Two approaches to describing the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy E_{S} is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function Z_{S} is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation E_{S}=-d(ln Z_{S})/dbeta is used to obtain the mean system energy. Employing both classical and quantum-mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.
Two approaches to describing the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy E_{S} is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function Z_{S} is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation E_{S}=-d(ln Z_{S})/dbeta is used to obtain the mean system energy. Employing both classical and quantum-mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.
For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble.The second ensemble is then used.The results are supposed to be the same, at least in the thermodynamic limit.Is this actually the case?In this work, we consider a system of N noninteracting bosons distributed among few energy levels.We can calculate the canonical partition function in this case and deduce the canonical mean energy.We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles.We consider the case of a large number of particles.
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
For a small system at a low temperature, thermal fluctuation and quantum effect play important roles in quantum thermodynamics. Starting from micro-canonical ensemble, we generalize the Boltzmann–Gibbs statistical factor from infinite to finite systems, no matter the interactions between particles are considered or not. This generalized factor, similar to Tsallis’s q-form as a power-law distribution, has the restriction of finite energy spectrum and includes the nonextensivities of the small systems. We derive the exact expression for distribution of average particle numbers in the interacting classical and quantum nonextensive systems within a generalized canonical ensemble. This expression in the almost independent or elementary excitation quantum finite systems is similar to the corresponding ones obtained from the conventional grand-canonical ensemble. In the reconstruction for the statistical theory of the small systems, we present the entropy of the equilibrium systems and equation of total thermal energy. When we investigate the thermodynamics for the interacting nonextensive systems, we obtain the system–bath heat exchange and “uncompensated heat” which are in the thermodynamical level and independent on the detail of the system–bath coupling. For ideal finite systems, with different traps and boundary conditions, we calculate some thermodynamic quantities, such as the specific heat, entropy, and equation of state, etc. Particularly at low temperatures for the small systems, we predict some novel behaviors in the quantum thermodynamics, including internal entropy production, heat exchanges between the system and its surroundings and finite-size effects on the free energy.
Schrödinger suggested that thermodynamical functions cannot be based on the gratuitous allegation that quantum-mechanical levels (typically the orthogonal eigenstates of the Hamiltonian operator) are the only allowed states for a quantum system [E. Schrödinger, Statistical Thermodynamics (Courier Dover, Mineola, 1967)]. Different authors have interpreted this statement by introducing density distributions on the space of quantum pure states with weights obtained as functions of the expectation value of the Hamiltonian of the system. In this work we focus on one of the best known of these distributions and prove that, when considered in composite quantum systems, it defines partition functions that do not factorize as products of partition functions of the noninteracting subsystems, even in the thermodynamical regime. This implies that it is not possible to define extensive thermodynamical magnitudes such as the free energy, the internal energy, or the thermodynamic entropy by using these models. Therefore, we conclude that this distribution inspired by Schrödinger's idea cannot be used to construct an appropriate quantum equilibrium thermodynamics.
Abstract Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.
Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of a mutually interacting subsystem and a heat bath, and assume that the whole system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the ``hypothesis of equal weights for eigenstates,'' we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (instantaneous) quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value. We present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses.