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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Eigenstate thermalization hypothesis ： ウィキペディア英語版 Eigenstate thermalization hypothesis The Eigenstate Thermalization Hypothesis (or ETH) is a set of ideas which purports to explain when and why an isolated quantum mechanical system can be accurately described using equilibrium statistical mechanics. In particular, it is devoted to understanding how systems which are initially prepared in far-from-equilibrium states can evolve in time to a state which appears to be in thermal equilibrium. The phrase "eigenstate thermalization" was first coined by Mark Srednicki in 1994, after similar ideas had been introduced by Josh Deutsch in 1991. The principal philosophy underlying the eigenstate thermalization hypothesis is that instead of explaining the ergodicity of a thermodynamic system through the mechanism of dynamical chaos, as is done in classical mechanics, one should instead examine the properties of matrix elements of observable quantities in individual energy eigenstates of the system. == Statement of the ETH == Suppose that we are studying an isolated, quantum mechanical many-body system. In this context, "isolated" refers to the fact that the system has no (or at least negligible) interactions with the environment external to it. If the Hamiltonian of the system is denoted $\backslash hat$, then a complete set of basis states for the system is given in terms of the eigenstates of the Hamiltonian, : $$ \hat |E_ \rangle = E_|E_ \rangle , where $|E\_\; \backslash rangle$ is the eigenstate of the Hamiltonian with eigenvalue $E\_$. We will refer to these states simply as "energy eigenstates." For simplicity, we will assume that the system has no degeneracy in its energy eigenvalues, and that it is finite in extent, so that the energy eigenvalues form a discrete, non-degenerate spectrum (this is not an unreasonable assumption, since any "real" laboratory system will tend to have sufficient disorder and strong enough interactions as to eliminate almost all degeneracy from the system, and of course will be finite in size). This allows us to label the energy eigenstates in order of increasing energy eigenvalue. Additionally, consider some other quantum-mechanical observable $\backslash hat$, which we wish to make thermal predictions about. The matrix elements of this operator, as expressed in a basis of energy eigenstates, will be denoted by : $$ A_ \equiv \langle E_ | \hat | E_ \rangle . We now imagine that we prepare our system in an initial state for which the expectation value of $\backslash hat$ is far from its value predicted in a microcanonical ensemble appropriate to the energy scale in question (we assume that our initial state is some superposition of energy eigenstates which are all sufficiently "close" in energy). The Eigenstate Thermalization Hypothesis says that for an arbitrary initial state, the expectation value of $\backslash hat$ will ultimately evolve in time to its value predicted by a microcanonical ensemble, and thereafter will exhibit only small fluctuations around that value, provided that the following two conditions are met: # The diagonal matrix elements $A\_$ vary smoothly as a function of energy, with the difference between neighboring values, $A\_\; -\; A\_$, becoming exponentially small in the system size. # The off-diagonal matrix elements $A\_$, with $\backslash alpha\; \backslash neq\; \backslash beta$, are much smaller than the diagonal matrix elements, and in particular are themselves exponentially small in the system size. These conditions can be written as :$$ A_ \simeq \overline\delta_+\sqrt}}R_, where $\backslash overline$ and $\backslash overline$ are smooth functions of energy, $\backslash mathcal=e^$ is the many-body Hilbert space dimension, and $R\_$ is a random variable with zero mean and unit variance. Conversely if a quantum many-body system satisfies the ETH, the matrix representation of any local operator in the energy eigen basis is expected to follow the above ansatz. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Eigenstate thermalization hypothesis」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.