Statistical mechanics is often taken to be the paradigm of a successful inter-theoretic reduction, which explains the high-level phenomena by using the fundamental theories of physics together with some auxiliary hypotheses. The central idea of statistical mechanics is that in order to account for the thermodynamic regularities only partial information about the microstates is needed, concerning some macrovariables of them, that is, concerning the fact that they belong to certain macrostates in which all the microstates share the same macrovariables. An important example of a thermodynamic property, which needs to be explained in quantum statistical mechanics, is that of being in thermal equilibrium. The thermodynamic law of approach to equilibrium says that systems, prepared in non-equilibrium states, will invariably evolve to equilibrium states fixed by the constraints on these systems, and remain in them indefinitely.
Statistical mechanics is often taken to be the paradigm of a successful inter-theoretic reduction, which explains the high-level phenomena by using the fundamental theories of physics together with some auxiliary hypotheses. The central idea of statistical mechanics is that in order to account for the thermodynamic regularities only partial information about the microstates is needed, concerning some macrovariables of them, that is, concerning the fact that they belong to certain macrostates in which all the microstates share the same macrovariables. An important example of a thermodynamic property, which needs to be explained in quantum statistical mechanics, is that of being in thermal equilibrium. The thermodynamic law of approach to equilibrium says that systems, prepared in non-equilibrium states, will invariably evolve to equilibrium states fixed by the constraints on these systems, and remain in them indefinitely.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
The author presents his experience in teaching at a graduate level the quantal exposition of a new non-statistically based paradigm of physics and thermodynamics. This paradigm, called the Unified Quantum Theory of Mechanics and Thermodynamics, applies to all systems large or small (including one particle systems) either in a state of thermodynamic (i.e. stable) equilibrium or not in a state of thermodynamic equilibrium. It uses as its primitives inertial mass, force, and time and introduces the laws of thermodynamics in the most unambiguous and general formulations found in the literature. Starting with a precise definition of system and of state followed by statements and corollaries of the laws of thermodynamics, the thermodynamic formalism is developed without circularity and ambiguity. In this quantal exposition of the new paradigm, a brief review of the formalism of thermodynamics as a general science not limited to stable equilibrium and large (macroscopic) systems as well as a very brief summary of the three prevalent formalisms in classical physics are presented followed by a presentation and development of solutions for a number of elementary problems in quantum physics (e.g., a particle in a box, a harmonic oscillator, a rigid rotor, etc.). These solutions and the maximum entropy principle are then used in a constrained optimization to develop the canonical and grand canonical distributions for Fermi-Dirac and Bose-Einstein types of particles, i.e. for fermions and bosons. This is done without the use of analogies between statistical and thermodynamic results and without additional hypotheses such as the ergodic hypothesis of statistical mechanics. These distributions are then employed under various assumptions (i.e. the Boltzmann, constant-potential, point-particle, and continuous eigenvalue-spectrum approximations) to derive the corresponding thermodynamic property expressions for perfect, semi-perfect (ideal), and Sommerfeld gases as well as for mixtures of ionized and dissociated gases. In a similar fashion but with a change from a single- to a multi-particle partition function and with the addition of various inter-particle potentials for two-particle interactions (e.g., the Lennard-Jones potential, the square-well potential, etc.), expressions for the thermodynamic properties of dense gases are developed and presented.An initial version of this paper was published inJuly of 2006 in the proceedings of ECOS’06, AghiaPelagia, Crete, Greece.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
The circumstances under which a system reaches thermal equilibrium, and how to derive this from basic dynamical laws, has been a major question from the very beginning of thermodynamics and statistical mechanics. Despite considerable progress, it remains an open problem. Motivated by this issue, we address the more general question of equilibration. We prove, with virtually full generality, that reaching equilibrium is a universal property of quantum systems: almost any subsystem in interaction with a large enough bath will reach an equilibrium state and remain close to it for almost all times. We also prove several general results about other aspects of thermalization besides equilibration, for example, that the equilibrium state does not depend on the detailed microstate of the bath.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].