Abstract The quantum equilibrium distribution, ϱQm, of an arbitrary number, m, of momentum or displacement components is determined for atoms that are part of a polyatomic molecule or a solid. This is shown to be a multidimensional Gaussian distribution. Two cases are considered: (1) the motion of the system as a whole is given, (2) it is in itself determined by the statistical equilibrium conditions. In the first case we obtain distributions for the vibrational momenta and displacements and in the second for the total momenta, including the momenta of vibrational, translational and rotational motions. The distributions for momenta and displacements of one particle and for the maximum number of linearly independent components of momenta and displacements of all particles of the system are considered as particular cases. It is shown that the averaging of any function, Fm, depending on an arbitrary number, m, of components of displacements or momenta of particles, over the canonical ensemble is reduced to the integration of this function weighted by ϱQm over all its arguments between infinite limits.
Abstract The quantum equilibrium distribution, ϱQm, of an arbitrary number, m, of momentum or displacement components is determined for atoms that are part of a polyatomic molecule or a solid. This is shown to be a multidimensional Gaussian distribution. Two cases are considered: (1) the motion of the system as a whole is given, (2) it is in itself determined by the statistical equilibrium conditions. In the first case we obtain distributions for the vibrational momenta and displacements and in the second for the total momenta, including the momenta of vibrational, translational and rotational motions. The distributions for momenta and displacements of one particle and for the maximum number of linearly independent components of momenta and displacements of all particles of the system are considered as particular cases. It is shown that the averaging of any function, Fm, depending on an arbitrary number, m, of components of displacements or momenta of particles, over the canonical ensemble is reduced to the integration of this function weighted by ϱQm over all its arguments between infinite limits.
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.
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)].
We consider non-equilibrium open statistical systems, subject to potentials and to external “heat baths” (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann’s classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials in momenta (the position-independent Hermite polynomialsHn’s). The moments of non-equilibrium classical distributions, implied by the Hn’s, fulfill a hierarchy: for long times, the lowest moment dominates the evolution towards thermal equilibrium, either with dissipation or without it (but under certain approximation). We revisit that hierarchy, whose solution depends on operator continued fractions. We review our generalization of that moment method to classical closed many-particle interacting systems with neither a hb nor ab initio dissipation: with initial states describing thermal equilibrium at T at large distances but non-equilibrium at finite distances, the moment method yields, approximately, irreversible thermalization of the whole system at T, for long times. Generalizations to non-equilibrium quantum interacting systems meet additional difficulties. Three of them are: (i) equilibrium distributions (represented through Wigner functions) are neither Gaussian in momenta nor known in closed form; (ii) they may depend on dissipation; and (iii) the orthogonal polynomials in momenta generated by them depend also on positions. We generalize the moment method, dealing with (i), (ii) and (iii), to some non-equilibrium one-particle quantum interacting systems. Open problems are discussed briefly.
A general method which permits the derivation of the equations which describe the approach to equilibrium correct to an arbitrary finite order in the coupling constant is presented. This method is applied in the present paper to normal modes interacting through three-phonon processes. In a subsequent paper the method will be applied to interacting particles. The distribution function is first Fourier-analyzed with respect to the angle variables. All Fourier components, except the distribution function of action variables, describe correlations among the normal modes. The formal solution for the Fourier components is studied in the limiting case of a very large number of degrees of freedom N → ∞, and for large times by means of a diagram technique. Each component ρ3n can be split into 2 parts: ρ3n′ and ρ3n″; one (ρ′) due to ``scattering'' of the normal modes satisfies diagonal differential equations. The other (ρ3n″) contains the direct interaction between the normal modes involved in the corresponding correlation. It is completely determined by the functions ρ3n′. The study of this set of equations enables us to study the approach to equilibrium.
The probability distribution of the total momentum P is studied in N-particle interacting homogeneous quantum systems at positive temperatures. Using Galilean invariance we prove that in one dimension, the asymptotic distribution of P/N is normal at all temperatures and densities, and in two dimensions, the tail distribution of P/N is normal. We introduce the notion of the density matrix reduced to the center of mass and show that its eigenvalues are N times the probabilities of the different eigenvalues of P. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the single-particle and the total momentum. The P = 0 ensemble is shown to be equivalent to the canonical ensemble. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that in interacting quantum systems above one dimension, in infinite space, the total momentum is finite with a nonzero probability at all temperatures and densities. In solids, this probability is 1, and in a crystal, it is distributed on a lattice. Since it is less than 1 in two dimensions, we conclude that a 2D system is always in a fluid phase, so if the hexatic phase existed classically, it would be destroyed by quantum fluctuations. For a superfluid, we conjecture that the total momentum is zero with a nonzero probability and otherwise its distribution is continuous. We define a macroscopic wave function based on the density matrix reduced to the center of mass. We discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter, and argue that Landau’s criterion cannot explain superfluidity and its breakdown at positive temperatures. We also comment on the relation between superfluidity and Bose-Einstein condensation.
Abstract A system of 4 point masses (without spin) is considered. The potential energy is assumed to be a function of the distances between the particles only. It is possible to express the kinetic energy relative to the center of mass as the kinetic energy of 3 "reduced" masses. The different possibilities for that procedure are discussed for systems with an arbitrary number of particles. The positions of the 3 reduced masses are described by 9 coordinates. 6 of them, the "internal" coordinates, are needed to determine the internal structure of the system, and 3, the "external" coordinates, for its orientation in space. The classical Hamiltonian in such a set of coordinates is derived. In a first step the Hamiltonian is expressed in terms of linear and angular momenta. Then these quantities are determined as functions of the coordinates and the conjugate momenta by infinitesimal trans-formations. From the classical Hamiltonian we get the quantum mechanical one by the well-known translation rule. It can be expressed in terms of the linear and angular momentum operators quite analogous to the classical case. For the solution of the time-independent Schrödinger equation the rotational symmetry of the system is used. The wave function is expanded in terms of the eigenfunctions of the total angular momentum. This leads to a system of coupled differential equations for the expansion-coefficients which depend on the 6 internal coordinates only. The further symmetries of the system (reflection and particle exchange) appear in symmetry properties of these functions. Differences and similarities in the solution theory for the 4-body-problem with respect to the 3-body-problem are discussed.