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Sharp energy determination of macroscopic isolated bodies in pure states

Vı́ctor Romero-Rochı́n

Published May 3, 2018• 0 citations
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Abstract

We show that a very large class of quantum pure states of isolated macroscopic bodies, which are superpositions of energy eigenstates within arbitrary finite energy intervals, have sharply peaked energy distributions, with their width relative to the average, scaling between $\sim N^{-1}$ and $\sim N^{-1/2}$, with $N \gg 1$, the number of atoms conforming the body. The implication of this statement is that closed systems in those states are {\it microcanonical} in the sense that, in their evolution and relaxation to equilibrium, visit only energy eigenstates very near to the mean energy. Since thermodynamics accurately describes processes of macroscopic bodies and requires that closed systems have constant energy, we argue that these pure states are typical of those systems. The main assumption beneath the energy sharpness is that the isolated body can reach thermal equilibrium if left unaltered.

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AIP conference proceedings•Jan 1, 2011•15 citations
We show that a very large class of quantum pure states of isolated macroscopic bodies, which are superpositions of energy eigenstates within arbitrary finite energy intervals, have sharply peaked energy distributions, with their width relative to the average, scaling between ∼N−1\sim N^{-1}∼N−1 and ∼N−1/2\sim N^{-1/2}∼N−1/2, with N≫1N \gg 1N≫1, the number of atoms conforming the body. The implication of this statement is that closed systems in those states are {\it microcanonical} in the sense that, in their evolution and relaxation to equilibrium, visit only energy eigenstates very near to the mean energy. Since thermodynamics accurately describes processes of macroscopic bodies and requires that closed systems have constant energy, we argue that these pure states are typical of those systems. The main assumption beneath the energy sharpness is that the isolated body can reach thermal equilibrium if left unaltered.
We argue that a very large class of quantum pure states of isolated macroscopic bodies have sharply peaked energy distributions, with their width relative to the average scaling between ∼N−1\sim N^{-1}∼N−1 and ∼N−1/2\sim N^{-1/2}∼N−1/2, with N≫1N \gg 1N≫1, the number of atoms conforming the body. Those states are dense superpositions of energy eigenstates within arbitrary finite or infinite energy intervals that decay sufficiently fast. The sharpness of the energy distribution implies that closed systems in those states are {\it microcanonical} in the sense that only energy eigenstates very near to the mean energy contribute to their thermodynamic evolution. Since thermodynamics accurately describes processes of macroscopic bodies and requires that closed systems have constant energy, our claim is that these pure states are typical of macroscopic systems. The main assumption beneath the energy sharpness is that the isolated body can reach thermal equilibrium if left unaltered. We argue that such a self-sharpness of the energy in macroscopic bodies indicates that the First Law of Thermodynamics is statistical in character.
We argue that a very large class of quantum pure states of isolated macroscopic bodies have sharply peaked energy distributions, with their width relative to the average scaling between ∼N−1\sim N^{-1}∼N−1 and ∼N−1/2\sim N^{-1/2}∼N−1/2, with N≫1N \gg 1N≫1, the number of atoms conforming the body. Those states are dense superpositions of energy eigenstates within arbitrary finite or infinite energy intervals that decay sufficiently fast. The sharpness of the energy distribution implies that closed systems in those states are {\it microcanonical} in the sense that only energy eigenstates very near to the mean energy contribute to their thermodynamic evolution. Since thermodynamics accurately describes processes of macroscopic bodies and requires that closed systems have constant energy, our claim is that these pure states are typical of macroscopic systems. The main assumption beneath the energy sharpness is that the isolated body can reach thermal equilibrium if left unaltered. We argue that such a self-sharpness of the energy in macroscopic bodies indicates that the First Law of Thermodynamics is statistical in character.
We study the nature of and approach to thermal equilibrium in isolated quantum systems. An individual isolated macroscopic quantum system in a pure or mixed state is regarded as being in thermal equilibrium if all macroscopic observables assume rather sharply the values obtained from thermodynamics. Of such a system (or state) we say that it is in macroscopic thermal equilibrium (MATE). A stronger requirement than MATE is that even microscopic observables (i.e., ones referring to a small subsystem) have a probability distribution in agreement with that obtained from the micro‐canonical, or equivalently the canonical, ensemble for the whole system. Of such a system we say that it is in microscopic thermal equilibrium (MITE). The distinction between MITE and MATE is particularly relevant for systems with many‐body localization (MBL) for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. However, if we consider superpositions of energy eigenfunctions (i.e., typical wave functions ψ) in an energy shell, then for generic macroscopic systems, including those with MBL, most ψ are in both MATE and MITE. We explore here the properties of MATE and MITE and compare the two notions, thereby elaborating on ideas introduced in . image
We consider the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system, determined by its wave function, must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may, however, not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevant for systems with many-body localization for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. We note, however, that for generic macroscopic systems, including those with MBL, most wave functions in an energy shell are in both MATE and MITE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point.
In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+δE. The thermal equilibrium macro‐state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to Heq. We argue that for “typical” Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.