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.     

Classical statistical mechanics and Landau damping

We study the retarded response function in scalar φ⁴-theory at finite temperature. We find that in the high-temperature limit the imaginary part of the self-energy is given by the classical theory to leading order in the coupling. In particular the plasmon damping rate is a purely classical effect to leading order, as shown by Aarts and Smit. The dominant contribution to Landau damping is given by the propagation of classical fields in a heat bath of non-interacting fields.     

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 preferable basis in quantum mechanics

A way to specify the preferable basis which determines the ensemble of universes in the many-worlds conception of quantum theory is proposed. This way is based on a consideration of the classical limit in quantum mechanics. The specified basis is shown to be necessary and sufficient for comparison of theory with observations.     

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 non-commutative phase space

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space. The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity. First, new Poisson brackets have been defined in non-commutative phase space. They contain corrections due to the non-commutativity of coordinates and momenta. On the basis of this new Poisson brackets, a new modified second law of Newton has been obtained. For two cases, the free particle and the harmonic oscillator, the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys. Rev. D, 2005, 72: 025010). The consistency between both methods is demonstrated. It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space, but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.     

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.     

Classical mechanics in non-commutative phase space

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space.The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity.First,new Poisson brackets have been defined in non-commutative phase space.They contain corrections due to the non-commutativity of coordinates and momenta.On the basis of this new Poisson brackets,a new modified second law of Newton has been obtained.For two cases,the free particle and the harmonic oscillator,the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys.Rev.D,2005,72:025010).The consistency between both methods is demonstrated.It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space.but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.     

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.     

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.     

Elementary particle scattering and statistical quasi-particles in quantum statistical mechanics

Previous work relating the thermodynamic potential to elementary particle S-matrix elements is generalized and rederived directly from the expressions for the diagrams of many body theory. The divergent physical region poles are shown to introduce energy derivatives of mass shell delta functions which tend to shift the energies of the scattering particles away from the elementary particle mass shell. These shifted energies are related to the statistical quasiparticle energies introduced by Balian and De Dominicis. The work of these authors is generalized to show that to all orders in the coupling strengths the many body diagrams for any system described by a relativistic or non-relativistic field theory can be summed to give: (1) the entropy and the statistical average of a non-spontaneously broken, conserved charge in terms of ideal gas-like formulae involving statistical quasi-particle energies; (2) the thermodynamic potential in terms of diagonal matrix elements of products of transition amplitudes wherein the energies of all external particles and the energy arguments of all ideal gas occupation numbers are the statistical quasi-particle energies.     

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.