On measures,convex cones,and foundations of thermodynamics II. Thermodynamic systems

In this two-part paper, a theory of non-equilibrium thermodynamic systems (with memory) is developed. Emphasis is laid upon the possibility of presenting non-equilibrium thermodynamics deductively starting from the basic laws in a form which is capable of a direct experimental verification. In the present second part mathematical conditions corresponding to statements of the second law due to Carnot, Clausius, Kelvin, and Planck are formulated, and some new mathematical forms of the first law are exhibited. In this way, similarities and differences between the two laws are revealed. As consequences of the appropriate versions of the first and the second laws there is proved the existence of Joule's mechanical equivalent of a unit of heat, the energy, the absolute temperature scale, and entropy. The absolute temperature and entropy are defined in and away from equilibrium and satisfy the entropy inequality for processes which are not restricted by the requirement to be quasi-static. The central concept of the theory is the heat distribution measure which contains all the necessary information about the way heat is gained by the system during the process. This approach suggests an interpretation of the absolute temperature different from the one given by Carathéodory (absolute temperature=integrating denominator): here the reciprocal function of the absolute temperature scale gives a functional which separates the set of all heat distribution measures corresponding to cyclic processes from the convex cone of positive measures.I wish to express my deep thanks to Dr. J. Kratochvil CSc. for his encouragement and for numerous discussions on the topic of the paper.     

On statistics and thermodynamics I

The paper is an attempt to develop the statistics of thermal equilibrium by incorporating fluctuation in the thermodynamic theory of measurement.The author wishes to thank U.G.C. for financial assistance. He also thanks the referee for his valuable suggestions in revising the paper.     

On the second law of thermodynamics I. General framework

In this paper a general framework for discussing the classical statements of the second law of thermodynamics is developed. The thermodynamic systems with which the theory deals need not obey the first law and can undergo general (not necessarily quasi-static) processes. By using the formalism of heat distribution measures introduced in previous papers of the author, the classical verbal statements are converted into meaningful mathematical conditions. These conditions can be put into a general form which is the same for all the classical statements. The main result of the paper is an abstract theorem which shows that the general condition leads to one or two inequalities for cyclic processes. In the subsequent part of the paper the abstract theorem is applied to the specific conditions corresponding to the classical statements of the second law. The number of the corresponding inequalities depends on the condition in question, but in each case these inequalities are generalization of the Clausius inequality to which they reduce if the first law holds. By comparing the inequalities corresponding to various statements of the second law also the relations among the statements are established in the second part of the paper.I wish to thank Dr. Jan Kratochvil, DrSc for a number of helpful suggestions concerning a previous draft of the paper.     

Relativistic thermodynamics of fluids. I

In 1953, Stueckelberg and Wanders derived the basic laws of relativistic linear nonequilibrium thermodynamics for chemically reacting fluids from the relativistic local conservation laws for energy-momentum and the local laws of production of substances and of non-negative entropy production by the requirement that the corresponding currents (assumed to depend linearly on the first derivatives of the state variables) should not be independent. Generalizing their method, we determine the most general allowed form of the energy-momentum tensor T^αβ and of the corresponding rate of entropy production under the same restriction on the currents. The problem of expressing this rate in terms of thermodynamic forces and fluxes is discussed in detail; it is shown that the number of independent forces is not uniquely determined by the theory, and several possibilities are explored. A number of possible new cross effects are found, all of which persist in the Newtonian (low-velocity) limit. The treatment of chemical reactions is incorporated into the formalism in a consistent manner, resulting in a derivation of the law for rate of production, and in relating this law to transport processes differently than suggested previously. The Newtonian limit is discussed in detail to establish the physical interpretation of the various terms of T^αβ. In this limit, the interpretation hinges on that of the velocity field characterizing the fluid. If it is identified with the average matter velocity following from a consideration of the number densities, the usual local conservation laws of Newtonian nonequilibrium thermodynamics are obtained, including that of mass. However, a slightly different identification allows conversion of mass into energy even in this limit, and thus a macroscopic treatment of nuclear or elementary particle reactions. The relation of our results to previous work is discussed in some detail.     

String tension and thermodynamics with tree level and tadpole improved actions

We calculate the string tension, deconfinement transition temperature and bulk thermodynamic quantities of the SU(3)SU(3) gauge theory using tree level and tadpole improved actions. Finite temperature calculations have been performed on lattices with temporal extent N_t=3N_\tau=3 and 4. Compared to calculations with the standard Wilson action on this size lattices we observe a drastic reduction of the cut-off dependence of bulk thermodynamic observables at high temperatures. In order to test the influence of improvement on long-distance observables at T_cT_c we determine the ratio T_c/?{s}T_c/\sqrt{\sigma} For all actions, including the standard Wilson action, we find results which differ only little from each other. We do, however, observe an improved asymptotic scaling behaviour for the tadpole improved action compared to the Wilson and tree level improved actions.     

Fluids with highly directional attractive forces. I. Statistical thermodynamics

A new formulation of statistical thermodynamics is derived for classical fluids of molecules that tend to associate into dimers and possibly highers-mers due to highly directional attraction. A breakup of the pair potential into repulsive and highly directionally attractive parts is introduced into the expansion of the logarithm of the grand partition function in fugacity graphs. The bonding by the directional attraction is used to classify the graphs and to introduce a topological reduction which results in the replacement of the fugacity by two variables: singlet density and monomer density ₀. Results for the thermodynamic functions as functionals of and ₀ are given in the form of graph sums. Pair correlations are analyzed in terms of a new matrix analog of the direct correlation function. It is shown that the low-density limit is treated exactly, while major difficulties arise when the Mayer expansion, which employs onlyp, is used. The intricate resummations required for the Mayer expansion are illustrated for the case where dimers are the only association products.Supported by the NSF under Grant Nos. CHE-81-14968 and CHE-82-11236 and by the U.S. Air Force under Grant No. AFOSR 82-0016A.     

Rigorous information-theoretic derivation of quantum-statistical thermodynamics. I

In previous publications we have criticized the usual application of information theory to quantal situations and proposed a new version of information-theoretic quantum statistics. This paper is the first in a two-part series in which our new approach is applied to the fundamental problem of thermodynamic equilibrium. Part I deals in particular with informational definitions of equilibrium and the identification of thermodynamic analogs in our modified quantum statistics formalism.Work supported by a grant from Research Corporation to J. L. P.     

Concepts of stability and symmetry in irreversible thermodynamics. I

Concepts of stability and symmetry in irreversible thermodynamics are developed through the analysis of system energy flows. The excess power function, derived from a local energy conservation equation, is shown to yield necessary and sufficient stability criteria for linear and nonlinear irreversible processes. In the absence of symmetry-destroying external forces, the linear range may be characterized by a set of phenomenological coefficient symmetries relating coupled forces and displacements, velocities, and accelerations, whereas rotational phenomena in nonlinear processes may be characterized by skew-symmetric components of the phenomenological coefficients. A physical interpretation of the nature of the skew-symmetric parts is given and the variational principle of minimum dissipation of energy is related to a stability criterion.     

Systems with separable many-particle interactions. I

In this paper we give an exact evaluation of the free energy per particle for systems with separable many-particle interactions described by a hamiltonian of the type ? = ∑_kT(k) + NP (N^-1 ∑_kV(k)), where P is an arbitrary polynomial. In the proof use is made of a fundamental theorem due to Bogoliubov Jr. for ferromagnetic quadratic operators. The free energy can be obtained from a trial hamiltonian, which is linear in the operators T and V.     

A unified quantum theory of mechanics and thermodynamics. Part I. Postulates

A unified axiomatic theory that embraces both mechanics and thermodynamics is presented in three parts. It is based on four postulates; three are taken from quantum mechanics, and the fourth is the new disclosure of the existence of quantum states that are stable (Part I). For nonequilibrium and equilibrium states, the theory provides general original results, such as the relation between irreducible density operators and the maximum work that can be extracted adiabatically (Part IIa). For stable equilibrium states, it shows for the first time that the canonical and grand canonical distributions are the only stable distributions (Part IIb). The theory discloses the incompleteness of the equation of motion of quantum mechanics not only for irreversible processes but, more significantly, for reversible processes (Part IIb). It establishes the operational meaning of an irreducible density operator and irreducible dispersions associated with any state, and reveals the relationship between such dispersions and the second law (Part III).     

Thoughts on the dynamics of foundations,or what I believe

Unification of physical theories is an ambiguous idea. Gravitation is perhaps distinct from other interactions. Arguments for a unification can be found, either from an analysis of the nature of time, or from the canonicity of most formalisms used in physics. Space-time continuum requires a physics of fields, but not necessarily of quantized fields. Various notions of space as used in physics show that these spaces—including Newtonian space—are never real spaces, but inventions of the mind not only useful, but necessary for the development of many physical theories. Time is, on the other hand, real.     

The laws of relativistic thermodynamics: I. The macroscopic quantities

The physical quantities which occur in the laws of relativistic thermodynamics are defined as statistical expressions of relativistic kinetic theory. The role of the hydrodynamic velocity field is discussed.     

On the pre‐metric foundations of wave mechanics I: massless waves

The mechanics of wave motion in a medium are founded in conservation laws for the physical quantities that the waves carry, combined with the constitutive laws of the medium, and define Lorentzian structures only in degenerate cases of the dispersion laws that follow from the field equations. It is suggested that the transition from wave motion to point motion is best factored into an intermediate step of extended matter motion, which then makes the dimension‐codimension duality of waves and trajectories a natural consequence of the bicharacteristic (geodesic) foliation associated with the dispersion law. This process is illustrated in the conventional case of quadratic dispersion laws, as well as quartic ones, which include the Heisenberg–Euler dispersion law. It is suggested that the contributions to geodesic motion from the non‐quadratic nature of a dispersion law might represent another source of quantum fluctuations about classical extremals, in addition to the diffraction effects that are left out by the geometrical optics approximation.     

Universality of One-Dimensional Fermi Systems,I. Response Functions and Critical Exponents

The critical behavior of one-dimensional interacting Fermi systems is expected to display universality features, called Luttinger liquid behavior. Critical exponents and certain thermodynamic quantities are expected to be related among each other by model-independent formulas. We establish such relations, the proof of which has represented a challenging mathematical problem, for a general model of spinning fermions on a one dimensional lattice; interactions are short ranged and satisfy a positivity condition which makes the model critical at zero temperature. Proofs are reported in two papers: in the present one, we demonstrate that the zero temperature response functions in the thermodynamic limit are Borel summable and have anomalous power-law decay with multiplicative logarithmic corrections. Critical exponents are expressed in terms of convergent expansions and depend on all the model details. All results are valid for the special case of the Hubbard model.