The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focusing specifically on whether they satisfy or violate the zeroth, first, and second laws of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples.
The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focusing specifically on whether they satisfy or violate the zeroth, first, and second laws of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples.
Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics.This tutorial focuses on the derivation of the first and second law for isolated and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies.The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat, and temperature.These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of states and processes.The consistency with previous results is demonstrated.The framework applies to multiple baths including particle transport and accounts for processes with, e.g., a changing temperature of the bath, which is determined microscopically.An integral fluctuation theorem for entropy production is satisfied.In summary, this tutorial introduces a consistent and versatile framework to understand and apply the laws of thermodynamics in the quantum regime and beyond.
To eliminate exceptions to the second law due to fluctuation phenomena, etc., the following changes are necessary. First we must state the laws of thermodynamics in terms of Gibbsian ensembles rather than of individual systems. Second we must sharpen our conception of the experimental procedure required for producing states of thermodynamic equilibrium. It will not suffice to isolate the system and wait. We must have the positive stirring action of thermal contact with a thermostat. Using the new definition of thermal equilibrium we can place statistical mechanics on a more satisfactory footing by showing that the thermostat experimental procedure leads to the canonical ensemble and maximizes v. Neumann's microscopic entropy. (Details of this proof appear in succeeding paper). Difficulties with his macroscopic definition of entropy are avoided by showing that his original microscopic definition is satisfactory when used in conjunction with a proper operational definition of thermodynamic equilibrium. In conclusion the second law of thermodynamics is derived from quantum statistics on the basis of the microscopic definition of entropy.
We study the time evolution of eleven microscopic entropy definitions (of Boltzmann-surface, Gibbs-volume, canonical, coarse-grained-observational, entanglement and diagonal type) and three microscopic temperature definitions (based on Boltzmann, Gibbs or canonical entropy). This is done for the archetypal nonequilibrium setup of two systems exchanging energy, modeled here with random matrix theory, based on numerical integration of the Schrödinger equation. We consider three types of pure initial states (local energy eigenstates, decorrelated and entangled microcanonical states) and three classes of systems: (A) two normal systems, (B) a normal and a negative temperature system and (C) a normal and a negative heat capacity system. We find: (1) All types of initial states give rise to the same macroscopic dynamics. (2) Entanglement and diagonal entropy sensitively depend on the microstate, in contrast to all other entropies. (3) For class B and C, Gibbs-volume entropies can violate the second law and the associated temperature becomes meaningless. (4) For class C, Boltzmann-surface entropies can violate the second law and the associated temperature becomes meaningless. (5) Canonical entropy has a tendency to remain almost constant. (6) For a Haar random initial state, entanglement or diagonal entropy behave similar or identical to coarse-grained-observational entropy.
The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value—their equilibrium value. This ultimately raises the question: ? For classical systems, one introduces the assumption of a low-entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the of observables. While the statistical behavior of observable expectation values is well established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends toward its equilibrium value in the course of the system’s unitary evolution. These results also support recent findings that question the necessity of nonintegrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations, which expose the underlying reversible evolution in accordance with the derived bounds.
Deriving the laws of thermodynamics from an underlying microscopic picture is a central quest of statistical mechanics. The present article focuses on the derivation of the first and second law of thermodynamics for closed and open quantum systems, where such foundational questions in addition become practically relevant for emergent nanotechnologies. Recent progress is presented in a self-contained way, with an emphasis on general ideas and useful tools instead of particular applications. We identify limits and conceptual shortcomings of previous approaches. Then, we propose a novel, unifying perspective, which starts from a microscopic definition of nonequilibrium thermodynamic entropy. The change of this entropy is identified with the entropy production, which satisfies a fluctuation theorem for a large class of initial states. Moreover, this entropy production can be naturally separated into a quantum and a classical component. Within our framework, we introduce the notions of recoverable work and remaining heat. In the case of a weakly perturbed ideal thermal bath this approach is in quantitative agreement with previous ones, thus ensuring their thermodynamic consistency.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.