Quantum Corrections in Thermodynamics

• Quantum corrections in thermodynamics are modifications to classical laws arising from quantum coherence, entanglement, and gravitational effects, typically captured by ℏ-expansions.
• They alter key observables such as entropy, free energy, and work, leading to refined bounds on efficiency in settings from quantum heat engines to black hole thermodynamics.
• These corrections impact diverse regimes—including weak/strong coupling, finite-time processes, and nanoscale systems—providing actionable insights for quantum information and error correction.

Quantum corrections on thermodynamics encompass the modifications to classical thermodynamic laws, equations of state, and macroscopic observables that arise due to quantum statistical, informational, and gravitational effects. These corrections manifest in diverse systems, including small quantum systems, quantum fields in curved spacetime, black holes, and quantum information processes. At the foundational level, they become relevant when characteristic action scales approach Planck’s constant, thermal wavelengths are comparable to system size, or when quantum coherence and entanglement play critical roles. Below, major aspects of quantum corrections to thermodynamics are systematically detailed, emphasizing the breadth of physical settings and the rigorous mathematical frameworks by which these corrections are derived.

1. Quantum Entropy Corrections and the Quantum–Classical Correspondence

Central to quantum corrections in thermodynamics is the relationship between the quantum von Neumann entropy and its classical analog, the Gibbs entropy. The quantum entropy for a state ρ̂ is

$$S_q = -\operatorname{Tr}[\hat{\rho} \ln \hat{\rho}].$$

Through the phase-space (Wigner function) formulation, such as the Weyl–Wigner mapping, the quantum entropy can be systematically expanded in powers of ℏ: $$S_q = S_q^{(0)} + i\hbar S_q^{(1)} + (i\hbar)^2 S_q^{(2)} + \cdots$$ with

$$S_q^{(0)} = -\int \frac{dx\,dp}{2\pi\hbar} W^{(0)} \ln W^{(0)}$$

corresponding to the classical Gibbs entropy, and S_q^{(2)} containing explicit quantum corrections via phase-space derivatives and Moyal products (Qiu et al., 2019).

For thermal equilibrium states, all odd-order terms in ℏ vanish, making ℏ^2 corrections the leading quantum effect for entropy and hence all thermodynamic quantities (e.g., free energy, internal energy). The expansion ensures quantum–classical correspondence: as ℏ → 0, the classical thermodynamics is recovered.

This ℏ-expansion can be directly transferred to quantum heat engines, such as the Carnot engine, where quantum corrections to net work extraction relate to entropy differences at different temperatures and states: $$\mathcal{W}_q = (T_h - T_l) [S_q(B) - S_q(A)] = \mathcal{W}_q^{(0)} + (i\hbar)^2 \mathcal{W}_q^{(2)} + \cdots$$ (Qiu et al., 2019).

2. Quantum Corrections in Weak- and Strong-Coupling Thermodynamics

Macroscopic thermodynamic laws, such as the second law and the maximum extractable work, require significant modification when the system is quantum and strongly coupled to thermal reservoirs. In the weak-coupling limit, maximal work extraction follows the non-equilibrium free energy difference: $$W^{\text{(weak)}} = F(\rho_S, H_S) - F(\omega_\beta(H_S), H_S)$$ with $F(\rho, H) = \operatorname{Tr}(\rho H) - T S(\rho)$. Under strong coupling, the system equilibrates to the reduced state of the global thermal state $\omega_\beta(H_S + V + H_B)$, not to the local Gibbs state. The actual extractable work then includes strictly positive correction terms

$$W = W^{\text{(weak)}} - \Delta F^{(\text{irr})} - \Delta F^{(\text{res})}$$

where

$$\Delta F^{(\text{irr})} = F(\rho^{(0)}, H^{(1)}) - F(\omega_\beta(H^{(1)}), H^{(1)}),$$

and

$$\Delta F^{(\text{res})} = F(\omega_\beta(H^{(N)}), H^{(0)}) - F(\omega_\beta(H^{(0)}), H^{(0)}),$$

with the corrections arising at order g^2 in the coupling strength g, and involving generalized covariance (Kubo–Mori) terms (Perarnau-Llobet et al., 2017).

Corrections to heat dissipation, Carnot efficiency, and even the optimal cycle power similarly appear, imposing stricter upper bounds on performance metrics than in classical (or weak-coupling) scenarios. A paradigmatic example is the quantum Brownian motion model, where corrections to all thermodynamic quantities can be explicitly computed and shown to conform to these predictions.

3. Quantum Corrections in Black Hole and Gravitational Thermodynamics

Quantum gravitational effects introduce subdominant—but crucial—corrections to the Bekenstein–Hawking entropy–area law. Two principal forms of corrections are encountered:

Logarithmic Corrections: $S = S_0 + \alpha \log(S_0 T_H^2)$

as derived from microcanonical density-of-states and saddle-point path integral treatments, where S_0 is the classical area–law term and α encapsulates details of the underlying quantum theory or gravitational sector (Upadhyay, 2017, Mele et al., 2021).

Exponential (Non-Perturbative) Corrections: $S = S_0 + \eta\, e^{-S_0}$

arising from nonperturbative quantum effects, such as instanton contributions or microstate counting in certain quantization schemes (Pourhassan, 2020, Pourhassan et al., 2020, Pourhassan et al., 2022, Pourhassan et al., 12 Mar 2024, Soroushfar et al., 16 Apr 2024). These exponential corrections are negligible for macroscopic black holes but dominate at Planck-scale horizons, and their sign and magnitude can significantly alter stability and remnant formation.

Such corrections directly affect thermodynamic potentials (free energies, internal energy), phase structure, and stability. For instance, in Schwarzschild or Reissner–Nordstrøm scenarios, specific heat can become positive at small radii under exponential corrections—for 4D cases, leading potentially to black hole remnants; in 5D, instability persists even with corrections (Pourhassan, 2020).

Path-integral approaches to Euclidean gravity additionally reveal large logarithmic corrections near extremality, associated with zero-mode fluctuations localized at the AdS₂ throat, producing terms such as

$\log Z \sim 3 \log T$

and logarithmic entropy corrections, consistent with dimensional reduction to the Schwarzian effective theory (Banerjee et al., 2023).

4. Quantum Thermodynamics in Information and Error Correction

Quantum error correction (QEC), when analyzed entropically, enforces compatibility with the second law by requiring that any decrease in system entropy is compensated by increased environmental entropy as per Landauer’s principle. In each QEC cycle, the retained entropy balance reads: $$\Delta S_{\text{tot}} = S(\rho) - S(\rho') + H(\{p_k\}) \geq 0$$ where $H(\{p_k\})$ is the Shannon entropy associated with syndrome measurement outcomes (Cafaro et al., 2013). For exact-QEC, syndrome measurements correspond to orthogonal state discrimination (maximal information gain), but in approximate-QEC, non-orthogonality reduces both information gain and correction fidelity, increasing overall entropy production.

A more refined thermodynamical–informational bridge is established by interpreting QEC as a quantum heat engine equipped with feedback control. Here, a triple trade-off is proven between error-correction fidelity $F_e$, thermodynamic efficiency η, and the efficacy of the measurement process: $$\mathcal{H}(F_e) + (1 - F_e) \ln(d^2 - 1) \geq \frac{Q_{\text{input}}}{k_B T_h} \frac{\eta - \eta_C}{1 - \eta_C} + \sum_x p_X(x)\left[ -\ln \mathcal{E}(\mathcal{N}_{|x}; \sigma_1^x) \right]$$ demonstrating that super-Carnot efficiencies demand suboptimal correction fidelity or less-efficient measurement (Danageozian et al., 2021).

Thermodynamic costs of measurements and erasure are bounded in terms of the Groenewold information gain, tightly connecting the energy flows and quantum informational properties of the system.

5. Quantum Corrections in Work Fluctuations and Nonequilibrium Thermodynamics

Quantum fluctuations introduce corrections to the full distribution of work and its fluctuation–dissipation relation (FDR). In a path-integral formalism, the quantum work functional, defined along Feynman trajectories, systematically yields the moments of the work distribution: $$W_\nu[x] = W_{\text{cl}}[x] + \frac{i\nu}{2} W_q^{(1)}[x] - \frac{\nu^2}{3!} W_q^{(2)}[x] + \cdots$$ with the first quantum correction,

$$W_q^{(1)}[x] = -i\hbar \int_0^\tau dt\, \dot{x}(t) \dot{\lambda}_t \frac{\partial^2 V(\lambda_t, x(t))}{\partial \lambda_t \partial x(t)}$$

(Funo et al., 2017). These corrections are nonvanishing in protocols involving noncommuting controls and coherent evolution, directly measurable in quantum work statistics experiments (Onishchenko et al., 2022).

At high temperature, quantum corrections to the FDR vanish (e.g., $\mathcal{Q} = 0$), but at low temperature or slow driving (adiabatic limit), genuine quantum corrections persist and are fundamentally linked to quantum coherence and friction.

Nonequilibrium thermodynamic frameworks further allow for the assessment of quantum work distributions in gravitational systems, such as evaporating AdS black holes, where nonperturbative entropy corrections modify fluctuation relations (Jarzynski equality), internal energy, and transition probabilities between states (Pourhassan et al., 2022).

6. Finite-Time Corrections and Emergence of Irreversibility

When quantum thermodynamic processes are performed in finite time, nonvanishing excess work or heat—captured via geometric thermodynamics—quantifies the irreversibility inherent in any protocol: $$W_{\text{diss}}^{*} = \frac{k_B T}{\tau} \mathcal{L}^2$$ with the thermodynamic length $\mathcal{L}[\gamma]$ computed along trajectories in parameter space via a metric derived from the Drazin inverse of the dynamical generator (Rolandi, 24 Oct 2024).

Finite-time corrections in information erasure are fundamentally bounded even in strong-coupling regimes: $$Q \geq k_B T \left(\ln 2 + a\,\frac{\tau_{\mathrm{Pl}}}{\tau} \right)$$ with $\tau_{\mathrm{Pl}} = \hbar / (k_B T)$ denoting the Planckian time. As τ → τ_{\mathrm{Pl}}, the quantum speed limit for thermalization is reached, enforcing an irreducible cost for rapid processes even with arbitrarily strong system–bath couplings. Thus, Planckian times set universal speed limits for quantum thermodynamic transformations.

Collective driving of entangled or correlated subsystems further allows for sub-extensive scaling of dissipated work, thus reducing per-subsystem irreversibility and approaching the reversible Carnot limit in large-scale mesoscopic engines.

7. Quantum Corrections in Quantum Gases and Confined Systems

For quantum gases in confined geometries and at low temperatures, finite phase-space cell volumes and quantum uncertainty modify the classical partition function. The single-particle quantum partition function acquires corrections involving statistical variances of momentum: $$\mathcal{Z}_Q = \sum_{n_1, n_2, n_3} \exp\left[-\beta\, E_{n}\right],\quad E_n = \frac{\langle p \rangle^2}{2m} + (2n+1)\frac{B_u}{m}$$ with corrections translating into modified expressions for entropy, internal energy, and notably, to the emergence of a tensor pressure: $$[P] = k_B T\, \mathbb{I} - [B_u] \coth(\beta m [B_u])$$ (Ravelonjato et al., 2023). These effects become pronounced for nanoscale systems or at temperatures where the de Broglie wavelength becomes comparable to system dimensions.

Conclusion

Quantum corrections to thermodynamics represent a diverse and rich set of phenomena bridging quantum theory, statistical mechanics, information theory, and gravitation. They alter foundational thermodynamic laws, observables, and operational limits, particularly in small, strongly coupled, non-equilibrium, or gravitationally extreme settings. Thermal fluctuations, quantum coherence, finite-time dynamics, and information-theoretic constraints are codified through nontrivial corrections (logarithmic, exponential, geometric) to entropy, free-energy landscapes, and macroscopic work. These results have broad implications, from the microscopic interpretation of black hole entropy within quantum gravity and the design of high-fidelity quantum information processors, to the ultimate energetic bounds for nanoscale engines and erasure operations, and the emergent irreversibility arising from quantum foundations.