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.
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.
Abstract We say of an isolated macroscopic quantum system in a pure state
ψ ψ that it is in macroscopic thermal equilibrium (MATE) if
ψ ψ lies in or close to a suitable subspace
Heq H eq of Hilbert space. It is known that every initial state
ψ0 ψ 0 will eventually reach and stay there most of the time (“thermalize”) if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation
HθfF H θ fF of the Hamiltonian
H0fF H 0 fF of
N≫1 N ≫ 1 free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of
H0fF H 0 fF . Here, we first point out that also for degenerate Hamiltonians all
ψ0 ψ 0 thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for
H0fF H 0 fF . Inspired by the fact that there is one eigenbasis of
H0fF H 0 fF for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given
H0 H 0 that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of
H0 H 0 lie in MATE. We also show that, as a consequence, after adding a small generic perturbation,
H=H0+λV H = H 0 + λ V with
λ≪1 λ ≪ 1 , for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.
Thermal equilibrium: In a system which is in thermal equilibrium, the long-time averages of the macroscopic quantities are time-independent (e.g. temperature, pressure, chemical potential).The probability distribution from which the average is taken is also timeindependent. Micro-states: If the generalized coordinate and momenta of each particle maybe determined, then that would be called a micro-state.On the other hand total momenta would be a macro-state of the system. Accessible Micro-states: If the total energy of the system is E, no individual particle can have a kinetic energy greater than that.Therefore some micro-states are not accessible to the system. If we increase E, then more micro-states will be accessible to the particle. Now, how all microscopic states become equally likely is an area of active current research. There is no general proof, because although this assumption is an extremely successful one in statistical mechanics, it doesn’t always happen. A lot of modern research tests its limits: like, if you begin with an isolated system in a highly non-equilibrium state, will it actually evolve such that a description in terms of something remarkable like a micro-canonical ensemble eventually becomes appropriate ?
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 and
∼N−1/2, with
N≫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.