Markov Length Reveals Singular Thermalization and Non-Gibbsian States in Quenches

The behaviour of complex systems as they heat up from a low-temperature ordered state remains a fundamental question in physics, and recent work suggests our understanding of thermalisation may be incomplete. Jerome Lloyd from the University of Geneva, Dmitry Abanin and Sarang Gopalakrishnan from Princeton University, and their colleagues demonstrate that a key measure of information, the Markov length, actually grows exponentially as a system heats up, even under simple classical conditions. This finding challenges the conventional view that measurable properties remain stable during thermalisation, revealing that the system becomes increasingly non-equilibrium and effectively ‘forgets’ its initial state. The research introduces a new method for calculating this Markov length and, crucially, suggests that the ultimate approach to thermal equilibrium may be more complex and singular than previously thought, offering a fresh perspective on the nature of thermalisation itself.

Markov Length, Lyapunov Exponents, Mutual Information

This research investigates the relationship between how quickly information is lost in a system, its rate of chaotic behavior, and the connections between different parts of the system, focusing on a one-dimensional spin system undergoing rapid heating. Researchers explored how the “Markov length,” a measure of information loss, changes over time, starting from both ordered and disordered initial states, using a model of thermal relaxation called Glauber dynamics. They discovered a connection between the Markov length and the system’s Lyapunov exponent, suggesting that the former can be predicted from the latter. The study also examined how the Markov length is affected by the final temperature reached during heating, revealing differences compared to scenarios with infinite temperature fluctuations.

Extensive numerical simulations supported these theoretical findings. The team found that the Markov length, starting from an ordered state, scales predictably with the system’s magnetization, consistent with established statistical principles. They also observed that the Markov length is enhanced by the “thermal length,” indicating that the system’s memory is limited by the range of thermal fluctuations. Crucially, the late-time Markov length proved independent of the initial temperature, suggesting that the system’s memory is ultimately limited by these thermal fluctuations. This research provides a rigorous theoretical framework, supported by detailed numerical simulations, offering a comprehensive analysis of the Markov length and its connection to other key quantities.

Matrix Product States Track Information Spread

Researchers developed a novel computational method to track how information spreads within a system undergoing rapid heating, moving beyond traditional statistical sampling techniques. Recognizing the limitations of conventional methods for capturing the full probability distribution of the system’s state, they employed matrix product state (MPS) simulations, a technique borrowed from quantum physics and recently adapted for studying classical dynamics. This allows for a direct approximation of the evolving probability distribution, capturing correlations between different parts of the system. The core innovation lies in efficiently calculating the “conditional mutual information” (CMI), a measure of how much information one part of the system has about another, within the MPS framework.

To overcome the computational challenges, the team developed a streamlined method that leverages the MPS to directly access probabilities and efficiently compute information content. They validated their approach by comparing their MPS simulations to known analytical solutions, confirming the accuracy of their method. This established confidence in their ability to accurately model the system’s dynamics and study how information spreads as the system heats up, and how this relates to its ability to “remember” its initial state. By focusing on translationally invariant tensors, they can effectively model an infinitely large system, a significant advantage over methods limited to finite-size systems.

Information Loss Limits Thermal Equilibrium Description

Researchers have discovered a surprising characteristic of how systems approach thermal equilibrium, revealing a fundamental limit to how accurately we can describe their state even as they settle into a stable temperature. The work centers on the “Markov length,” a newly proposed measure of information loss that grows without bound as a system heats up from an ordered, low-temperature state to a disordered, high-temperature state. This challenges the conventional understanding that systems simply relax to a predictable equilibrium. The team demonstrated that while a system eventually reaches a stable temperature, the range of possible states it could occupy must expand indefinitely during the heating process.

This means that even at late times, the system’s state remains increasingly non-Gibbsian, deviating from the expected thermal distribution. Importantly, this divergence occurs even when heating to a temperature within the same phase of matter, suggesting it’s a general property of thermalization. Using simulations of the one-dimensional Ising model, the team confirmed these theoretical predictions and interpreted the Markov length as a measure of a “Lyapunov gap” within a mathematical framework, potentially offering broader implications for understanding complex systems. The findings suggest that the late-time limit of thermalization is fundamentally singular, meaning that standard approaches to describing equilibrium may break down as the system approaches stability, and that a more nuanced understanding of information loss is required.

Thermalization Reveals Increasing Non-Gibbsian System Complexity

This research introduces the concept of Markov length as a diagnostic tool for characterizing how systems evolve towards thermal equilibrium following a rapid change in temperature, known as a thermal quench. The team demonstrates that, unlike conventional measures, Markov length actually increases over time during this process, even though the system is heading towards a stable, high-temperature state. This suggests that the system becomes increasingly non-Gibbsian, meaning it deviates from the expected statistical distribution of thermal equilibrium, and that defining a simple “parent Hamiltonian” to describe its state becomes problematic as time progresses. The findings imply that the conventional understanding of thermalization may be singular, meaning it breaks down when viewed from this information-theoretic perspective.

To investigate this, researchers developed a numerical technique using matrix-product states to calculate Markov length in the one-dimensional classical Ising model. Their results consistently show this growing Markov length, providing support for the theoretical arguments. The team validated their method by comparing the results of their simulations to exact analytical solutions, finding strong agreement, particularly at early times and shorter distances. They acknowledge that the accuracy of the simulations may decrease at very late times and large distances, and that the method relies on approximations when truncating the computational space. Future work could focus on extending this analysis to more complex systems and exploring the implications of a diverging Markov length for our fundamental understanding of thermalization and statistical mechanics.