We analyze the so-called classical limit of the quantum-mechanical canonical partition function. In order to do that, we define accurately the density matrix for symmetrized and antisymmetrized wave functions only (Bose-Einstein and Fermi-Dirac), and find an exact relation between them and the density matrix for non symmetrized functions (Maxwell-Boltzmann). Our results differ from the generally assumed in a numerical factor N!, for which we suggest a physical interpretation. We derive as well the reverse (and also exact) relation, to find the canonical partition function for non-symmetrized wave functions in terms of the corresponding function for fermions and bosons.
We analyze the so-called classical limit of the quantum-mechanical canonical partition function. In order to do that, we define accurately the density matrix for symmetrized and antisymmetrized wave functions only (Bose-Einstein and Fermi-Dirac), and find an exact relation between them and the density matrix for non symmetrized functions (Maxwell-Boltzmann). Our results differ from the generally assumed in a numerical factor N!, for which we suggest a physical interpretation. We derive as well the reverse (and also exact) relation, to find the canonical partition function for non-symmetrized wave functions in terms of the corresponding function for fermions and bosons.
In the classical mechanical limit in which quantum cells of volume (2πħ)2 are reduced to points in phase space, the Bose and Fermi distributions are shown to approach the Boltzmann distribution. This limit is a continuous limit and therefore does not alter the indistinguishability of bosons (fermions) which is defined in terms of a discontinuous permutation symmetry of the density operator and Hamiltonian for the system. This means that the classical particles which correspond to the identical boson (fermions) remain indistinguishable. In the same limit, the partition function for N identical bosons approaches (2πħ)−3N (N!)−1 f. f d3r1 d3p1…d3rN d3pN exp (–βH) Zc. The two factors, (2πħ)−3N and (N!)−1 which are often added in an ad hoc manner in many books on statistical mechanics, are thus derived from first principles. In the classical statistical limit in which the number density n and the reciprocal temperature β become small, the quantum statistical distributions also approach the Boltzmann distribution. This is shown by examining the chemical potential μ as a function of n and β. The classical approximation is valid if the thermal de Broglie wavelength is much shorter than the interparticle distance irrespective of any interparticle interaction.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.
In a previous work [Phys. Rev. E 48, 4263 (1993)] we have derived a nonlinear one-dimensional kinetic equation for the distribution function of particles obeying an exclusion principle. In the present work, on the same grounds, we extend this kinetics to D-dimensional continuous or discrete space, in order to study the distribution function of particles obeying a generalized exclusion-inclusion Pauli principle (EIP). This exclusion or inclusion principle is introduced into the classical transition rates by means of an inhibition or an enhancement factor, which contains a parameter \ensuremath{\kappa}, whose values range between -1 and +1 and can balance the effect of the full or partial validity of EIP. After deriving the kinetic equation we obtain a general expression of the stationary distribution function, depending on the value we give to the parameter \ensuremath{\kappa}. When we limit ourselves to Brownian particles, we derive exactly for \ensuremath{\kappa}=-1 the Fermi-Dirac (FD) distribution, for \ensuremath{\kappa}=0 the Maxwell-Boltzmann distribution, and for \ensuremath{\kappa}=1 the Bose-Einstein (BE) distribution. When \ensuremath{\kappa} assumes an intermediate value, except zero, between the extreme values -1 and +1, we obtain statistical distributions different from the FD and BE ones. We attribute to the parameter \ensuremath{\kappa} the meaning of the degree of indistinguishability of identical particles, the degree of antisymmetrization, or the symmetrization of the wave function of the particle system.
Based on the density operator's coherent state representation [Phys. Lett.A252, 281 (1999)], we extend the well-known Bose distribution's partition function Ξ [Formula: see text] for ideal photon gas [Formula: see text] to that of an assembly of photon gas interacting with nonlinear medium described by the Hamiltonian [Formula: see text], where A=(a1,a2,…,an), Ω is a symmetric matrix, and we report that the corresponding partition function is [Formula: see text] As a result, the generalized Bose distribution and thermodynamic functions are derived.