Hybrid thermalization in the large $N$ limit

TL;DR

The paper develops and analyzes a semi-holographic, two-sector framework in the large $N$ limit where perturbative and non-perturbative degrees of freedom interact through democratic effective metrics. It establishes thermodynamic and statistical consistency for global thermal equilibrium, showing it is the unique maximum-entropy state in the microcanonical ensemble, while allowing pseudo-equilibria with distinct sector temperatures. The authors prove that typical high-energy non-equilibrium states relax to global equilibrium, and they demonstrate this both analytically (via entropy extremization) and numerically (via a BRSSS-based hybrid dynamics for homogeneous relaxation), revealing emergent conformality and universal entropy scaling at large energy. This work provides a principled bridge between perturbative and holographic descriptions of thermalization, with potential implications for quark-gluon plasma dynamics and quantum statistical interpretations of large-$N$ thermalization.

Abstract

Semi-holography provides a formulation of dynamics in gauge theories involving both weakly self-interacting (perturbative) and strongly self-interacting (non-perturbative) degrees of freedom. These two subsectors interact via their effective metrics and sources, while the full local energy-momentum tensor is conserved in the physical background metric. In the large $N$ limit, the subsectors have their individual entropy currents, and so the full system can reach a pseudo-equilibrium state in which each subsector has a different physical temperature. We first complete the proof that the global thermal equilibrium state, where both subsectors have the \textit{same} physical temperature, can be defined in consistency with the principles of thermodynamics and statistical mechanics. Particularly, we show that the global equilibrium state is the unique state with maximum entropy in the microcanonical ensemble. Furthermore, we show that in the large $N$ limit, a \textit{typical} non-equilibrium state of the full isolated system relaxes to the global equilibrium state when the average energy density is large compared to the scale set by the inter-system coupling. We discuss quantum statistical perspectives.

Figures (6)

• Figure 1: The isolated system, $\mathcal{F}$, is composed of two subsystems, $\mathcal{U}$ and $\tilde{\mathcal{U}}$, which individually satisfy their own conservation equations \ref{['Eq:WI']} in their respective background metric. The two subsystems are coupled and interact via deformations of their metric, given by \ref{['Eq:metriccoup']}.
• Figure 2: Pseudo-equilibrium states for two 3-dimensional conformal subsystems coupled via the democratic effective metric coupling. Explicitly, $\varepsilon = 2 P = n_1 T_1^3$, $\tilde{\varepsilon} = 2 \tilde{P} = n_2 T_2^3$ and we set $\gamma'=2$, $\gamma=1$, $n_1=10$, $n_2=1$. Left: subsystem entropies and the full system entropy as functions of $T_1$. Right: full system entropy and global equilibrium measure ($a T_1- \tilde{a} T_2 = T_a - T_b$) as functions of $T_1$. Note that the total entropy maximizes precisely when the global equilibrium condition is satisfied.
• Figure 3: Evolution of the full system entropy density $\mathcal{S}$ (left) and the subsystem entropy densities $s_1$ and $s_2$ (right) for fixed energy density $\mathcal{E}_0 = 0.991$ and anisotropy $A1$ given in \ref{['tab:Tab1']}. The inset plot in the left figure shows the evolution of the subsystem entropy density at early time. Here we have fixed $\gamma = 1$ and $\tilde{\gamma} = -2$. We note that although the full entropy density reaches very close to the global equilibrium value, the subsystem entropy densities are away from the global equilibrium value. The final state is a pseudo-equilibrium state. We can explicitly check that the final state is typically much closer to the global equilibrium state for larger values of $\mathcal{E}_0$.
• Figure 4: Total energy density as a function of $T_0$ for typical initial state with $\delta = 0$ for the three cases. For the three different anisotropies in \ref{['tab:Tab2']}, the curves overlap.
• Figure 5: Dependence of the total diagonal anisotropy $\Pi_d = \Pi_{22} = -\Pi_{33}$ (left) and off-diagonal anisotropy $\Pi_{od} = \Pi_{23} = \Pi_{32}$ (right) of the system as a function of the energy density of the full system for different initial anisotropy in \ref{['tab:Tab1']}.