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.
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.
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
A system, or a set of systems, is said to be in thermal equilibrium when no net flow of heat energy occurs between its constituent parts when they are connected by a path permeable to heat (a diathermal wall).1 This condition is the endpoint of a spontaneous process: if two bodies at different temperatures are brought into thermal contact, energy will flow from the higher-temperature object to the lower-temperature object until this flow ceases and the equilibrium state is achieved.3 Within a single system, thermal equilibrium is characterized by a temperature that is both spatially uniform (the same at all points) and temporally constant (unchanging over time).4 This uniformity of macroscopic properties—such as temperature and pressure—is what defines the state of the system.1 It is important to distinguish this from the broader state of thermodynamic equilibrium. Thermal equilibrium (uniform temperature) is a necessary component, but full thermodynamic equilibrium also requires mechanical equilibrium (uniform pressure) and diffusive equilibrium (uniform chemical potential).6 Systems can achieve thermal equilibrium without achieving full thermodynamic equilibrium.6 The Zeroth Law of Thermodynamics: The "Discovery" of Temperature The concept of temperature, while intuitive, requires a formal physical principle to exist as a measurable, fundamental property. This principle is the Zeroth Law of Thermodynamics. Although it was formulated and named after the First and Second Laws, its foundational role was recognized as a logical prerequisite for the entire structure of thermodynamics.8 The law states: If system A is in thermal equilibrium with system C, and system B is in thermal equilibrium with system C, then systems A and B are in thermal equilibrium with each other.1 In this construction, system C plays the role of a thermometer.13 The profound implication of this law is mathematical: it establishes thermal equilibrium as a transitive relation 9, or more formally, an equivalence relation.14 In mathematics, any equivalence relation partitioning a set of objects implies the existence of a common, scalar property shared by all members of a given equivalence class. The Zeroth Law thus proves the existence of a scalar function of state—a property we call temperature—that is common to all systems in mutual thermal equilibrium.9 The law is, therefore, the fundamental basis for defining temperature and building consistent, reproducible temperature scales.1 This axiomatic foundation also provides the logical origin for one of temperature's most critical characteristics: it is an intensive property. An intensive property is one that does not depend on the size or amount of the system.15 If a large system 'A' is in internal thermal equilibrium 6, any two parts of it, subsystems 'A1' and 'A2', must also be in thermal equilibrium with each other. By the Zeroth Law, a thermometer 'C' that is in thermal equilibrium with the whole system 'A' must also be in thermal equilibrium with 'A1' and 'A2' individually.13 The thermometer, therefore, gives the same reading for all three. This demonstrates that the temperature of a system in thermal equilibrium is the same as the temperature of any part of it.16
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
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.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
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 and ∼N−1/2, with N≫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.