A condensed survey is given of the results of a previous investigation on quantum concepts and principles developed from the anti-Gibbs postulate of thermodynamic continuity. The same postulate now also yields major parts of quantum statistics, such as the equality of various occupation number sets without permutation of individual particles, not however the alternative "either Bose or Fermi statistics." A distinction must be made between a coherent redistribution of particles without entropy change, and an incoherent redistribution leading to a new thermodynamic equilibrium. The latter is obtained from the former by a process of diffusion from separate level groups to a totality of levels.
A condensed survey is given of the results of a previous investigation on quantum concepts and principles developed from the anti-Gibbs postulate of thermodynamic continuity. The same postulate now also yields major parts of quantum statistics, such as the equality of various occupation number sets without permutation of individual particles, not however the alternative "either Bose or Fermi statistics." A distinction must be made between a coherent redistribution of particles without entropy change, and an incoherent redistribution leading to a new thermodynamic equilibrium. The latter is obtained from the former by a process of diffusion from separate level groups to a totality of levels.
The basic concepts and rules of quantum mechanics are shown to be immediate consequences, on the basis of simple reasoning, of a postulate of continuity in the domain of macroscopic thermodynamics. The continuity principle leads first to the concept of a fractional likeness between two states in general, as opposed to the classical alternative of either like or unlike. Then all possible states of a system can be arranged in sets of mutually unlike or “orthogonal” states, with members A and B of different sets interconnected by fractional likeness factors q(A, B) between zero and unity. These factors q are identical with the relative intensities in a splitting effect of a state into components, ruled by probability relations and by the basic principle of quantum statistics of not counting permutations of individual particles. The problem of relation between the various fractional likenesses, or probabilities of transition under analysis, is a purely mathematical problem and is solved by the introduction of probability amplitudes necessarily subject to a matrix law of multiplication, i.e., to the principle of superposition. Thus, starting from the continuity postulate of thermodynamics, quantum theory in its general outline is obtained by simple reasoning. All further details, such as the duality of waves and particles, the uncertainty relation, and the mechanics of special systems, need only one more bit of empirical information, namely, the symmetric and periodic form of the amplitude ψ of fractional likeness between a state of given coordinate and a state of given momentum.
The proper definition of entropy is fundamental to the relationship between statistical mechanics and thermodynamics. It also plays a major role in the recent debate about the validity of the concept of negative temperature. In this paper, I analyze and calculate the thermodynamic entropy for large but finite quantum mechanical systems. A special feature of this analysis is that the thermodynamic energy of a quantum system is shown to be a continuous variable, rather than being associated with discrete energy eigenvalues. Calculations of the entropy as a function of energy can be carried out with a Legendre transform of thermodynamic potentials obtained from a canonical ensemble. The resultant expressions for the entropy are able to describe equilibrium between quantum systems having incommensurate energy-level spacings. This definition of entropy preserves all required thermodynamic properties, including satisfaction of all postulates and laws of thermodynamics. It demonstrates the consistency of the concept of negative temperature with the principles of thermodynamics.
The statistical problem of the distribution of s quanta of equal energy ε0 and total energy E among N distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by p(κ)=(N−1N+s−1)(N−2N+s−κ−2);κ=0,1,2,⋯,s In this context, the special case (N=4, s=4) presented indicates that the alternative concept of most probable state is not valid for finite values of s and N. In the present article we derive alternatively p(κ) by distributing s quanta over N particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; sεo=E∼ fixed, where the energy of the particles becomes continuous, p(κ) transforms to the Boltzmann law P(ε)=⟨ε⟩1e−⟨ε⟩ε;0≤ε<+∞ where ⟨ε⟩=E/N. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.
The author presents his experience in teaching at a graduate level the quantal exposition of a new non-statistically based paradigm of physics and thermodynamics. This paradigm, called the Unified Quantum Theory of Mechanics and Thermodynamics, applies to all systems large or small (including one particle systems) either in a state of thermodynamic (i.e. stable) equilibrium or not in a state of thermodynamic equilibrium. It uses as its primitives inertial mass, force, and time and introduces the laws of thermodynamics in the most unambiguous and general formulations found in the literature. Starting with a precise definition of system and of state followed by statements and corollaries of the laws of thermodynamics, the thermodynamic formalism is developed without circularity and ambiguity. In this quantal exposition of the new paradigm, a brief review of the formalism of thermodynamics as a general science not limited to stable equilibrium and large (macroscopic) systems as well as a very brief summary of the three prevalent formalisms in classical physics are presented followed by a presentation and development of solutions for a number of elementary problems in quantum physics (e.g., a particle in a box, a harmonic oscillator, a rigid rotor, etc.). These solutions and the maximum entropy principle are then used in a constrained optimization to develop the canonical and grand canonical distributions for Fermi-Dirac and Bose-Einstein types of particles, i.e. for fermions and bosons. This is done without the use of analogies between statistical and thermodynamic results and without additional hypotheses such as the ergodic hypothesis of statistical mechanics. These distributions are then employed under various assumptions (i.e. the Boltzmann, constant-potential, point-particle, and continuous eigenvalue-spectrum approximations) to derive the corresponding thermodynamic property expressions for perfect, semi-perfect (ideal), and Sommerfeld gases as well as for mixtures of ionized and dissociated gases. In a similar fashion but with a change from a single- to a multi-particle partition function and with the addition of various inter-particle potentials for two-particle interactions (e.g., the Lennard-Jones potential, the square-well potential, etc.), expressions for the thermodynamic properties of dense gases are developed and presented.An initial version of this paper was published inJuly of 2006 in the proceedings of ECOS’06, AghiaPelagia, Crete, Greece.
In the mid-19th century, both the laws of mechanics and thermodynamics were known, and both appeared fundamental. This was changed by Boltzmann and Gibbs, who showed that thermodynamics can be *derived*, by applying mechanics to very large systems, and making simple statistical assumptions about their behavior. Similarly, when Quantum Mechanics (QM) was first discovered, it appeared to require two sets of postulates: one about the deterministic evolution of wavefunctions, and another about the probabilistic measurement process. Here again, the latter is derivable from the former: by applying unitary evolution to large systems (apparatuses, observers and environment), and making simple assumptions about their behavior, one can derive all the features of quantum measurement. We set out to demonstrate this claim, using a simple and explicit model of a quantum experiment, which we hope will be clear and compelling to the average physicist.
A consensus seems to have developed that the Gibbs paradox in classical thermodynamics (the discontinuous drop in the entropy of mixing when the mixed gases become equal to each other) is unmysterious: in any actual situation, two gases can be separated or not, and the associated harmless discontinuity from “yes” to “no” is responsible for the discontinuity. By contrast, the Gibbs paradox in statistical physics continues to attract attention. Here, the problem is that standard calculations in statistical mechanics predict a non-vanishing value of the entropy of mixing even when two gases of the same kind are mixed, in conflict with thermodynamic predictions. This version of the Gibbs paradox is often seen as a sign that there is something fundamentally wrong with either the traditional expression S=klnW or with the way W is calculated. It is the aim of this article to review the situation from the orthodox (as opposed to information theoretic) standpoint. We demonstrate how the standard formalism is not only fully capable of dealing with the paradox, but also provides an intuitively clear picture of the relevant physical mechanisms. In particular, we pay attention to the explanatory relevance of the existence of particle trajectories in the classical context. We also discuss how the paradox survives the transition to quantum mechanics, in spite of the symmetrization postulates.