Generalized Carnot Efficiency

• Generalized Carnot efficiency is a framework that extends the classical bound by relaxing equilibrium and reversibility assumptions.
• It incorporates corrections from finite-time operations, multiple reservoirs, low-dissipation dynamics, and even quantum and relativistic effects to describe realistic engine performance.
• Practical models use parameters like heat capacity asymmetry and effective temperatures to bridge the gap between ideal Carnot limits and achievable efficiencies in real systems.

Generalized Carnot efficiency denotes extensions of the classical Carnot expression $\eta_C=1-T_c/T_h$ that arise when the standard assumptions of two equilibrium reservoirs, quasistatic reversibility, and working-medium universality are relaxed. In the arXiv literature, the term covers several distinct constructions: a one-parameter efficiency family for realistic thermal engines, generalized reversible bounds for multiple baths, finite-time and low-dissipation corrections, system-specific quantum efficiencies, effective-temperature formulations for tunable or moving reservoirs, and fluctuation-based bounds for small engines (Ponmurugan, 2019, Izumida, 2022, Xiao et al., 2015). The unifying theme is that the Carnot result remains the reference point, but the operational efficiency acquires explicit dependence on asymmetry, irreversibility, spectra, or nonequilibrium resources.

1. Classical benchmark and the rationale for generalization

For a cyclic engine operating reversibly between a hot bath at temperature $T_h$ and a cold bath at temperature $T_c<T_h$, the Carnot theorem gives

$$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$

In the two-bath setting this is the familiar upper bound, while in a multiple-bath setting it is replaced by a generalized reversible efficiency defined from the heat exchanges with all baths (Izumida, 2022).

Within linear irreversible thermodynamics, the same benchmark appears through the thermoelectric figure of merit. For coupled particle and energy transport, the maximum efficiency can be written as

$$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$

so Carnot efficiency is reached if and only if $ZT\to\infty$ (Benenti et al., 2013). In that paper, generic systems with exactly one relevant conserved quantity are shown to reach Carnot efficiency in the thermodynamic limit because the ballistic contribution to the Onsager matrix becomes rank one, driving $ZT$ to diverge.

The need for generalized formulations follows directly from the fact that many practically relevant engines are not reversible and often involve finite-rate driving, several reservoirs, engineered baths, or working media whose spectral properties matter explicitly. A generalized Carnot efficiency is therefore not a single universal formula; it is a family of constructions that modify either the bound itself or the route by which the Carnot benchmark is approached.

2. Heat-capacity asymmetry and the realistic thermal-engine formula

A concrete generalized efficiency was derived for a realistic model of thermal heat engines in which the working substance exchanges heat in two stages and has different effective heat capacities in the hot- and cold-contact stages (Ponmurugan, 2019). The resulting efficiency is

$$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$

where $\gamma\equiv C_s^c/C_s^h$ is the ratio of the working-substance heat capacities during exchange with the cold and hot reservoirs.

The model assumes two-step heat exchange with local equilibrium. In stage 1, during a time $\tau_1$, the working substance absorbs heat from the hot bath at $T_h$0 and its temperature falls from $T_h$1 to an intermediate local-equilibrium temperature $T_h$2. In stage 2, during a time $T_h$3, it releases heat from $T_h$4 to the cold bath at $T_h$5. Imposing zero net entropy production of the working substance over the two stages gives

$T_h$6

Treating the engine thereafter as an effectively reversible cycle between $T_h$7 and $T_h$8 yields $T_h$9 and hence the generalized efficiency formula above.

Its limiting cases interpolate between established benchmarks. For the symmetric-capacity case $T_c<T_h$0, one obtains $T_c<T_h$1 and therefore

$T_c<T_h$2

which is exactly the Curzon–Ahlborn efficiency. In the asymmetric limit $T_c<T_h$3, $T_c<T_h$4 and the formula approaches the Carnot efficiency,

$T_c<T_h$5

The paper also gives the small-$T_c<T_h$6 expansion in terms of $T_c<T_h$7,

$T_c<T_h$8

Empirically, data from a wide range of practical thermal power plants were fitted by plotting $T_c<T_h$9 against $$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$0. The best straight-line fit has slope

$$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$1

The same work states that real engines behave as if $$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$2, significantly larger than the symmetric-capacity case $$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$3, so that practical machines lie well below the Curzon–Ahlborn bound while still following the same one-parameter form.

3. Finite-time, low-dissipation, and irreversible extensions

For finite-time Carnot cycles in the low-dissipation regime, the heat exchanges are expanded as

$$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$4

with entropy production on each isothermal branch scaling inversely with the contact time. Maximizing power with respect to the hot and cold durations yields the efficiency at maximum power $$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$5 and the universal bounds

$$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$6

while symmetric dissipation $$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$7 recovers the Curzon–Ahlborn value $$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$8 (Esposito et al., 2010). In a minimally nonlinear irreversible heat-engine model based on extended Onsager relations with a quadratic dissipation term, the same upper bound $$\eta=\frac{W}{Q_h}\le 1-\frac{T_c}{T_h}\equiv \eta_C.$$9 emerges under tight coupling (Izumida et al., 2011).

A related but more general linear-response formulation for engines coupled to multiple baths gives

$$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$0

where

$$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$1

and the irreversible loss is a positive Onsager-quadratic functional of the protocol. An optimized form introduces the thermodynamic length $$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$2 and yields

$$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$3

(Izumida, 2022). In this formulation, the Carnot theorem appears as the special case of reversible quasistatic operation between two baths.

Finite power near the Carnot limit has been studied explicitly in an underdamped Brownian Carnot cycle. In the limit $$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$4 and for small $$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$5, the irreversible works vanish as $$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$6, and the efficiency becomes

$$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$7

while the work remains $$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$8 and the power stays finite for fixed cycle time (Miura et al., 2020). The same model satisfies the trade-off relation

$$\eta_{\max}=\eta_C\frac{\sqrt{ZT+1}-1}{\sqrt{ZT+1}+1},$$9

The claim that Carnot efficiency requires strict reversibility is also not universal across the literature. In the Feynman–Smoluchowski ratchet, one paper shows that $ZT\to\infty$0 can occur with positive entropy production provided

$ZT\to\infty$1

so that the irreversible entropy production grows sublinearly compared with the heat throughput (Lee et al., 2016). By contrast, for a generalized Carnot cycle with realistic engine-bath interactions, work-optimal operation near maximal efficiency leads to long relaxation times and thus vanishing power, whereas purposefully designed interactions can make Carnot efficiency achievable at large power (Allahverdyan et al., 2013). These results delimit, rather than eliminate, the power–efficiency tension.

4. Quantum generalizations and the loss of universality

In few-particle quantum systems, a quantum analog of the Carnot cycle can be built from two quantum adiabatic steps and two isothermal steps. Using the minimum work principle, the thermally isolated strokes are uniquely selected to be quantum adiabatic, and the optimized efficiency is

$ZT\to\infty$2

In general, this optimized efficiency is lower than $ZT\to\infty$3, depends separately on $ZT\to\infty$4 and $ZT\to\infty$5, and is system-specific through the full spectrum $ZT\to\infty$6 (Xiao et al., 2015). The optimization is governed by force-matching conditions involving the generalized force operator $ZT\to\infty$7. A notable exception occurs when the spectrum scales uniformly,

$ZT\to\infty$8

in which case $ZT\to\infty$9 and $ZT$0.

A distinct zero-temperature construction considers a pure-state quantum-mechanical Carnot engine whose efficiency is determined purely by the structure of the energy spectrum (Abe, 2012). The paper emphasizes nonuniversality: the efficiency depends explicitly on the confining potential, which plays the role of the working material, and this is attributed to the absence of a second-law-like principle in pure-state quantum mechanics. Illustrative examples show this dependence directly. For an infinite square well,

$ZT$1

whereas for a harmonic oscillator one finds

$ZT$2

For homogeneous spectra $ZT$3, the efficiency takes a simple universal form, but for a 3D Morse potential the result is more complicated and cannot be cast into the same two-level expression.

Quantum generalization can also preserve a Carnot-like structure by replacing physical temperatures with effective temperatures. For temperature-tunable baths such as squeezed baths in the high-temperature limit, the effective temperature is

$ZT$4

and the generalized Carnot limit becomes

$ZT$5

Under low dissipation, the efficiency at maximum power obeys

$ZT$6

and, under left-right symmetry and weak dissipation, reduces to the generalized Curzon–Ahlborn expression

$ZT$7

(Liu et al., 2017). This is a generalized Carnot efficiency in the strict sense of retaining the Carnot form while renormalizing the bath temperatures.

5. Multiple baths, fluctuations, and informational resources

For cyclic engines in contact with several baths, the Clausius inequality yields a reversible-efficiency bound in terms of the heats exchanged with baths above and below a reference temperature $ZT$8: $ZT$9 When only two baths are present and $$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$0, this reduces to the ordinary Carnot result (Izumida, 2022). The same framework identifies the irreversible correction explicitly from linear-response theory and the fluctuation–dissipation theorem, with Onsager coefficients written as equilibrium time-correlation integrals.

For finite-time small engines, thermodynamic uncertainty relations yield both lower and upper bounds on efficiency in terms of fluctuations. Writing the entropy production as

$$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$1

one obtains the lower bound

$$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$2

and, in terms of the generalized precision

$$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$3

the upper bound

$$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$4

(Monnai, 2023). In the quasi-static limit, $$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$5 and $$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$6, so both bounds reduce to $$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$7.

A further generalization replaces one thermal branch by an informational one. In a quantum information engine, the cold isothermal stroke is replaced by an information isochore consisting of measurement and feedback at fixed Hamiltonian. Under a low-dissipation hot isotherm, the work and efficiency satisfy

$$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$8

Optimizing the power gives

$$\eta = 1-\left(\frac{T_c}{T_h}\right)^{1/\delta}, \qquad \delta=1+\frac{1}{\gamma},$$9

with bounds

$\gamma\equiv C_s^c/C_s^h$0

(Fadler et al., 2023). The paper states that replacing the information-stroke time $\gamma\equiv C_s^c/C_s^h$1 by a cold-bath coupling time recovers the standard two-bath low-dissipation results, so the generalized Carnot structure extends beyond purely thermal reservoirs.

6. Relativistic motion and effective-temperature Carnot bounds

Relativistic motion introduces generalized Carnot bounds by modifying the spectra seen by the working medium. In a three-level maser with one moving bath modeled by Unruh–DeWitt coupling, the maximum efficiency in the weak-coupling, high-temperature regime is bounded by

$\gamma\equiv C_s^c/C_s^h$2

where $\gamma\equiv C_s^c/C_s^h$3 is the rapidity of the cold bath (Pandit et al., 19 Aug 2025). Since $\gamma\equiv C_s^c/C_s^h$4 and tends to $\gamma\equiv C_s^c/C_s^h$5 as $\gamma\equiv C_s^c/C_s^h$6, the nonrelativistic limit recovers the ordinary Carnot efficiency. The same result can be written as

$\gamma\equiv C_s^c/C_s^h$7

so the generalized Carnot bound is again a Carnot expression in terms of an effective temperature. The paper further reports that many configurations surpass the ordinary Carnot limit while none crosses $\gamma\equiv C_s^c/C_s^h$8, and that positive work can be extracted even for $\gamma\equiv C_s^c/C_s^h$9 when motion alone acts as the thermodynamic resource.

A related construction considers a two-qubit SWAP engine with a moving working medium. Relativistic motion yields frequency-dependent effective temperatures $\tau_1$0 defined from the ratio of excitation and de-excitation rates, and the entropy production over one cycle implies the generalized bound

$\tau_1$1

(Moustos et al., 15 Aug 2025). In the stationary limit $\tau_1$2, one has $\tau_1$3 and $\tau_1$4. The same work states that $\tau_1$5 can exceed the standard Carnot value whenever the moving cold qubit sees $\tau_1$6 or the moving hot qubit sees $\tau_1$7, and that the efficiency at maximum power can surpass both the Curzon–Ahlborn value and the standard Carnot efficiency while still respecting the generalized second law.

Taken together, these relativistic results sharpen a recurring theme in generalized Carnot theory: once the reservoirs or the working medium are driven outside the equilibrium assumptions entering the standard second-law proof, the relevant bound remains Carnot-like in form but must be written with effective temperatures or modified resource parameters rather than with rest-frame bath temperatures alone.