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.
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 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.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
According to the first and second laws of thermodynamics and the definitions of work and heat, microscopic expressions for the non-equilibrium entropy production have been achieved. Recently, a redefinition of heat has been presented in [Nature Communications volume 8, Article number: 2180 (2017)]. Since thermal operations play an important role in the resource theory of thermodynamics, it would be very interesting to find out the effect of the above-mentioned definition on the expression of the entropy production for these kind of operations. This is one of the aims of the present paper. Using the new definition of heat, it is shown that the entropy production is the same as the mutual information between a system and a bath both for thermal operations and, if the system-bath initial state is factorized, for entropy-preserving operations. It is also discussed that how one can recognize the type of the correlation between a system and a bath through knowledge of the initial state of the system only. It is shown that if the initial state of a system is diagonal in the energy basis, the thermal operations cannot create a quantum correlation between the system and the bath, however, if the system initial state is coherent Gibbs state, there cannot be classical correlation due to the thermal operations.
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.
The quantum entropy is usually defined using von Neumann's formula, which measures lack of information and vanishes for pure states. In contrast, we obtain a formula for the entropy of a pure state as it is measured from thermodynamic experiments, solely from the self-entanglement of the wave function, and find strong numerical evidence that the two are in agreement for nonintegrable systems, both for energy eigenstates and for states that are obtained at long times under the evolution of more general initial conditions. This is an extension of Boltzmann's hypothesis for classical systems, relating microscopic motion to thermodynamics.
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.