📚 What is the Eigenstate Thermalization Hypothesis (ETH)?
The Eigenstate Thermalization Hypothesis (ETH) is a central concept in quantum statistical mechanics. It addresses how individual quantum systems, governed by the time-dependent Schrödinger equation, can exhibit thermodynamic behavior typically associated with large ensembles of particles. In essence, it provides a mechanism for thermalization in isolated quantum systems without resorting to external heat baths or ensemble averaging.
📜 History and Background
The development of ETH was driven by the need to reconcile quantum mechanics with the success of statistical mechanics in describing macroscopic systems. Traditional statistical mechanics relies on ensemble averaging, assuming that systems are in contact with a heat bath. However, many modern experiments deal with highly isolated quantum systems, such as ultracold atoms in optical lattices. ETH emerged as a way to understand thermalization in these isolated systems.
- ⚛️ Early work in the 1950s and 60s, particularly by John von Neumann, explored the foundations of quantum ergodicity.
- 🔥 In the 1990s, Marcos Rigol, Mark Srednicki, and others formalized the Eigenstate Thermalization Hypothesis.
- 🧪 Subsequent experimental and theoretical studies have validated and refined the ETH, establishing it as a cornerstone of quantum statistical mechanics.
🔑 Key Principles of ETH
ETH is based on several crucial assumptions about the structure of the Hamiltonian's eigenstates in quantum systems:
- 📊 Eigenstate Structure: Observables' expectation values in individual energy eigenstates depend smoothly on the energy and resemble thermal averages.
- 🔢 ETH Ansatz: For a generic few-body observable $O$, its matrix elements in the energy eigenbasis can be written as:
$O_{mn} = \overline{O}(\overline{E})\delta_{mn} + e^{-S(\overline{E})/2}f_{O}(E_m, E_n)R_{mn}$, where $\overline{E} = (E_m + E_n)/2$, $S(\overline{E})$ is the thermodynamic entropy at energy $\overline{E}$, $f_{O}(E_m, E_n)$ is a smooth function of $E_m$ and $E_n$, and $R_{mn}$ is a random variable with zero mean and unit variance.
- 🌡️ Diagonal Ensemble: The long-time average of any observable is equivalent to its microcanonical average if the initial state satisfies ETH.
- 🌌 Implications: This implies that even though the system is evolving unitarily and coherently, local observables behave as if the system has thermalized.
🌍 Real-world Examples
ETH has profound implications and applications across various areas of physics:
- 🧊 Ultracold Atoms: Experiments with ultracold atoms in optical lattices provide a testing ground for ETH. Observing the momentum distribution after a quench reveals whether the system thermalizes according to ETH predictions.
- ⚫ Black Holes: ETH is connected to the black hole information paradox. The thermalization of black holes, as described by ETH, suggests that information is not lost but scrambled in a complex way.
- 💻 Quantum Chaos: ETH is closely linked to quantum chaos. Systems exhibiting quantum chaos typically satisfy ETH, indicating that their energy eigenstates are highly complex and thermal-like.
- 🔬 Condensed Matter Physics: In condensed matter systems, ETH explains the behavior of interacting electrons in metals and insulators, where thermalization occurs even in the absence of external heat baths.
💡 Conclusion
The Eigenstate Thermalization Hypothesis is a cornerstone of modern quantum statistical mechanics. It provides a framework for understanding how isolated quantum systems can thermalize and exhibit thermodynamic behavior. By elucidating the structure of energy eigenstates, ETH connects the microscopic quantum world with the macroscopic world described by statistical mechanics, offering insights into various phenomena from ultracold atoms to black holes.