arXiv:1805.01384

Sharp energy self-determination of macroscopic quantum bodies in pure states, as a validation of the First Law of Thermodynamics

Vı́ctor Romero-Rochı́n

Published May 3, 2018• 0 citations

Abstract

DOI: 10.48550/arxiv.1805.01384

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arXiv:1805.01384

Sharp energy self-determination of macroscopic quantum bodies in pure states, as a validation of the First Law of Thermodynamics

Vı́ctor Romero-Rochı́n

Published May 3, 2018• 0 citations

arXiv HTML PDF

Abstract

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

\sim N^{-1}

and

\sim N^{-1/2}

, with

N \gg 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.

DOI: 10.48550/arxiv.1805.01384

Related papers

Sharp Energy Self-Determination of Macroscopic Quantum Bodies in Pure States, as a Validation of the First Law of Thermodynamics

Vı́ctor Romero-Rochı́n

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 $\sim N^{-1}$ and $\sim N^{-1/2}$ , with $N \gg 1$ , 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 $\sim N^{-1}$ and $\sim N^{-1/2}$, with $N \gg 1$, 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.

Jul 14, 20210 citations

Sharp energy determination of macroscopic isolated bodies in pure states

Vı́ctor Romero-Rochı́n

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.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.

May 3, 20180 citations

Approach to thermal equilibrium of macroscopic quantum systems

Sheldon Goldstein, Joel L. Lebowitz, Christian Mastrodonato +2 more

We consider an isolated macroscopic quantum system. Let $\mathcal{H}$ be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between $E$ and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy $E$ corresponds to a subspace ${\mathcal{H}}_{eq}$ of $\mathcal{H}$ such that $\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}$ is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to ${\mathcal{H}}_{eq}$ . We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times $t$ . This result is closely related to von Neumann's quantum ergodic theorem of 1929.We consider an isolated macroscopic quantum system. Let $\mathcal{H}$ be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between $E$ and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy $E$ corresponds to a subspace ${\mathcal{H}}_{eq}$ of $\mathcal{H}$ such that $\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}$ is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to ${\mathcal{H}}_{eq}$. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times $t$. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

arXiv:0911.1724Physical Review EJan 7, 2010152 citations

The approach to thermal equilibrium and "thermodynamic normality" --- An observation based on the works by Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi in 2009, and by von Neumann in 1929

Hal Tasaki

We treat the problem of the approach to thermal equilibrium by only resorting to quantum dynamics of an isolated macroscopic system. Inspired by the two important works in 2009 and in 1929, we have noted that a condition we call "thermodynamic normality" for a macroscopic observable guarantees the approach to equilibrium (in the sense that a measurement of the observable at time $t$ almost certainly yields a result close to the corresponding microcanonical average for a sufficiently long and typical $t$ ). A crucial point is that we make no assumptions on the initial state of the system, except that its energy is distributed close to a certain macroscopic value. We also present three (rather artificial) models in which the thermodynamic normality can be established, thus providing concrete examples in which the approach to equilibrium is rigorously justified. Note that this kind of results which hold for ANY initial state are never possible in classical systems. We are thus dealing with a mechanism which is peculiar to quantum systems. The present note is written in a self-contained (and hopefully readable) manner. It only requires basic knowledge in quantum physics and equilibrium statistical mechanics.We treat the problem of the approach to thermal equilibrium by only resorting to quantum dynamics of an isolated macroscopic system. Inspired by the two important works in 2009 and in 1929, we have noted that a condition we call "thermodynamic normality" for a macroscopic observable guarantees the approach to equilibrium (in the sense that a measurement of the observable at time $t$ almost certainly yields a result close to the corresponding microcanonical average for a sufficiently long and typical $t$). A crucial point is that we make no assumptions on the initial state of the system, except that its energy is distributed close to a certain macroscopic value. We also present three (rather artificial) models in which the thermodynamic normality can be established, thus providing concrete examples in which the approach to equilibrium is rigorously justified. Note that this kind of results which hold for ANY initial state are never possible in classical systems. We are thus dealing with a mechanism which is peculiar to quantum systems. The present note is written in a self-contained (and hopefully readable) manner. It only requires basic knowledge in quantum physics and equilibrium statistical mechanics.

arXiv:1003.5424Mar 29, 20108 citations

Typical pure states and the analysis of nonequilibrium processes of mesoscopic systems

Takaaki Monnai, Ayumu Sugita

Journal of Statistical Mechanics Theory and ExperimentMay 1, 20162 citations

Macroscopic and microscopic thermal equilibrium

Sheldon Goldstein, David A. Huse, Joel L. Lebowitz +1 more

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 . imageWe 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

arXiv:1610.02312Annalen der PhysikFeb 9, 201745 citations

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

\sim N^{-1}

and

\sim N^{-1/2}

, with

N \gg 1

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

\sim N^{-1}

and

\sim N^{-1/2}

, with

N \gg 1

\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.

We consider an isolated macroscopic quantum system. Let

\mathcal{H}

be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between

E

and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy

E

corresponds to a subspace

{\mathcal{H}}_{eq}

\mathcal{H}

such that

\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}

is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to

{\mathcal{H}}_{eq}

. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times

t

. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

t

almost certainly yields a result close to the corresponding microcanonical average for a sufficiently long and typical

t

). A crucial point is that we make no assumptions on the initial state of the system, except that its energy is distributed close to a certain macroscopic value. We also present three (rather artificial) models in which the thermodynamic normality can be established, thus providing concrete examples in which the approach to equilibrium is rigorously justified. Note that this kind of results which hold for ANY initial state are never possible in classical systems. We are thus dealing with a mechanism which is peculiar to quantum systems. The present note is written in a self-contained (and hopefully readable) manner. It only requires basic knowledge in quantum physics and equilibrium statistical mechanics.