amount of heat? This is the question that keeps engineers and physicists up at night. Efficiency is the holy grail because it means getting more done with less fuel, which saves money and protects the planet. To understand the answer, we need to look at how these engines run. They operate in cycles, heating up a gas to make it expand (doing work) and then cooling it down to reset. The big question is: does the specific "personality" of the gas matter? Does it matter if the gas behaves like a bouncy ball, a stretched rubber band, or a vibrating spring?

This paper by Gregory Behrendt and Sebastian Deffner dives right into that mystery. They asked a simple but profound question: If you want the highest possible efficiency for a heat engine, what kind of working material should you use? While many people might instinctively shout "Ideal Gas!", the authors discovered that the answer is actually much broader and more surprising.

The researchers didn't just guess; they built a mathematical "super-model" that could describe many different types of materials at once. They imagined a formula where the energy of the system depends on its temperature and volume in a specific, flexible way. Using this master formula, they tested three famous engine designs: the Stirling, Otto, and Brayton cycles. These are the blueprints for engines ranging from old-fashioned steam engines to modern jet turbines.

What they found is that the "ideal gas" is indeed a champion, but it's not the only one. The secret to maximum efficiency isn't about being a gas at all; it's about how the material's energy changes with heat. The authors proved mathematically that the most efficient engines are those where the energy of the working medium is linearly related to the temperature.

To use a playful analogy, imagine temperature as the speed of a runner. In a "linear" system, if you double the runner's speed, you double their energy. This is true for the ideal gas, but it's also true for other things that aren't gases at all! The paper shows that classical harmonic oscillators (think of a weight bouncing on a spring) and even rubber bands (which stretch and snap back) follow this same linear rule. If you built an engine using a rubber band as your "fuel" instead of air, and you ran it through a Stirling, Otto, or Brayton cycle, you would get the exact same top-tier efficiency as you would with the best possible gas.

The authors were very careful to rule out some common misconceptions. They showed that for the Stirling cycle, the efficiency depends only on this temperature relationship and doesn't care about how the material's volume changes. For the Otto and Brayton cycles, the math is a bit more complex, but the result is the same: the peak efficiency is achieved when the energy scales linearly with temperature. They also noted that while there are mathematical solutions that look even more efficient, those solutions require physical conditions that don't exist in the real world (like infinite exponents in their equations), so we can safely ignore them.

In short, this paper doesn't just confirm that the ideal gas is great; it reveals a whole new neighborhood of materials that are equally great. It tells us that if we want to build the most efficient heat engines possible, we shouldn't just look for better gases. We should look for any system—whether it's a spring, a rubber band, or a gas—that keeps its energy perfectly in step with its temperature. The ideal gas is just one member of a very exclusive club of "perfectly linear" materials, and they all share the crown of maximum efficiency.

=== DRAFT ===
Technical Summary: Ideal Heat Engine Cycles at Maximal Efficiency – The Ideal Gas and Beyond

Problem Statement
The paper addresses a fundamental question in thermodynamics: given a specific heat engine cycle, what is the optimal working medium that yields the highest possible efficiency? While standard pedagogical treatments often assume an ideal gas as the working medium due to the simplicity of the resulting formulas, the authors note that this assumption obscures the fact that efficiencies for ideal cycles can vary significantly depending on the fundamental thermodynamic relations of the medium. Previous literature on endoreversible cycles (e.g., Curzon-Ahlborn efficiency) has shown that efficiency at maximal power depends on both the cycle type and the working medium's fundamental relation, yet a comprehensive answer regarding the optimal medium for ideal cycles has been lacking.

Methodology
To investigate this, the authors adopt a general thermodynamic framework rather than restricting analysis to specific substances. They posit a general Helmholtz potential, $F(T, v, N)$ , that depends polynomially on molar volume ( $v$ ) and temperature ( $T$ ):
$F(T, v, N) = -f(n, m) N v^n T^m$
where $n > 0$ and $f(n, m)$ is a collective constant. The formalism is applied to derive general mechanical and caloric equations of state, with specific examples (like the photonic gas) utilizing $m > 1$ , while the analysis explicitly extends to the limit $m \to 1$ to recover the ideal gas and identify the optimal case. This ansatz allows for the derivation of general mechanical and caloric equations of state. The authors then apply this general potential to derive exact analytical expressions for the efficiencies of three classic ideal cycles:

Stirling Cycle: Comprising two isothermal and two isochoric strokes.
Otto Cycle: Comprising two isentropic and two isochoric strokes.
Brayton Cycle: Comprising two isentropic and two isobaric strokes.

The efficiency for each cycle is calculated as $\eta = -W_{tot}/Q_{in}$ , utilizing the derived expressions for work and heat transfer based on the general potential.

Key Contributions and Results
The primary contribution of the work is the derivation of closed-form efficiency expressions for the three cycles as functions of the exponents $n$ and $m$ , and the subsequent identification of the conditions for maximal efficiency.

Stirling Cycle: The derived efficiency is $\eta_S = \frac{1}{m} [1 - (T_c/T_h)^m]$ . The authors find that $\eta_S$ is a monotonically decreasing function of $m$ . The maximum efficiency is achieved when $m=1$ , at which point the efficiency equals the Carnot efficiency. Notably, the Stirling efficiency is independent of the volume exponent $n$ .
Otto Cycle: The derived efficiency is $\eta_O = 1 - \kappa^{n/(m-1)}$ , where $\kappa$ is the compression ratio. Maximizing this efficiency requires maximizing the exponent $n/(m-1)$ . For any finite $n$ , the maximum is approached as $m \to 1$ . While $n \gg 1$ also maximizes the term, the authors note this limit is unphysical for gaseous systems.
Brayton Cycle: The derived efficiency is $\eta_B = 1 - C^{n/(1-m-n)}$ , where $C$ is the compressor ratio. Similar to the Otto cycle, the efficiency is maximized in the limit $m \to 1$ (and unphysically in the limit $n \gg 1$ ). In the $m \to 1$ limit, the efficiency converges to $1 - C^{-1}$ .

Synthesis of Findings
Across all three cycles, the analysis reveals a universal condition for maximal efficiency: the working medium must possess a caloric equation of state that is linear in temperature.

This condition corresponds to $m=1$ in the general polynomial ansatz.
While the ideal gas satisfies this condition (obtained in the limit $n \to 0$ with $m=1$ ), it is not unique. The authors explicitly identify that classical harmonic oscillators and phenomenological models of rubber bands also possess caloric equations of state linear in temperature and thus achieve the same maximal efficiency.
Conversely, systems like the photonic gas (where $m=4$ ) yield lower efficiencies for these cycles.

Significance and Scope
The paper claims to resolve the question of which working medium is optimal for ideal heat engine cycles. The significance lies in demonstrating that the "ideal gas" is not the sole or inherently superior choice; rather, any system with a linear temperature dependence in its internal energy (caloric equation of state) is optimal.

The authors maintain a modest scope, noting that their analysis is entirely framed in terms of thermodynamic notions for ideal (reversible) cycles. They suggest that these insights likely extend to endoreversible regimes, as those systems also operate through successions of local equilibrium states. The paper concludes by identifying future directions, such as analyzing more complex cycles, employing more generalized fundamental relations, treating finite-time thermodynamic processes, and deriving fundamental relations that account for quantum effects which might lead to performance beyond the polynomial equations of state discussed.

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