ABS>It is shown that methods applied earlier to classical systems may be extended to quantum systems of particles interacting by a two-body potential. In particular, using the canonical ensemble, for a large class of potentials, the existence of a limit for the free energy per particle when the system becomes infinite was proven. (auth)
ABS>It is shown that methods applied earlier to classical systems may be extended to quantum systems of particles interacting by a two-body potential. In particular, using the canonical ensemble, for a large class of potentials, the existence of a limit for the free energy per particle when the system becomes infinite was proven. (auth)
The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, ...) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra, produces precisely those interacting particle systems studied in classical statistical mechanics. Moreover, in many interesting cases, the underlying classical process "drives" the quantum one, at least as far as ergodicity or convergence to equilibrium are concerned. Thus many deep results concerning classical systems can be directly applied to carry information on the corresponding quantum system. The thermodynamic limit itself is obtained thanks to a technique (the four-semigroup method, new even in the classical case) which reduces the infinitesimal structure of a stochastic flow to that of four semigroups canonically associated to it (Chap. 1). Simple and effective methods to analyze qualitatively the ergodic behavior of quantum Markov semigroups are discussed in Chap. 2. Powerful estimates used to control the infinite volume limit, ergodic behavior and the spectral gap (Gaussian, exponential and hypercontractive bounds, classical and quantum logarithmic Sobolev inequalities, ...) are discussed in Chap. 3
A class O of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class O. A "limit L" is defined as the limiting case which arises of the number of particles N and the volume V of the system are both allowed to tend to infinity so as to keep the ratio VN a finite and nonzero constant. It is shown that if one wishes to calculate the mean value of an extensive variable, as averaged over a canonical or over a grand canonical ensemble, for a system of class O in the limit L, one can apply the continuous spectrum approximation directly to the particle quantum states (excepting, possibly, the states which belong to the lowest energy level) without using a limiting process.
The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, …) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra
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.
In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit L," i.e., in the limit in which the number of particles N and the volume V of the system are allowed to tend to infinity so as to keep VN a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit L the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.
The original formulation of the Bohr–van Leeuwen (BvL) theorem states that, in a uniform magnetic field and in thermal equilibrium, the magnetization of an electron gas in the classical Drude–Lorentz model vanishes identically. This stems from classical statistics which assign the canonical momenta all values ranging from -∞ to ∞ that makes the free energy density magnetic-field-independent. When considering a classical (Maxwell–Boltzmann) interacting electron gas, it is usually admitted that the BvL theorem holds upon condition that the potentials modeling the interactions are particle-velocities-independent and do not cause the system to rotate after turning on the magnetic field. From a rigorous viewpoint, when treating large macroscopic systems, one expects the BvL theorem to hold provided the thermodynamic limit of the free energy density exists (and the equivalence of ensemble holds). This requires suitable assumptions on the many-body interactions potential and on the possible external potentials to prevent the system from collapsing or flying apart. Starting from quantum statistical mechanics, the purpose of this paper is to give, within the linear-response theory, a proof of the BvL theorem in the semiclassical limit when considering a dilute electron gas in the canonical conditions subjected to a class of translational invariant external potentials.