Exponential Corrected Entropy

• Exponential corrected entropy is a framework that introduces exponential terms to classical entropy formulas, capturing nonperturbative quantum and statistical effects.
• It modifies key physical laws in black hole thermodynamics and cosmology, leading to new insights such as entropy decay rates and bouncing cosmological models.
• The concept also underpins robust operational techniques in quantum information, such as privacy amplification and error exponent analyses derived from advanced functional inequalities.

Exponential Corrected Entropy refers to corrections to entropy formulas or entropy dynamics involving exponential terms, either as explicit corrections to the entropy–area relation in gravitational systems, as exponential rates in entropy decay for stochastic or quantum dynamical processes, or as precise exponential asymptotics in operational tasks such as entropy smoothing and privacy amplification. The term appears across diverse subfields, including quantum gravity, black hole thermodynamics, quantum information theory, kinetic theory, and cosmological model building. Precise mathematical forms and operational meanings of “exponential correction” depend on the context, yet commonly embody nonperturbative or leading-order quantum/statistical effects that alter entropy’s behavior or impart new robustness and universality properties.

1. Quantum Gravity and Black Hole Thermodynamics: Exponential Corrections to the Entropy–Area Law

In quantum gravitational settings, the classical Bekenstein–Hawking entropy formula $S = A/4$ (in Planck units, where $A$ is the horizon area) is subject to quantum corrections. While leading corrections are often logarithmic in $A$ (i.e., $S = A/4 + \alpha \ln A + \cdots$), nonperturbative effects can produce genuinely exponential terms: $S = \frac{A}{4} + \alpha \, e^{-A/4}$ as shown in (Ökcü et al., 19 Jul 2024), where $\alpha$ is a small parameter governing the strength of the correction. Such exponentially small terms are negligible for macroscopic black holes (large $A$), but dominate as $A\to 0$, suggesting nontrivial quantum microstructure effects.

The explicit incorporation of exponential corrections is motivated by statistical fluctuations and microstate counting in quantum gravity frameworks, leading to modified expressions for the entropy that, in turn, affect gravitational thermodynamics and cosmic evolution.

2. Modified Friedmann Equations and Cosmological Dynamics

The exponential corrected entropy modifies the thermodynamic derivation of the Friedmann equations for Friedmann–Robertson–Walker (FRW) cosmologies. Employing the first law of thermodynamics at the apparent horizon and the exponential-area correction (see (Ökcü et al., 19 Jul 2024)), the standard Einstein–Hilbert cosmological equations acquire correction terms: $S = \frac{1}{4}A + \alpha e^{-A/4}$

$$\frac{dE}{dt} = T_h \frac{dS}{dt} + W \frac{dV}{dt}$$

Pushing this through for the cosmological apparent horizon, one obtains modified Friedmann equations (cf. (8),(10) in (Ökcü et al., 19 Jul 2024)): $$-(2 r_A^2)^{-1} - \frac{\alpha}{2 r_A^2}(e^{-\pi r_A^2} - e^{-\pi r_A^2} Ei(-\pi r_A^2)) = -\frac{4\pi}{3} \rho$$

$$(\dot{H} - \frac{k}{a^2})(1 - \alpha e^{-\pi r_A^2}) = -4\pi(\rho + p)$$

with $r_A = 1/\sqrt{H^2 + k/a^2}$ the apparent horizon radius and $Ei(z)$ the exponential integral. The correction manifests as $H$-dependent terms which are negligible at late times (large $r_A$) but significant in the early universe (small $r_A$).

These modified equations lead to a variety of new physical phenomena, including:

• Enhanced deceleration parameter $q$ due to the corrections.
• Persistent validity of the generalised second law (GSL) of thermodynamics.
• The natural emergence of a bouncing cosmology at high energy densities, allowing nonsingular evolution (i.e., the avoidance of the classical Big Bang singularity).

Numerical analyses demonstrate that, for certain parameter ranges and nonzero spatial curvature ($k = \pm1$), these corrections guarantee the existence of a bounce—a minimum scale factor $a_c$ at which $\dot{a}=0$ and $\ddot{a} > 0$, see (Ökcü et al., 19 Jul 2024) Tables 1,2.

3. Black Hole Thermodynamics: Microcanonical Effects and Stability

Within black hole thermodynamics, exponential corrections to entropy, $S = S_0 + e^{-S_0}$, where $S_0 = A/4$, alter both equilibrium and stability properties (Pourhassan, 2020, Pourhassan et al., 2022). For large black holes, the correction is negligible, but as the black hole evaporates and $S_0$ decreases, the exponential term influences the final stages:

• Modifies the canonical partition function, shifting internal energy $U$, free energies, and specific heat $C$.
• Stabilizes small 4D Schwarzschild and Schwarzschild-AdS black holes by rendering the specific heat positive at small $A$ ((Pourhassan, 2020), Eq. (6)), an effect not present in higher dimensions (e.g., 5D Schwarzschild black holes remain unstable even with corrections).

Generalizations to charged and AdS black holes demonstrate that exponential corrections can trigger changes in the phase structure (first-order phase transitions, shifting/remnant states), and affect the equation of state, driving it to a leading-order virial form in the small black hole regime (Pourhassan et al., 2022).

To accommodate the correction in thermodynamic laws, the first law is adjusted: $dM = T dS_0 [1 - e^{-S_0}]$ along with generalized relations for electric charge ($\Phi dQ$), pressure-volume ($V dP$) terms, and extra work contributions.

4. Exponential Entropy Decay: Quantum Markov Semigroups and Kinetic Theory

Exponential corrected entropy also denotes entropy dynamics that exhibit exponential decay along the flow of a quantum (or classical) Markov semigroup, and in kinetic models such as the Kac and Boltzmann equations.

In quantum settings, (Wirth, 12 May 2025) establishes the equivalence: $$\beta\, D(\psi \Vert E_{*}(\psi)) \leq I_{*}(\psi \Vert E_{*}(\psi)) \;\Longleftrightarrow\; D(P_{t*}(\psi) \Vert E_{*}(\psi)) \leq e^{-\beta t}\, D(\psi \Vert E_{*}(\psi))$$ where $D$ is the relative entropy, $E_{*}$ the conditional expectation, and $I_{*}$ the entropy production (Fisher information). The decay rate $\beta$ is determined by a modified logarithmic Sobolev inequality (MLSI). Additional intertwining criteria for GNS-symmetric quantum Markov semigroups provide practical tools for proving these inequalities and thereby demonstrating rapid mixing and decoherence in quantum systems, even in the infinite-dimensional von Neumann algebra case.

In kinetic theory, analogous results are seen in the Kac master equation (Bonetto et al., 2017), where entropy relative to thermal equilibrium decays exponentially with an explicit, particle-number independent rate: $S(f(t)) \leq e^{-p t/2} S(f_0)$ for a suitable choice of collision kernel $p(\theta)$. The robustness of this decay is further ensured by correlation inequalities (Brascamp–Lieb, hypercontractive estimates), and the specific correction is tied to the definition of entropy relative to a reservoir (thermal bath).

5. Exponential Rate Analysis in Quantum Information: Privacy Amplification and Smoothing

In quantum information theory, “exponential corrected entropy” often refers to exact asymptotic exponents for error terms in operational tasks, such as smoothing the max-relative entropy or privacy amplification with quantum side information.

(Li et al., 2021) derives that the smoothing error for the max-relative entropy between $\rho$ and $\sigma$ scales as: $$\lim_{n \to \infty} -\frac{1}{n} \log \epsilon(\rho^{\otimes n} \| \sigma^{\otimes n}, nr) = \frac{1}{2} \sup_{s \geq 0} s[r - D_{1+s}(\rho\|\sigma)]$$ where $D_{1+s}$ is the sandwiched Rényi divergence. This exponent precisely characterizes the trade-off between the approximation error and the rate $r$ in the quantum setting.

For privacy amplification, the insecurity (distinguishability from ideal uniformity) decays as: $$\lim_{n\to\infty} -\frac{1}{n} \log (\text{insecurity}) = \frac{1}{2} \max_{0 \leq s \leq 1} s[H_{1+s}(X|E)_{\rho} - R]$$ for rates $R$ exceeding a critical value, where $H_{1+s}$ is the sandwiched Rényi conditional entropy. Here, the “correction” is an exact exponential rate, operationally meaningful and tightly matched between achievability and converse for $R \geq R_\mathrm{critical}$.

In both smoothing and privacy amplification, these exponents are governed by optimizations over Rényi information quantities and showcase the modern understanding that entropy correction is not merely an additive term, but can and should be characterized by precise exponential rates.

6. Mathematical Structures Underlying Exponential Corrections

The emergence of exponential corrections and exponential decay rates is, in many cases, underpinned by functional inequalities of logarithmic Sobolev type and de Bruijn identities (see (Wirth, 12 May 2025)), by operator convexity and intertwining conditions for noncommutative semigroups, or by the analysis of “error balls” and large deviation rates in information-theoretic settings (Li et al., 2021).

For gravitational/cosmological applications, the exponential correction to entropy modifies the equations of state and dynamical evolution via nonpolynomial terms, often requiring advanced techniques from the theory of special functions (exponential integrals, Lambert W function, etc.) to analyze their physical consequences.

In all settings, the robust feature is that exponential corrections—whether as entropy decay rates, as nonperturbative terms in entropy formulas, or as error exponents—mark a transition beyond leading-order (polynomial, linear, or logarithmic) behaviors and encode essential information about fluctuations, stability, and quantum (or stochastic) effects.

7. Broader Impact and Open Directions

The concept of exponential corrected entropy unifies disparate topics—quantum gravity, information theory, statistical mechanics, and quantum dynamics—by emphasizing the operational and dynamical significance of exponential terms in entropy. In gravitational theory, such corrections are crucial for modeling quantum effects in the early universe and in black hole evaporation, naturally supporting viable bouncing cosmologies and potentially resolving classical singularities. In quantum information, precise exponential exponents rigorously capture the trade-offs between security, error, and resources in protocols such as privacy amplification and hypothesis testing.

Further developments may include:

• Deeper understanding of the universality and limitations of exponential decay exponents in noncommutative and infinite-dimensional systems.
• Investigation of the role of exponentially corrected entropy in complex quantum many-body systems and nonequilibrium dynamics.
• Application of exponential entropy corrections in the search for quantum gravity–induced phenomenology in cosmology and black hole physics.

In all contexts, exponential corrected entropy demarcates the regime where quantum effects, fluctuations, and nonclassical phenomena exert a non-negligible, often dominant, influence on entropy evolution and physical observables.