Thermodynamic Inequalities Research Articles

Non-compact QED 3 coupled to a four-fermi interaction

We present preliminary numerical results for the three dimensional non-compact QED with a weak four-fermion term in the lattice action. Approaches based on Schwinger-Dyson studies, arguments based on thermodynamic inequalities and numerical simulations lead to estimates of the critical number of fermion flavors (below which chiral symmetry is broken) ranging from $N_{fc}=1$ to $N_{fc}=4$. The weak four-fermion coupling provides the framework for an improved algorithm, which allows us to simulate the chiral limit of massless fermions and expose delicate effects.

Heating the Solar Corona

A typical temperature for the quiet solar corona is ~ 1.5 x 106K, whereas the photosphere – the likely source of the thermal energy – has a temperature less than 6 × 103 K. Although many theories have been advanced to explain why the corona is so much hotter than the photosphere, this old problem remains unsolved. However, there is a mechanism based on second-order transport that may provide the answer, or at least part of the answer. This process, described by the author in Thermodynamic inequalities in gases and magnetoplasmas, John Wiley & Sons Ltd, 1996, causes heat to be transported across strong magnetic fields up temperature gradients.

Thermodynamics of Superfluidity

New, superfluid specific additive integral of motion is found. This facilitates investigation of general thermodynamic equilibrium conditions for superfluid. The analysis is performed in an extended space of thermodynamic variables containing (along with the usual thermodynamic coordinates such as pressure and temperature) superfluid velocity and momentum density. The equilibrium stability conditions lead to thermodynamic inequalities which replace the Landau superfluidity criterion at finite temperatures.

Two fluid model: new aspects

We investigate general thermodynamic stability conditions for the superfluid. This analysis is performed in an extended space of thermodynamic variables containing (along with the usual thermodynamic coordinates such as pressure and temperature) superfluid velocity and momentum density. The stability conditions lead to thermodynamic inequalities which replace the Landau superfluidity criterion at finite temperatures.

Thermodynamic inequalities in a superfluid

We investigate general thermodynamic stability conditions for the superfluid. This analysis is performed in an extended space of thermodynamic variables containing (along with the usual thermodynamic coordinates such as pressure and temperature) superfluid velocity and momentum density. The stability conditions lead to thermodynamic inequalities which replace the Landau superfluidity criterion at finite temperatures.

Rectification of thermodynamic inequalities

The Clausius inequality is converted to an equality in which a positive quantity with units of energy, termed the deficit function, is used to track entropy production and other effects of irreversible processes. This inequality rectification procedure is shown to yield particularly transparent new derivations of several important results based on the Second Law, with some consequences that have apparently not been previously recognized. For example, the Helmholtz free energy of any system is shown to decrease towards a minimum as the result of any spontaneous processes which begin and end at the same temperature and involve no work exchange, without the need to invoke the stricter requirements of fixed system volume and/or chemical composition.

Sign of refractive index and group velocity in left-handed media

We argue that the widely spread opinion that the left-handed media (LHM) are characterized by a negative refractive index n − is misleading. Since n does not enter into Maxwell's equations and boundary conditions, any medium may be described by both positive n and negative n −=− n. Two thermodynamic inequalities are presented, that make a difference between the LHM and the regular media (RM). The first one reads that the group velocity is positive in the RM and negative in the LHM. The second one is that the product Re( n)Im( n) is positive in the RM and negative in the LHM. Both inequalities are invariant with respect to the change n→ n −. However, to use n − one should change some traditional electrodynamics definitions.

Fluctuations in the presence of fields: phenomenological Gaussian approximation and a class of thermodynamic inequalities.

The fluctuations of thermodynamic systems in the presence of the fields are considered. The approach is of phenomenological nature and developed in a Gaussian approximation. The cases of a magnetizable continuum in a magnetoquasistatic field, as well as the so called discrete systems are used to exemplify the study. In the latter case one finds that the fluctuation estimators depend both on the intrinsic properties of the system and on the characteristics of the environment. Following earlier ideas of one of the authors we present a class of thermodynamic inequalities for the systems investigated in this paper. In the case of two variables these inequalities are nonquantum analogs of the well known quantum Heisenberg "uncertainty" relations. In this context, the fluctuation estimators support the idea that Boltzmann's constant k has the signification of a generic indicator of stochasticity for thermodynamic systems.

Why is it so difficult to simulate entropies, free energies, and their differences?

The classical 19th century thermodynamic inequalities of Clausius and Helmholtz are applied to the calculation of entropy and free energy changes by computer simulation. The irreversibility of finite-time thermodynamic paths is exploited to obtain upper and lower bounds on these quantities. Schrödinger's microscopic interpretation of heat and work provides the basis for a literal implementation of the key historical concepts on the computer using the Monte Carlo algorithm of Metropolis. Coupling schemes, paths, and reference states are variationally optimized to improve the convergence of the simulated properties, and a newly introduced variational flexibility, metric scaling, is overviewed. Reasons for expecting limiting power laws for the convergence are outlined.

Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states)

Plasmas consisting exclusively of particles with a single sign of charge (e.g., pure electron plasmas and pure ion plasmas) can be confined by static electric and magnetic fields (in a Penning trap) and also be in a state of global thermal equilibrium. This important property distinguishes these totally unneutralized plasmas from neutral and quasineutral plasmas. This paper reviews the conditions for, and the structure of, the thermal equilibrium states. Both theory and experiment are discussed, but the emphasis is decidedly on theory. It is a huge advantage to be able to use thermal equilibrium statistical mechanics to describe the plasma state. Such a description is easily obtained and complete, including for example the details of the plasma shape and microscopic order. Pure electron and pure ion plasmas are routinely confined for hours and even days, and thermal equilibrium states are observed. These plasmas can be cooled to the cryogenic temperature range, where liquid and crystal-like states are realized. The authors discuss the structure of the correlated states separately for three plasma sizes: large plasmas, in which the free energy is dominated by the bulk plasma; mesoscale plasmas, in which the free energy is strongly influenced by the surface; and Coulomb clusters, in which the number of particles is so small that the canonical ensemble is not a good approximation for the microcanonical ensemble. All three cases have been studied through numerical simulations, analytic theory, and experiment. In addition to describing the structure of the thermal equilibrium states, the authors develop a thermodynamic theory of the trapped plasma system. Thermodynamic inequalities and Maxwell relations provide useful bounds on and general relationships between partial derivatives of the various thermodynamic variables.

Thermal equilibria and thermodynamics of trapped plasmas with a single sign of charge

Plasmas consisting exclusively of particles with a single sign of charge (e.g., pure electron plasmas and pure ion plasmas) can be confined by static electric and magnetic fields (e.g., in a Penning trap) and also be in a state of global thermal equilibrium. This important property distinguishes these totally un-neutralized plasmas from neutral and quasineutral plasmas. This paper reviews the conditions for and structure of the thermal equilibrium states and then develops a thermodynamic theory of the trapped plasmas. Thermodynamics provides hundreds of general relations (Maxwell relations) between partial derivatives of thermodynamic variables with respect to one another. Thermodynamic inequalities place general and useful bounds on various quantities. General and relatively simple expressions are provided for fluctuations of the thermodynamic variables. In practice, trapped plasmas are often made to evolve through a sequence of thermal equilibrium states through the slow addition (or subtraction) of energy and angular momentum (say, by laser cooling and torque beams). A thermodynamic approach to this late time transport describes the evolution through coupled ordinary differential equations for the thermodynamic variables, which is a huge reduction in complexity compared to the partial differential equations typically required to describe plasma transport. These evolution equations provide a theoretical basis for the dynamical control of the plasmas.

On the stability for linear viscoelastic solids

The stability and asymptotic behaviour of solutions to the dynamical initial past-history problem within the theory of linear viscoelasticity is investigated. First, a family of free energies is introduced as a Liapunov functional via the energy method. Then, emphasizing the role of thermodynamic inequalities, the asymptotic stability property is derived by means of some original a priori integral estimates of the solution.

General kinetic description of relativistic mixed neutrinos

We derive a general Boltzmann-type collision integral for mixed neutrinos interacting with each other and with a medium. Our treatment is fully relativistic in that antineutrino degrees of freedom are included. This collision integral allows one to account for the simultaneous effects of neutrino oscillations in a medium and for the effects of collisions. Our results generalizes previous attempts of unify the first- and second-order interaction effects in a single self-consistent equation. Most importantly, our equation includes effects non-linear in the neutrino density matrices (or occupation numbers) such as Pauli blocking of neutrino final states or neutrino refraction in a medium of neutrinos. We apply the definition of the entropy of a non-equilibrium Fermi gas to the case of mixed neutrinos, and we prove that our collision integrals obey the relevant thermodynamic inequalities.

On electromagnetic waves in isotropic media with dielectric relaxation

In some previous papers one of us discussed dielectric relaxation phenomena from the point of view of nonequilibrium thermodynamics. If the theory is linearized one may derive a dynamical constitutive equation (relaxation equation) which has the form of a linear relation among the electric field E, the polarization P, the first derivatives with respect to time of E and P and the second derivative with respect to time of P. The Debye equation for dielectric relaxation in polar liquids and the De Groot-Mazur equation (obtained by these authors with the aid of methods which are also based on nonequilibrium thermodynamics) are special cases of the more general equation of which the structure has been described above. It is the purpose of the present paper to investigate the propagation and damping of electromagnetic waves. We consider the case in which the dielctric relaxation may be described by the above mentioned relaxation equation derived by one of us, the case in which the Debye equation may be used and the case in which one may apply the De Groot-Mazur equation. We derive solutions of the relaxation equations which also satisfy Maxwell's equations. We limit ourselves to plane waves of a single frequency in isotropic homogeneous linear media with vanishing electric conductivity. It is also assumed that the media are at rest. From thermodynamic arguments several inequalities are derived for the coefficients which occur in the relaxation equations. Explicit expressions are given for the complex wave vector, the complex dielectric permeability and for the phase velocity of the waves. All these quantities are functions of the frequency ω of the waves. For ω→0 the complex permeability e(compl) → e(eq) where e(eq) is the equilibrium value of the permeability for static fields. If ω→∞ we find e(compl) → 1 except for the case of the Debye equation. This is due to the fact that a part of the polarization changes in a reversible way in media for which the Debye equation holds.

A note on the thermodynamic inequalities

The authors propose the use of different symbols used in calculating actual energy and equilibrium energy.

Thermodynamic inequalities for percolation

In this paper we describe the percolation analogues of the Gibbs and Helmholtz potentials and use these quantities to prove some general inequalities concerning the critical exponents of percolation processes.

Thermodynamic Inequalities in Continuum Mechanics

Thermodynamic Inequalities in Continuum Mechanics

Stochastic coupling and thermodynamic inequalities

Let μ1 and μ2 be thermodynamic Gibbs measures on ℝm and ℝn, respectively. Diffusions are constructed having μ1, and μ2 as invariant measures. These diffusions are then coupled; inequalities between expectations of certain random variables on the two spaces result.

Quantum Heisenberg ferromagnets and stochastic exclusion processes

Let H be the isotropic spin‐s quantum ferromagnetic Heisenberg Hamiltonian associated with a finite volume in Zd. In an appropriate representation, the restriction of −H to the N‐spin wave sector −HN can be regarded as the generator of a Markov semigroup exp(−tHN). For s= 1/2 the corresponding process XN(t) is identified as a simple N‐particle exclusion process; for higher spin it is a more complicated exclusion process involving different species of particles. A Feynman–Kac, Cameron–Martin perturbation formula for exp(−tHN) is developed in the s= 1/2 case, and the formula is used to obtain some thermodynamic correlation inequalities.

Shock waves in relativistic magnetohydrodynamics under general assumptions

We present a rigorous theory of magnetohydrodynamical shock waves in the framework of a given curved space–time, under general assumptions corresponding both to a plasma and to a condensed medium. The results can be of use in astrophysics. We prove the timelike character of the wavefronts, the main thermodynamic inequalities, the relative location of the speeds of the shock waves with respect to the magnetosonic and Alfvén speeds, and show some existence and uniqueness theorems. In particular, we show that there can exist initial states giving slow shocks, but no weak shocks.