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.
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.
Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e., the wavefunction, of an isolated system is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger equation. On the other hand a mixed state, i.e., a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.
Despite early developments on the foundations of quantum mechanics concern the wave function, quantum statistics has been developed with the density matrix formalism, leading to very important results in explaining molecular observations. Only recently, several authors argued a new interpretation by focusing on the wave function representing the quantum state of an isolated system, showing how a single wave function can exhibit statistical properties and generate the same results expected in the standard quantum statistical framework. Starting from these results, investigation on the foundations of quantum statistical mechanics has gained recently a renewed interest. As a matter of fact, the possibility of studying single molecule properties as well as the need of a better understanding of the effect of quantum dynamics, in order to develop new nanoscaled materials suitable to quantum computing tasks, have opened new intriguing questions leading to quantum statistical approaches far from being well understood and accepted. In this framework the behaviour of a single realization of quantum systems has gained a central role in the description of molecular systems. Furthermore, in recent years, an increasing number of studies has been presented on quantum dynamics through the numerical solution of the Schrödinger equation for systems of interacting components. These studies demonstrates that Quantum Dynamics Simulations could be a practicable route. In order to study phenomena such as dissipation, relaxation and thermalization, the focus has to be moved from isolated molecules to modular systems made of mutual interacting components, with model Hamiltonians possessing a sufficiently low dimensional representation. An important issue concerns the rules to be employed for the choice of the initial quantum state for the simulation of isolated systems. As long as one considers molecular degrees of freedom interacting with a (model) environment, there are no reasons to select a particular quantum state for the overall system and, therefore, a random choice has to be performed amongst a well defined statistical ensemble of pure states. Furthermore, one would like to operate a choice assuring that the simulation of the system is in a well defined thermal state with given temperature. This necessarily calls for a statistical description like for classical systems. A useful parametrization of the wave function will be presented in order to highlight the most appropriate variable for the statistical analysis. In particular some of them, called phases, retain all the dynamical information whereas the others, called populations, are the constants of motion. The latter, in particular are very important because they describe the equilibrium properties strictly related to the thermodynamic description. Since the dynamics of wave function does not supply any information about populations, the definition of a probability distribution on these variables is required. Different probability distributions on populations have been proposed, only on the bases of reasonable assumptions and they validation has been performed only with a posteriori considerations. In particular Fresch and Moro have demonstrated how the agreement with thermodynamics can be employed to discriminate different probability distributions on pure states. This had led the uniform ensembles to be, up to now, the most self-consistent models for quantum pure states. However, in this thesis I will highlight a drawback of the uniform distribution ensemble that can be described as follow: if we bring into contact two systems, even through a perturbative interaction, we are not able to describe the equilibrium properties after the interaction within the uniform distribution statistics, since the uniform character is lost. It represents a severe shortcoming of the statistical ensemble from a methodological point of view, since closed systems can be always considered as the result of interaction among previously isolated systems. On the other hand this drawback introduces a further requirement of a different nature that can be used for the definition of a new statistical ensemble. In this work I intend to find and characterize a statistical ensemble for populations that overcomes the drawbacks of the uniform distribution of pure states. The invariance of the thermal state in the coupling of identical systems will be used as a guideline in the definition af a new probability distribution on populations. Such an ensemble for pure states, called Thermalization Resilient Ensemble, provides a convenient framework for treating the interactions between quantum systems, as long as the structure of the statistical distribution is preserved and the identification of thermodynamic properties is assured. In perspective it should be the privileged statistical ensemble to implement Quantum Dynamics Simulations. Once the average properties of the Thermalization Resilient Ensemble have been introduced, I will obtain a probability distribution on pure states with the use of a geometrical analysis on the Hilbert space. The surface elements of an ellipsoidal manifold will be related to the probability density on populations. As a matter of fact the explicit form of the probability distribution is a prerequisite in order to perform Quantum Dynamical Simulations. However the results obtained through the geometrical analysis cannot be easily extended to systems with unbounded energy spectrum and an alternative strategy has been developed. A scaling algorithm on the basis of the uniform statistical ensemble will be described and this allows a well defined sampling of a probability distribution with desired averages. In this framework I demonstrate the emergence of thermodynamic behavior in the limit of macroscopic systems. In the last part of the thesis I consider the dynamical features of the thermalization experiment. Two identical systems, initially at different temperature, will be brought in interaction and the analysis of the final equilibrium state will be performed for two different generic forms of the interaction Hamiltonian, highlighting how the statistical approach can be very useful in the definition of the equilibrium in complex quantum systems.
Statistical mechanics reveals that the properties of a macroscopic physical system emerge as an average over an ensemble of statistically independent microscopic subsystems, each occupying a specific microstate. In the study of quantum systems, these microstates can be chosen to correspond to the pure state wave functions of individual quantum systems. However, the physical principles that govern the distribution of a pure state wave function ensemble, even under conditions of thermal equilibrium, are not well established. For instance, the canonical Boltzmann distribution cannot be applied to wave functions because they lack a definite energy. In this paper we present a maximum entropy principle for the quantum wave function ensemble at thermal equilibrium, the so-called Scrooge ensemble. We highlight that a constraint on the energy expectation value, or even the shape of the associated eigenstate distribution, fails to yield a valid equilibrium state. We find that in addition to these constraints, one must also constrain the measurement entropy to be equal to the Rényi divergence of the ensemble with respect to the Gibbs state, indicating that the Rényi divergence may have uninvestigated physical importance to thermal equilibrium in quantum systems.
Statistical mechanics reveals that the properties of a macroscopic physical system emerge as an average over an ensemble of statistically independent microscopic subsystems, each occupying a specific microstate. In the study of quantum systems, these microstates can be chosen to correspond to the pure state wavefunctions of individual quantum systems. However, the physical principles that govern the distribution of a pure state wavefunction ensemble, even under conditions of thermal equilibrium, are not well established. For instance, the canonical Boltzmann distribution cannot be applied to wavefunctions because they lack a definite energy. In this manuscript, we present a maximum entropy principle for the quantum wavefunction ensemble at thermal equilibrium, the so-called Scrooge ensemble. We highlight that a constraint on the energy expectation value, or even the shape of the associated eigenstate distribution, fails to yield a valid equilibrium state. We find that in addition to these constraints, one must also constrain the measurement entropy to be equal to the Rényi divergence of the ensemble with respect to the Gibbs state, indicating that the Rényi divergence may have uninvestigated physical importance to thermal equilibrium in quantum systems.
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.
Equilibrium thermodynamics emerges because a vast number of microscopic states become indistinguishable at the level of local observables. We start from the observation that a nonequilibrium quantum state can also be represented by an ensemble of pure states that minimizes the average energy variance. We show that varying the statistical weights within this optimal family organizes state space into ``minimum-variance leaves'', which generalize the space of stationary states underlying equilibrium. We observe thermodynamic compression on the leaves, which in the simplest scenario manifests itself in the indistinguishability of states with the same energy density. We call this ``leaf typicality''; at equilibrium it reduces to eigenstate thermalization or its generalizations. Numerical tests support leaf typicality on some leaves and indicate a breakdown of its stronger forms on others, suggesting a rich thermodynamic organization beyond equilibrium. These results open a route to a unified statistical mechanics of equilibrium and nonequilibrium quantum states.