Thermalization and Mpemba-like patterns in effective temperature dynamics of strongly coupled dissipative quantum chaotic systems

TL;DR

The paper addresses how Mpemba-like crossings (MPCs) manifest in temperature dynamics of strongly coupled, quantum chaotic systems, using the SYK model coupled to thermal baths and its gravity dual as a testbed. It employs Schwinger-Dyson equations and Keldysh real-time dynamics to track the time-dependent inverse temperature $\beta(t)$ defined through low-frequency Green's functions, revealing MPCs and temperature oscillations only at sufficiently strong system-bath coupling $V$, while Lindblad dissipative dynamics fail to reproduce these anomalies. The study shows that MPCs arise from nonequilibrium statistics and a dynamical imbalance between fast scrambling and slow energy relaxation, with bath biases in the two-bath setup raising the coupling threshold for crossings; these effects are not captured by Lindblad models. The results have implications for nonequilibrium black hole thermodynamics and early-universe cosmology, suggesting that dynamically driven thermalization anomalies could influence Hawking-like radiation spectra in strongly coupled gravitational systems. Overall, the work indicates a potentially universal nonequilibrium mechanism for MPCs in chaotic quantum matter and provides a bridge between quantum chaos, holography, and gravitational thermodynamics.

Abstract

Anomalous thermalization, particularly the crossings of temperature trajectories from different initial states termed Mpemba crossings (MPCs), have intrigued scientists for decades. While recent studies in quantum systems suggest that initial conditions play a decisive role in its emergence, they offer limited insight into MPCs in complex, highly nonequilibrium systems. In this study, we investigate temperature dynamics in the strongly coupled, quantum chaotic Sachdev-Ye-Kitaev (SYK) model, which is dual to the low-energy dynamics of 2D dilaton gravity. Our findings reveal a dynamically driven nonequilibrium mechanism underlying MPCs during rapid thermalization, with implications for gravitational systems. We explore quench dynamics in SYK systems under three conditions: coupling to a single SYK thermal bath, coupling to two thermal baths at different temperatures, and dissipative SYKs modeled by the Lindblad equation. We find that strong system-bath coupling induces oscillating effective temperatures and trajectory crossings in transient states due to nonequilibrium statistics, phenomena absent in quasi-static thermodynamics and Lindbladian SYKs. These MPCs highlight a unique feature of anomalous thermalization of strongly coupled quantum chaotic systems driven far from equilibrium. Besides, the results also provide qualitative insights into the nonequilibrium thermodynamics of black holes strongly interacting with their environment, such as primordial black holes in the early universe.

Figures (20)

• Figure 1: Diagrammatic illustration of MPEs and MPCs. The slanted black and blue curves are examples of temperature trajectories from two initial states. The region inside the dotted circle represents other unknown dynamics. The MPE is usually defined as the order exchange of the initial and the final states in the blue shaded regions along the x- and y-axes. The final state is at a certain critical temperature $T_c$, corresponding to a phase transition temperature such as the icing point or the temperature of the surrounding heat bath. The MPC refers to the crossing of the temperature trajectories during the dynamic process, indicated by the orange circled regions at the intersections of the two curves. MPCs can be viewed as an generalization of the MPE.
• Figure 2: The evolution of inverse temperature $\beta$ vs time $t$. The coupling between the system and the bath is set to be (a) $V=0.3J_0$, (b) $V=0.525J_0$, (c) $V=0.525J_0$ with the zoomed-in view of (b), where the dashed horizontal line indicating the bath temperature, and (d) $V=0.55J_0$ with an inset the zoomed-in view. For weak system-bath couplings, the thermalization processes behave as expected. The anomalous effects only exist for strong couplings. For all, $n=3,\, \beta_{\mathrm{bath}} J_0=0.5$, and $J_0$ is the reference coupling strength related to the SYK coupling $J$ by $J_0=2J$.
• Figure 3: Nonequilibrium statistics during cooling. (a) Energy distribution functions of the system when the effective temperature reaches the bath temperature for the $i$-th time, as indicated in Figure \ref{['fig:Fig_2']}(c). (b) Distance of the SYK system during thermalization from the asymptotic equilibrium state at the bath temperature. The high-frequency cutoff in the computation of the distance function is chosen to be $\omega/J_0=1.0$. (c) Distance function quantifying the deviation from the corresponding time-dependent equilibrium states at the same temperatures during cooling. Parameters are the same as in Figure \ref{['fig:Fig_2']}(c), and initial inverse temperature is $\beta_{\rm initial}J_0=4.0$. $J_0$ is the reference coupling strength and is set to be twice the SYK coupling, namely, $J_0=2J$.
• Figure 4: Mpemba crossings in the case of two different baths. (a) The full dynamics from equilibrium states to the steady states. The inset is a zoomed-in view of the dynamics. The parameters used in the numerical simulations are $V_1=V_2=0.5J_0,\, \beta_{bath1}J_0=4.4,\, \beta_{bath2}J_0=4.8$. (b) The threshold coupling $V_\mathrm{th}$ for the emergence of MPCs against the temperature bias between the two baths. For all, $n=3,\, J=0.5J_0.$
• Figure 5: (a) The effective temperature dynamics in the Lindblad description at the dissipative constant $\mu=0.1$. The SYK coupling is set to be $J/J_0=0.5$ and the interacting order is $q=4$. (b) The effective temperature dynamics in the exact calculation of SYKs coupled with baths at infinite temperature. The inset is a zoomed-in view of the dynamics. The parameters used are $J/J_0=0.5,\, V/J_0=0.525, \, n=3$.