Anti-Hawking-Page Transition

• Anti-Hawking-Page transition is a thermodynamic phenomenon in AdS black holes marked by a direct, sharp transition from large black holes to thermal radiation, bypassing the traditional unstable small black hole phase.
• Advanced stochastic models using generalized Langevin equations and non-Markovian friction kernels reveal how memory effects and evaporation alter classical Kramers kinetics.
• The Maxwell equal-area construction and noncommutative geometry frameworks provide insights into phase reconstruction and criticality, linking the transition to confinement/deconfinement phenomena in gauge theories.

The anti-Hawking-Page transition refers to a class of thermodynamic phenomena in AdS black hole systems where the decay of large black holes into thermal radiation (thermal AdS) exhibits features distinct from the canonical Hawking-Page transition. Unlike the standard scenario—where phase coexistence and unstable “small” black holes precede the jump to thermal AdS—the anti-Hawking-Page regime is marked either by a direct, sharp transition (as in Maxwell construction) or by a modified kinetic pathway due to non-Markovian effects, black hole evaporation, or noncommutative spacetime structure. These processes have significant implications for the phase diagram, kinetics, and microscopic interpretation of black hole thermodynamics.

1. Thermodynamic Landscape and Classical Hawking-Page Transition

The classical Hawking-Page transition, formulated for neutral Schwarzschild-AdS black holes, delineates a first-order phase transition between thermal AdS and a large stable black hole at temperature $T_{HP}=1/(\pi L)$, where $L$ is the AdS curvature radius. The off-shell Gibbs free energy $V(r)=G(r)=\frac12 r(1+r^2/L^2)-\pi T r^2$ is a function of horizon radius $r\equiv r_+$, with critical structure:

• For $T<T_{min} = \sqrt{3}/(2\pi L)$, $V(r)$ has a single minimum at $r=0$ (thermal AdS).
• For $T_{min}<T<T_{HP}$, there are two minima at $r=0$ and $r=r_l>0$ (“large BH”), separated by a barrier at $r=r_s$ (“small BH”).
• The HP transition at $T=T_{HP}$ marks the degeneracy of the two minima.

This landscape underpins the stochastic kinetics in subsequent sections.

2. Kinetic Pathways and Non-Markovian Effects

Stochastic decay from the large BH to thermal AdS under thermal fluctuation is modeled via a generalized Langevin equation (GLE) for the collective variable $r(t)$:

$$\begin{aligned} & \dot{r} = v, \ & \dot{v} = - V'(r) - \int_0^t \zeta(t-s)\,v(s)\,ds + \eta(t), \end{aligned}$$

where $\zeta(t)$ is the friction kernel and $\eta(t)$ is Gaussian colored noise obeying $$\langle \eta(t) \eta(s) \rangle = k_B T \zeta(|t-s|)$$. In the Markovian limit $(\zeta(t)=\zeta_0\delta(t))$, the dynamics reduce to standard Kramers theory. For non-Markovian (memory) kernels:

• Exponential kernel: $\zeta(t)=(\zeta/\gamma)e^{-|t|/\gamma}$, $\hat{\zeta}(\lambda)=\zeta/(1+\lambda\gamma)$.
• Oscillatory kernel: $$\zeta(t)=\zeta e^{-|t|/\gamma}[\cos(\omega t)+(1/(\omega \gamma))\sin(\omega|t|)]$$, $$\hat{\zeta}(\lambda)=\zeta\gamma(2+\gamma\lambda)/[\omega^2\gamma^2+(1+\gamma\lambda)^2]$$.

The Grote-Hynes rate for the long-time transition $\kappa$ is:

$$\kappa = \frac{\lambda_r}{\omega_s} \frac{\omega_l}{2\pi} e^{-\beta W},$$

where $\lambda_r$ solves $$\lambda = \omega_s^2/[\lambda + \hat{\zeta}(\lambda)]$$, $W$ is the barrier height, $\omega_l$ and $\omega_s$ are the well and barrier frequencies.

Kinetic Modifications:

• Exponential kernel: Increasing the bath memory time $\gamma$ monotonically accelerates barrier crossing, saturating at large $\gamma$ to the energy-diffusion regime $(\lambda_r\to\omega_s)$.
• Oscillatory kernel: $\kappa(\gamma)$ is nonmonotonic: for small $\gamma$ effective friction increases and kinetics slow; for large $\gamma$ it reverts to the accelerating regime, with a turnover around $\gamma\sim 1/\omega_s$. Raising the oscillation frequency $\omega$ increases $\kappa$.

This mechanism is essential for the anti-Hawking-Page transition, demonstrating that bath memory generically enhances stochastic decay from the large BH phase to thermal AdS (Li et al., 2022).

3. Reaction-Diffusion and Hawking Evaporation Effects

The kinetics of the anti-Hawking-Page transition are fundamentally altered when black hole evaporation is implemented as an absorbing boundary condition (\emph{evaporation-dominated regime}). On the free energy landscape $G(x;T)$ with $x\equiv r_+\in [0,\infty)$, the evolution is governed by the reaction-diffusion equation:

$$\frac{\partial P(x,t)}{\partial t} = D \frac{\partial}{\partial x} \left\{ e^{-\beta G(x)} \frac{\partial}{\partial x}\left[ e^{\beta G(x)} P(x,t)\right] \right\} - k(x)\,\delta(x-r_\ell)\,P(x,t),$$

where $D=T/\zeta$ is the diffusion coefficient, $k(x) = |\dot{M}/M|$ quantifies local evaporation rate, and $P(x,t)$ is the probability density.

Mean First Passage Time (MFPT):

Overdamped MFPT $\tau$ from $x_\ell$ (large BH) to $x_s$ (barrier top):

$$\tau = \int_{x_\ell}^{x_s} dy\, \frac{e^{\beta G(y)}}{D} \int_0^y dz\, e^{-\beta G(z)},$$

Without evaporation, $\tau$ scales as $\exp[\Delta G / T]$. With evaporation (reaction term), $\tau$ must be calculated numerically from the survival probability $\Sigma(t)$ and the first passage distribution $F_p(t)$.

Kinetic Turnover Phenomenon:

Numerical analysis reveals a turnover temperature $T_{turn}$, marking the regime in which evaporation dominates over thermal barrier crossing. For fixed $\zeta$, $\tau(T)$ first increases (barrier limited) then decreases (evaporation limited) at sufficiently high $T$. For fixed $T>T_{HP}$, $\tau$ grows with $\zeta$ for small $\zeta$ (diffusion limited), then saturates or decreases for large $\zeta$ (reaction limited), creating a “dynamical phase diagram” that separates free-energy-controlled from evaporation-controlled kinetics. The interplay determines the anti-Hawking-Page transition (Li et al., 2021).

4. Maxwell Equal-Area Law and Thermodynamic Reconstruction

The Maxwell equal-area construction applied to the $T$-$S$ plane (temperature-entropy curve) provides a rigorous method to eliminate the negative heat capacity (unstable small-BH) branch in Schwarzschild-AdS thermodynamics. The construction replaces the segment between $S_1$ and $S_2$ with a flat isotherm $T=T^*$, chosen so the areas above and below are equal:

$\int_{S_1}^{S_2} T(S)\,dS = T^*(S_2-S_1).$

This changes the conventional phase diagram:

• For $T<T^*$, pure radiation is the unique stable phase.
• At $T=T^*$, a family of black holes nucleates suddenly, all with the same free energy.
• For $T>T^*$, a single large, positive-heat-capacity black hole is globally stable.

The Maxwell temperature $$T^*=\frac{\sqrt{3}}{12\pi l}(2\sqrt{13}-1)\approx 0.2868\,l^{-1}$$ is below $T_{HP}$. The unstable small-BH regime ($T_{min} < T < T_{HP}$) is excised; the system transitions directly from radiation to large BH at $T^*$. This is termed the “anti-Hawking-Page” transition (Spallucci et al., 2013).

5. Noncommutative Geometry, Crossover, and Criticality

In noncommutative Schwarzschild-AdS spacetime, the point-mass source is replaced with a Gaussian profile ($$\rho_\theta(r) = \frac{M}{(4\pi\theta)^{3/2}}e^{-r^2/4\theta}$$). The spacetime metric is completely regular and the curvature singularity at $r=0$ is removed. The phase structure is altered:

• For small noncommutativity parameter $q\equiv \sqrt{\theta}/\ell < q^*\approx 0.18243$, the $T(r_h)$ curve exhibits two extrema (spinodal points) and a “swallowtail” in $F(T)$, signaling first-order transitions.
• At critical $q=q^*$, the swallowtail degenerates and a second-order critical point emerges, with typical mean-field exponents ($\alpha=0$, $\beta=1/2$, $\gamma=1$, $\delta=3$).
• For $q>q^*$, $F(T)$ is single-valued and the system undergoes a smooth crossover between small and large black holes, not a true phase transition.

Within AdS/CFT duality, $\theta/\ell^2$ corresponds to $N_f/N_c$ in the boundary gauge theory, controlling the transition from sharp confinement/deconfinement (first order) to crossover at finite $N_f/N_c$ (Nicolini et al., 2011).

6. Physical Interpretation, Implications, and Microscopic Insights

The anti-Hawking-Page transition, instantiated via kinetic turnover, non-Markovian memory, Maxwell equal-area construction, and noncommutative modifications, fundamentally alters the nucleation and decay processes of large AdS black holes:

• Thermodynamic Impact: The direct transition from thermal AdS to large BH at $T^*$ (Maxwell construction) eliminates the unstable regime, providing a more physical phase diagram consistent with positive heat capacity and stable nucleation.
• Kinetic Signatures: Memory effects and evaporation significantly speed up or dominate barrier crossing at high temperature or strong coupling, breaking canonical Kramers scaling law and providing tools for probing black hole microstructure via the friction parameter $\zeta$ and diffusion constant $D$.
• Gauge/Gravity Duality: The phase structure aligns with confinement/deconfinement crossover in boundary gauge theories, with bulk parameters ($\theta$, $\zeta$) mapping to microscopic properties ($N_f/N_c$, bath correlations).
• Experimental and Theoretical Relevance: A plausible implication is that in any setup simulating a “black hole in a box,” observable lifetimes deviate from classical predictions, asymptoting to evaporation timescales under strong bath coupling or high temperature.

The interplay between stochastic thermodynamics, quantum field “reaction” processes (evaporation), and microscopic bath correlations provides a comprehensive framework for understanding the anti-Hawking-Page transition as a central element of black hole phase dynamics and thermodynamics in AdS spacetimes.