Classical statistical mechanics of identical particles and quantum effects

The identification of the phase space ofN classical identical particles with the equivalence class of points is of crucial importance for statistical mechanics. We show that the refined phase space leads to the correct statistical mechanics for an ideal gas; moreover, Gibbs's paradox is resolved and the Third Law of Thermodynamics is recovered. The presence of both induced and stimulated transitions is shown as a consequence of the identity of the particles. Other results are the quantum contribution to the second virial coefficient and the Bose-Einstein condensation. Photon bunching and Hanbury Brown-Twiss effect are also seen to follow from the classical model. The only element of quantum theory involved is the notion of phase cells necessary to make the entropy dimensionless. Assuming the existence of the light quantum or the phonon hypothesis we could derive the Planck distribution law for blackbody radiation or the Debye formula for specific heats respectively.     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

Classical formulation of quantum mechanics

The concept of quantum state is given in terms of classical probability for position in squeezed and rotated classical reference frames in phase space. Stationary states and energy levels of the quantum system are obtained in a classical formulation of quantum mechanics. The positive probability density of the harmonic oscillator position is obtained by solving a new eigenvalue equation of standard quantum mechanics instead of the Schrödinger equation. The orthogonality and completeness relations are found for the eigendistributions.     

Classical mechanics in phase space revisited

A Bose type of classical Hamilton algebra, i.e., the algebra of the canonical formalism of classical mechanics, is represented on a linear space of functions of phase space variables. The symplectic metric of the phase space and possible algorithms of classical mechanics (which include the standard one) are derived. It is shown that to each of the classical algorithms there is a corresponding one in the phase space formulation of quantum mechanics.     

Action-angle variables in quantum statistical mechanics

Utilizing the facts (i) that the number of particles in the many-boson system is conserved and (ii) that the Hamiltonian is Hermitian, a new set of variables comprising action and angle variables has been introduced. These variables are conjugate in the mean and provide a rigorous approach to introducing phase variables for total-number-conserving many-boson systems.Lecture given at the Saha Institute of Nuclear Physics, Calcutta, June 1971.     

Graph reconstruction and quantum statistical mechanics

We study how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.     

Classical statistical mechanics of a one-dimensional system of molecules

The thermodynamic properties of a rectilinear assembly are solved taking into account the interaction of each molecule with any given number n near-by molecules. The result is a non-linear integral equation, which determines a distribution function. The connection between this distribution function and the Helmholtz free energy is derived. The method can apparently be extended to higher dimensional cases.     

Classical statistical mechanics of a lattice model of superradiance

A classical model of the interaction of N two-level atoms with the radiation field discretized on a lattice, having particular relevance for superradiance, is discussed. By evaluating the corresponding partition function the threshold condition for the occurrence of a second-order phase transition is derived. A comparison with previous work, where a finite number of field modes was considered, is also given.     

Classical particle dynamics in quantum space

It is suggested that if space-time is quantized at small distances, then even at the classical level particle motion in space is complicated and described by a nonlinear equation. In the quantum space the Lagrangian function or energy of the particle consists of two parts: the usual kinetic terms, and a rotation term determined by the square of the inner angular momentum-a torsion torque caused by the quantum nature of space. Rotational energy and rotational motion of the particle disappear in the limitl0, wherel the value of the fundamental length. In the free particle case, in addition to the rectilinear motion, the particle undergoes a rotation given by the inner angular momentum. Different possible types of particle motion are discussed. Thus, the scheme may shed light on the appearance of rotating or twisting, stochastic, and turbulent types of motion in classical physics and, perhaps, on the notion of spin in quantum physics within the framework of the quantum character of space-time at small distances.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

The BBGKY hierarchy in quantum statistical mechanics

It is shown that the infinite volume limit of the equilibrium reduced density matrices, shown by Ginibre to exist at low densities, satisfy the quantum time independent BBGKY hierarchy of equations. This extends analogous results for classical systems due to Gallavotti.Supported by AFOSR Contract Number F44620-71-C-0013.     

Relativistic quantum statistical mechanics in two-dimensional space-time

We construct for a boson field in two-dimensional space-time with polynomial or exponential interactions and without cut-offs, the positive temperature state or the Gibbs state at temperature 1/β. We prove that at positive temperatures i.e. β<∞, there is no phase transitions and the thermodynamic limit exists and is unique for all interactions. It turns out that the Schwinger functions for the Gibbs state at temperature 1/β is after interchange of space and time equal to the Schwinger functions for the vacuum or temperature zero state for the field in a periodic box of length β, and the lowest eigenvalue for the energy of the field in a periodic box is simply related to the pressure in the Gibbs state at temperature 1/β.