Operational Entropy for Solitonic Horizons

• The paper presents a novel framework that quantifies irreversible entropy production in solitonic event horizons using spectral partitioning in dispersive Kerr media.
• It employs Fourier analysis to decompose the optical field into coherent and incoherent components, tracking photon exchange and entropy growth as integrability is broken.
• Numerical simulations confirm that resonant radiation produces entropy growth exceeding the soliton’s decay, thereby validating a generalized second law in a Hamiltonian system.

Operational entropy for solitonic event horizons provides a rigorous framework for quantifying irreversibility and information flow in optical analogs of gravitational horizons. Specifically, it defines thermodynamic entropy for an event horizon analog formed by a propagating soliton in a dispersive Kerr medium, capturing the transition from coherent to incoherent spectral subsystems and enabling the application of a generalized second law in a system governed by Hamiltonian dynamics. This approach relies on partitioning the optical field’s spectrum, tracking the exchange of photon number and entropy between the soliton and its radiative continuum, and demonstrating strict entropy growth as integrability is broken. The following sections elaborate the core principles, definitions, mechanisms, and implications underlying this operational entropy construction (Oguz, 8 Jan 2026).

1. Mathematical Formulation and Spectral Partitioning

Solitonic event horizons in Kerr media are modeled by the dimensionless generalized nonlinear Schrödinger equation (GNLSE) in the soliton’s comoving frame: $$i\,\frac{\partial \psi}{\partial \xi} \;+\;\frac{1}{2}\,\frac{\partial^2\psi}{\partial\tau^2} \;-\;i\,\delta_{3}\,\frac{\partial^3\psi}{\partial\tau^3} \;+\;|\psi|^2\,\psi \;=\;0$$ where

$$\xi=\frac{z}{L_D},\quad \tau=\frac{T}{T_0},\quad \psi=\frac{A}{\sqrt{P_0}},\quad \delta_3=\frac{\beta_3}{6\,|\beta_2|\,T_0}$$

The dynamics depend critically on $\delta_3$: for $\delta_3=0$, the system is integrable and supports stable solitons; for $\delta_3\neq 0$, integrability is broken, inducing coupling between the soliton and radiation continuum.

In the spectral domain, the optical field is decomposed using the Fourier transform $$\tilde{\psi}(k, \xi) = \mathcal{F}[\psi(\tau, \xi)](k)$$, and the wavenumber axis is partitioned into:

• $\Omega_S$: a narrow band centered on the soliton’s coherent spectral peak,
• $\Omega_R$: the complementary, broad dispersive band where resonant (Cherenkov) radiation and incoherent modes reside.

This spectral bipartition is foundational to the operational entropy framework.

2. Operational Entropy: Coherent and Incoherent Subsystems

The entropy is defined by partitioning the field into coherent (“system”) and incoherent (“bath”) components, associating different entropic functionals with each:

• Horizon (soliton) entropy:

$$S_{\mathrm{hor}}(\xi) = \alpha \int_{\Omega_S} |\tilde{\psi}(k, \xi)|^{2} \, dk$$

This measures the photon number within $\Omega_S$, serving as a proxy for the coherent information capacity.

• Radiation (incoherent) entropy:

The normalized spectral probability density over $\Omega_R$,

$$p(k,\xi) = \frac{|\tilde{\psi}(k,\xi)|^2}{\int_{\Omega_R}|\tilde{\psi}(k',\xi)|^2\,dk'}$$

defines the classical Shannon entropy

$$S_{\mathrm{rad}}(\xi) = -\int_{\Omega_R} p(k, \xi) \ln p(k, \xi)\, dk$$

which quantifies the degree of mixing and spectral broadening in the radiation subsystem.

The total entropy is the sum: $$S_{\mathrm{tot}}(\xi) = S_{\mathrm{hor}}(\xi) + S_{\mathrm{rad}}(\xi)$$

This construction does not employ the Gibbs–von Neumann entropy for quantum states, but rather a classical, coarse-grained measure arising from the partition of spectral modes into coherent and incoherent sectors.

3. Integrability Breaking, Resonant Radiation, and Entropy Production

Entropy production in these systems is directly linked to the breaking of integrability by the third-order dispersion term $-i\delta_3 \partial_\tau^3\psi$ in the GNLSE. This term mediates coupling between the soliton (discrete spectral mode) and the continuum of linear waves in $\Omega_R$, with resonant energy transfer governed by a phase-matching condition: $$\omega'_{\mathrm{sol}}(k_{\mathrm{RR}}) = \left.\frac{d}{dk} \left(\tfrac{1}{2}k^2 - \delta_3 k^3\right)\right|_{k=k_{\mathrm{RR}}}$$ such that dispersive (Cherenkov) radiation at resonant wavenumbers $k_{\mathrm{RR}}$ is emitted.

This process irreversibly transfers photon number and energy from the coherent soliton to the radiation bath, causing $S_{\mathrm{hor}}$ to decrease and $S_{\mathrm{rad}}$ to increase. The sum $S_{\mathrm{tot}}$ monotonically increases, reflecting the fundamentally nonequilibrium thermodynamic character of the solitonic event horizon under integrability breaking.

4. Generalized Second Law: Monotonic Entropy Growth

Despite the system’s purely Hamiltonian evolution (total photon number conserved), the coarse-grained entropy defined above satisfies a generalized second law: $$\frac{dS_{\mathrm{tot}}}{d\xi} = \frac{d}{d\xi}[S_{\mathrm{hor}} + S_{\mathrm{rad}}] \geq 0$$ This reflects that any photon flux $\delta N$ lost by the coherent soliton yields a greater or equal increase of Shannon entropy in the radiation bath, due to the mixing effect within $\Omega_R$.

Heuristically:

• $dS_{\mathrm{hor}}/d\xi \leq 0$ (photon number diminishes in $\Omega_S$),
• $$dS_{\mathrm{rad}}/d\xi \geq |dS_{\mathrm{hor}}/d\xi|$$ (entropy in $\Omega_R$ increases more than the loss in $\Omega_S$),
• ensuring $dS_{\mathrm{tot}}/d\xi \geq 0$.

This statement is validated numerically and holds for a broad class of filter choices partitioning the spectrum.

5. Numerical Methods and Simulation Results

The operational entropy framework has been verified by direct simulation of the GNLSE using a symmetrized split-step Fourier method with adaptive step-size control to ensure photon number conservation (local error $\sim 10^{-6}$). A dynamic spectral filter $W_S(k)$, typically super-Gaussian, is used at every step to isolate $\Omega_S$.

Simulation parameters are chosen to match fiber supercontinuum experiments:

• $\beta_2 = -15$ ps$^2$/km, $\beta_3 = 0.1$ ps$^3$/km, $\gamma = 0.1$ W$^{-1}$km$^{-1}$
• $T_0 = 50$ fs, soliton order $N = 3.5$, dimensionless $\delta_3 \approx 0.01$

At each propagation step:

• $S_{\mathrm{hor}}(\xi)$, $S_{\mathrm{rad}}(\xi)$, and $S_{\mathrm{tot}}(\xi)$ are computed. Numerical results show $S_{\mathrm{hor}}$ decreases as the soliton fissions, but $S_{\mathrm{rad}}$ increases more than compensating, yielding a strictly monotonically increasing $S_{\mathrm{tot}}(\xi)$. No violations of $\Delta S_{\mathrm{tot}} \geq 0$ are observed over hundreds of dispersion lengths.

After significant fission events, the radiation spectrum exhibits an exponential tail,

$$|\tilde{\psi}(k)|^2 \propto \exp(-k/\kappa_{\mathrm{eff}})$$

indicative of an effective Boltzmann distribution with temperature $T_H \sim \hbar\kappa_{\mathrm{eff}}/2\pi k_B$, which emerges only with irreversible entropy production.

6. Nonequilibrium Thermodynamics and Conceptual Implications

Operational entropy endows optical event horizons with thermodynamic status, despite underlying Hamiltonian dynamics. Key aspects:

• Entropy arises via classical coarse-graining between coherent and incoherent spectral sectors, not via quantum statistical mixtures.
• Integrability breaking ($\delta_3\neq 0$) acts as an intrinsic entropy generator, analogous to horizon-induced mode coupling in gravitational analogs.
• The emergent Hawking-like temperature in the radiation spectrum has physical meaning only in the presence of irreversible entropy growth.

Notably, the spectral filter–based partition underlying $S_{\mathrm{hor}}$ and $S_{\mathrm{rad}}$ is heuristic; while different choices yield similar qualitative behavior, quantitative details may vary.

Quantum fluctuations and dissipative or Raman effects are omitted; all entropic changes arise from coherent Hamiltonian mode coupling. This suggests further extensions are necessary for fully quantum or dissipative scenarios.

7. Summary and Outlook

Operational entropy, as constructed for solitonic event horizons, provides a practical and rigorous metric for irreversibility and information flow in analog gravity systems governed by optical solitons. By defining entropies specific to coherent and incoherent spectral partitions, linking entropy production to integrability breaking and resonant radiation, and numerically demonstrating a strict generalized second law, the formalism elevates optical event horizons to bona fide nonequilibrium thermodynamic systems. This connection opens pathways for laboratory exploration of information-theoretic aspects of analog gravity, with potential extensions to quantum and dissipative regimes (Oguz, 8 Jan 2026).