Exact Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac Statistics

The exact Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) entropies and probabilistic distributions are derived by the combinatorial method of Boltzmann, without Stirling's approximation. The new entropy measures are explicit functions of the probability and degeneracy of each state, and the total number of entities, N. By analysis of the cost of a "binary decision", exact BE and FD statistics are shown to have profound consequences for the behaviour of quantum mechanical systems.Comment: 18 pages; 6 figures; accepted for publication by Physics Letters A, 13/5/0

Cost of s-fold Decisions in Exact Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac Statistics

The exact forms of the degenerate Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) entropy functions, derived by Boltzmann's principle without the Stirling approximation (Niven, Physics Letters A, 342(4) (2005) 286), are further examined. Firstly, an apparent paradox in quantisation effects is resolved using the Laplace-Jaynes interpretation of probability. The energy cost of learning that a system, distributed over s equiprobable states, is in one such state (an s-fold decision) is then calculated for each statistic. The analysis confirms that the cost depends on one's knowledge of the number of entities N and (for BE and FD statistics) the degeneracy, extending the findings of Niven (2005).Comment: 7 figures; 5 pages; REVTEX / TeXShop; paper from 2005 NEXT-Sigma-Ph

Hadron transverse momentum distributions in the Tsallis statistics with escort probabilities

The exact and approximate transverse momentum distributions of the Tsallis statistics with escort probabilities (the Tsallis-3 statistics) for the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics of particles have been derived. We have revealed that in the zeroth term approximation the Maxwell-Boltzmann transverse momentum distribution of the Tsallis-3 statistics exactly coincides with the classical phenomenological Tsallis distribution and the entropy of the system is equal to zero for all values of state variables. Thus, we have proven that the classical phenomenological Tsallis distribution in the framework of the Tsallis-3 statistics corresponds to the unphysical condition of zero entropy of the system. We have shown that the quantum phenomenological Tsallis distributions and the quantum Tsallis-like distributions used in high-energy physics are similar to the quantum transverse momentum distribution of the Tsallis-3 statistics obtained by introducing a mathematically inconsistent factorization approximation in the zeroth term approximation. We have found that the classical and quantum transverse momentum distributions in the zeroth term approximation and the quantum transverse momentum distributions in the factorization approximation of the zeroth term approximation are the same in the Tsallis-3, Tsallis-2 and $q$-dual statistics. The exact Maxwell-Boltzmann transverse momentum distribution of the Tsallis-3 statistics and the classical phenomenological Tsallis distribution have been compared and applied to describe the experimental spectra of the charged pions produced in the proton-proton collisions at high energies. We have revealed that the numerical results for the parameters of the classical phenomenological Tsallis distribution deviate essentially from the results of the Tsallis-3 statistics for all values of collision energy.Comment: 31 pages, 7 figures. arXiv admin note: text overlap with arXiv:1608.0188

The Thermal Abundance of Semi-Relativistic Relics

Approximate analytical solutions of the Boltzmann equation for particles that are either extremely relativistic or non-relativistic when they decouple from the thermal bath are well established. However, no analytical formula for the relic density of particles that are semi-relativistic at decoupling is yet known. We propose a new ansatz for the thermal average of the annihilation cross sections for such particles, and find a semi-analytical treatment for calculating their relic densities. As examples, we consider Majorana- and Dirac-type neutrinos. We show that such semi-relativistic relics cannot be good cold Dark Matter candidates. However, late decays of meta-stable semi-relativistic relics might have released a large amount of entropy, thereby diluting the density of other, unwanted relics.Comment: 22 pages, 5 figures. Comments and references adde

Generalized Measure of Entropy, Mathai's Distributional Pathway Model, and Tsallis Statistics

The pathway model of Mathai (2005) mainly deals with the rectangular matrix-variate case. In this paper the scalar version is shown to be associated with a large number of probability models used in physics. Different families of densities are listed here, which are all connected through the pathway parameter 'alpha', generating a distributional pathway. The idea is to switch from one functional form to another through this parameter and it is shown that basically one can proceed from the generalized type-1 beta family to generalized type-2 beta family to generalized gamma family when the real variable is positive and a wider set of families when the variable can take negative values also. For simplicity, only the real scalar case is discussed here but corresponding families are available when the variable is in the complex domain. A large number of densities used in physics are shown to be special cases of or associated with the pathway model. It is also shown that the pathway model is available by maximizing a generalized measure of entropy, leading to an entropic pathway. Particular cases of the pathway model are shown to cover Tsallis statistics (Tsallis, 1988) and the superstatistics introduced by Beck and Cohen (2003).Comment: LaTeX, 13 pages, title changed, introduction, conclusions, and references update

The thermodynamics for a hadronic gas of fireballs with internal color structures and chiral fields

The thermodynamical partition function for a gas of color-singlet bags consisting of fundamental and adjoint particles in both $U(N_c)$ and $SU(N_c)$ group representations is reviewed in detail. The constituent particle species are assumed to satisfy various thermodynamical statistics. The gas of bags is probed to study the phase transition for a nuclear matter in the extreme conditions. These bags are interpreted as the Hagedorn states and they are the highly excited hadronic states which are produced below the phase transition point to the quark-gluon plasma. The hadronic density of states has the Gross-Witten critical point and exhibits a third order phase transition from a hadronic phase dominated by the discrete low-lying hadronic mass spectrum particles to another hadronic phase dominated by the continuous Hagedorn states. The Hagedorn threshold production is found just above the highest known experimental discrete low-lying hadronic mass spectrum. The subsequent Hagedorn phase undergoes a first order deconfinement phase transition to an explosive quark-gluon plasma. The role of the chiral phase transition in the phases of the discrete low-lying mass spectrum and the continuous Hagedorn mass spectrum is also considered. It is found crucial in the phase transition diagram. Alternate scenarios are briefly discussed for the Hagedorn gas undergoes a higher order phase transition through multi-processes of internal color-flavor structure modification.Comment: 110 pages and 13 figures. Added references to the introductio

Chaos and Quantum Thermalization

We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey {\it Berry's conjecture}. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell--Boltzmann, Bose--Einstein, or Fermi--Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon {\it eigenstate thermalization}. We show that a generic initial state will approach thermal equilibrium at least as fast as $O(\hbar/\Delta)t^{-1}$, where $\Delta$ is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a new foundation for quantum statistical mechanics.Comment: 28 pages in Plain TeX plus 2 uuencoded PS figures (included); minor corrections only, this version will be published in Phys. Rev. E; UCSB-TH-94-1