Thermodynamic Potential In Terms Research Articles

Thermodynamically consistent versions of approximations used in modelling moist air

AbstractSome existing approaches to modelling the thermodynamics of moist air make approximations that breakthermodynamic consistency, such that the resulting thermodynamics does not obey the first and second laws or has other inconsistencies. Recently, an approach to avoid such inconsistency has been suggested: the use ofthermodynamic potentialsin terms of theirnatural variables, from which all thermodynamic quantities and relationships (equations of state) are derived. In this article, we develop this approach forunapproximatedmoist‐air thermodynamics and two widely used approximations: the constant‐ approximation and the dry heat capacities approximation. The (consistent) constant‐ approximation is particularly attractive because it leads to, with the appropriate choice of thermodynamic variable, adiabatic dynamics that depend only on total mass and are independent of the breakdown between water forms. Additionally, a wide variety of material from different sources in the literature on thermodynamics in atmospheric modelling is brought together. It is hoped that this article provides a comprehensive reference for the use of thermodynamic potentials in atmospheric modelling, especially for the three systems considered here.

Exclusion statistics for particles with a discrete spectrum

We formulate and study the microscopic statistical mechanics of systems of particles with exclusion statistics in a discrete one-body spectrum. The statistical mechanics of these systems can be expressed in terms of effective single-level grand partition functions obeying a generalization of the standard thermodynamic exclusion statistics equation of state. We derive explicit expressions for the thermodynamic potential in terms of microscopic cluster coefficients and show that the mean occupation numbers of levels satisfy a nesting relation involving a number of adjacent levels determined by the exclusion parameter. We apply the formalism to the harmonic Calogero model and point out a relation with the Ramanujan continued fraction identity and appropriate generalizations.

Phase field approach to mode-I fracture by introducing an eigen strain tensor: General theory

A thermodynamically consistent phase field theory for fracture is developed. An alternative approach to describe the reduction in the slope of the stress–strain curve and the reduction of the stresses to zero during damage is utilized by introducing transformation strain instead of the degradation function. The main requirements for the thermodynamic potential are presented and used to present the general energy terms. The analysis of the equilibrium solution is formulated which describes the desired stress–strain curves and for which the infinitesimal damage produces an infinitesimal change in the equilibrium elastic modulus. Analytical analysis of the equations is performed to calibrate phase field parameters for a general case. The explicit form of the Ginzburg-Landau equation is derived based on which the evolution of the order parameter is obtained. The importance of the analysis of the thermodynamic potential in terms of stress–strain curves is shown. The developed theory includes a broad spectrum of the shapes of stress–strain relationships.

A phenomenological constitutive theory for polycrystalline ferroelectric ceramics based on orientation distribution functions

A phenomenological constitutive theory for the ferro-electro-elastic response of polycrystalline ceramics with tetragonal perovskite structure is proposed. The state of a material point is characterized by an intrinsic polarization vector, an infinitesimal deformation tensor, and an internal variable representing the orientation distribution of the ferroelectric domains at that point. A class of convex thermodynamic potentials in terms of these state variables is posited, and constitutive relations within the framework of generalized standard materials are then derived. The functional form of the dissipation is selected in such a way that it effects an order reduction of the constitutive description whereby the infinite-dimensional internal variable is reduced to finite-dimensional internal variables representing polarization and deformation due to ferroelectric switching, preserving at the same time the generalized standard structure of the theory. By way of example, a specific set of constitutive functions is considered. The resulting constitutive relations are able to emulate most essential features of ferroelectric and ferroelastic behavior with minimal computational cost and, furthermore, generate stable predictions in contrast to earlier phenomenological theories.

Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses

A thermodynamically-consistent phase field approach for crack propagation which includes the following novel features is presented. (1) Scale dependency was included by relating the length scale to the number of cohesive interatomic planes at the crack tip. Because of this, the developed theory is applicable from the atomistic to the macroscopic scales. (2) The surface stresses (tension) are introduced by employing some geometrical nonlinearities even in small strain theory. They produce multiple contributions to the Ginzburg-Landau equation for crack propagation. (3) Crack propagation in the region with compressive closing stresses is eliminated by employing a stress-state-dependent kinetic coefficient in the Ginzburg-Landau equation. (4) The importance of analysis of the thermodynamic potential in terms of stress-strain curves is shown. The developed theory includes a broad spectrum of the shapes of stress-strain relationships. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The effect of the above novel features is analyzed numerically for various model problems.

Thermodynamic insight into stimuli-responsive behaviour of soft porous crystals

Knowledge of the thermodynamic potential in terms of the independent variables allows to characterize the macroscopic state of the system. However, in practice, it is difficult to access this potential experimentally due to irreversible transitions that occur between equilibrium states. A showcase example of sudden transitions between (meta)stable equilibrium states is observed for soft porous crystals possessing a network with long-range structural order, which can transform between various states upon external stimuli such as pressure, temperature and guest adsorption. Such phase transformations are typically characterized by large volume changes and may be followed experimentally by monitoring the volume change in terms of certain external triggers. Herein, we present a generalized thermodynamic approach to construct the underlying Helmholtz free energy as a function of the state variables that governs the observed behaviour based on microscopic simulations. This concept allows a unique identification of the conditions under which a material becomes flexible.

Phenomenological theory of phase diagrams. The case of Landau’s potentials with $$\mathcal{L} = 43m$$

Phase diagrams of the crystals undergoing phase transitions induced by irreducible representations with the $$\mathcal{L} = 43m$$ symmetry group have been constructed within the phenomenological theory of the second-order phase transitions. The equations of state of all phases induced by symmetry have been received, and the general conditions of their thermodynamic stability have been formulated. For the fourth-order expansion of thermodynamic potential in terms of the order parameter components, exhaustive results have been obtained: equations of phases stability boundaries and all possible types of phase diagrams have been found analytically. For the eighth-order expansion of the potential the variety of phase diagrams has been analyzed using computer calculations. For the first time the possibility of existence of phase diagrams described by this potential and containing one or several triple points, three-phase and four-phase regions is shown. Examples of the crystals undergoing phase transitions with considered symmetry of the order parameter have been given.

Phenomenological theory of multicritical points splitting in the phase diagrams. The case of Landau’s potentials with ℒ = 3m

The detailed research of phase diagrams is carried out for Landau’s thermodynamic potential with ℒ = 3m symmetry group. All possible types of phase diagrams for the sixth-order expansion of thermodynamic potential in terms of the order parameter components are received. Realization conditions for each type of phase diagrams are established, conditions of the multicritical point splitting are studied, and the equations for calculation of tricritical and triple points coordinates are obtained. Surfaces of order parameter components as functions of thermodynamic potential parameters are designed. Qualitative correspondence of phase diagrams theoretical calculations results and an experimental phase diagram is noted at the example of spinel structural type.

PAULI EXCLUSION PRINCIPLE

In terms of an exact equation for the thermodynamic potential due to interaction between two particles and based on Green's function method; we have derived the Landau expansion of the thermodynamic potentials in terms of the variation of the quasiparticle distribution function. We have also derived the expansion of the thermodynamic potential in terms of the variation of an exact single particle (not quasiparticles), this derivations lead to the relationship between the interaction function for two quasiparticles and the interaction energy between two particles as shown. Further, in terms of the four-point vertex part we are led to the Pauli exclusion principle.

Superconductivity in a Fermi liquid from repulsive interactions: The role of electron–phonon interaction

Based on our previously proven theorem that the interaction between a pair of quasiparticles in the normal Fermi liquid has an opposite sign to the interaction between particles, we consider pair correlation between a pair of quasiparticles when the interaction between particles is repulsive. For the convenience of statements, we have presented in this article once again the proof of the theorem in terms of an exact equation for the thermodynamic potential due to interaction between particles and based on the Green’s function method. Further, we have derived the Landau expansion of the thermodynamic potentials in terms of the variation of the quasiparticle distribution function. We have also derived the expansion of the thermodynamic potential in terms of the variation of an exact single particle (not quasiparticles), these derivations lead to the relationship between the interaction function for two quasiparticles and the interaction energy between two particles as shown.According to the proven theorem the interaction between a pair of quasiparticles is attractive in this case, the pairing – Cooper’s pairing between a pair of quasiparticles is possible. We solve the Bethe–Salpeter type equation for paring of two quasiparticles when both interactions – the Coulomb repulsive and electron–phonon interaction are present. We show that the electron–phonon interaction, in fact, leads to the pair breaking effect, in contrast to the common belief that electron–phonon interaction is the main mechanism for Cooper’s pair formation. We have calculated the transition temperature and the isotope effect on the transition temperature in terms of the Bethe–Salpeter equation.

Nonequilibrium Phase Transitions and a Nonequilibrium Critical Point from Anti–de Sitter Space and Conformal Field Theory Correspondence

We find novel phase transitions and critical phenomena that occur only outside the linear-response regime of current-driven nonequilibrium states. We consider the strongly interacting (3+1)-dimensional N = 4 large-N(c) SU(N(c)) supersymmetric Yang-Mills theory with a single flavor of fundamental N = 2 hypermultiplet as a microscopic theory. We compute its nonlinear nonballistic quark-charge conductivity by using the AdS/CFT correspondence. We find that the system exhibits a novel nonequilibrium first-order phase transition where the conductivity jumps and the sign of the differential conductivity flips at finite current density. A nonequilibrium critical point is discovered at the end point of the first-order regime. We propose a nonequilibrium steady-state analogue of thermodynamic potential in terms of the gravity-dual theory in order to define the transition point. Nonequilibrium analogues of critical exponents are proposed as well. The critical behavior of the conductivity is numerically confirmed on the basis of these proposals. The present work provides a new example of nonequilibrium phase transitions and nonequilibrium critical points.

On Maxwell’s Relations of Thermodynamics for Polymeric Liquids away from Equilibrium

To describe complex systems deeply in the nonlinear regime, advanced formulations of nonequilibrium thermodynamics such as the extended irreversible thermodynamics (EIT), the matrix model, the generalized bracket formalism, and the GENERIC (=general equation for the nonequilibrium reversible−irreversible coupling) formalism consider generalized versions of thermodynamic potentials in terms of a few, well-defined position-dependent state variables (defining the system at a coarse-grained level). Straightforward statistical mechanics considerations then imply a set of equalities for its second derivatives with respect to the corresponding state variables, typically known as Maxwell’s relations. We provide here direct numerical estimates of these relations from detailed atomistic Monte Carlo (MC) simulations of an unentangled polymeric melt coarse-grained to the level of the chain conformation tensor, under both weak and strong flows. We also report results for the nonequilbrium (i.e., relative to the quiesc...

Diagrammatic theory for the Anderson impurity model: Stationary property of the thermodynamic potential

We propose a diagrammatic theory around the atomic limit for the normal state of the Anderson impurity model. The new diagram method is based on Wick’s theorem for conduction electrons and a generalized Wick’s theorem for strongly correlated impurity electrons, which coincides with the definition of the Kubo cumulant. We prove a linked-cluster theorem for the mean of the evolution operator and obtain Dysontype equations for the one-particle propagators. The main element in these equations is the impurity electron correlation function, which contains the spin, charge, and pairing fluctuations of the system. We express the system thermodynamic potential in terms of the full propagator of conduction electrons and the correlation function. We establish that the thermodynamic potential is stationary under changes of the correlation function.

On thermodynamic potentials in linear thermoelasticity

The four thermodynamic potentials, the internal energy u= u( ϵ ij , s), the Helmholtz free energy f= f( ϵ ij , T), the Gibbs energy g= g( σ ij , T) and the enthalpy h= h( σ ij , s) are derived, independently of each other, by using the Duhamel–Neumann extension of Hooke's law and an assumed linear dependence of the specific heat on temperature. A systematic procedure is then presented to express all thermodynamic potentials in terms of four possible pairs of independent state variables. This procedure circumvents a tedious transition from one potential to another, based on the formal change of variables, and inversions of the stress–strain and entropy–temperature relations. The general results are applied to uniaxial loading paths under isothermal, adiabatic, constant stress, and constant strain conditions. An interplay of adiabatic and isothermal elastic constants in the expressions for exchanged heat along certain thermodynamic paths is indicated.

Use of legendre transforms in chemical thermodynamics: International Union of Pure and Applied Chemistry, Physical Chemistry Division, Commission on Thermodynamics

The fundamental equation of thermodynamics for the internal energy U may include terms for various types of work and involves only differentials of extensive variables. The fundamental equation for U yields intensive variables as partial derivatives of the internal energy with respect to other extensive properties. In addition to the terms from the combined first and second laws for a system involving PV work, the fundamental equation for the internal energy may involve terms for chemical work, gravitational work, work of electric transport, elongation work, surface work, work of electric and magnetic polarization, and other kinds of work. Fundamental equations for other thermodynamic potentials can be obtained by use of Legendre transforms that define these other thermodynamic potentials in terms of U minus conjugate pairs of intensive and extensive variables involved in one or more work terms. The independent variables represented by differentials in a fundamental equation are referred to as natural variables. The natural variables of a thermodynamic potential are important because if a thermodynamic potential can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be obtained by taking partial derivatives of the thermodynamic potential with respect to the natural variables. The natural variables are also important because they are held constant in the criterion for spontaneous change and equilibrium based on that thermodynamic potential. By use of Legendre transforms any desired set of natural variables can be obtained. The enthalpy H, Helmholtz energy A, and Gibbs energy G are defined by Legendre transforms that introduce P, T, and P and T together as natural variables, respectively. Further Legendre transforms can be used to introduce the chemical potential of any species, the gravitational potential, the electric potentials of phases, surface tension, force of elongation, electric field strength, magnetic field strength, and other intensive variables as natural variables. The large number of transformed thermodynamic potentials that can be defined raises serious nomenclature problems. Some of the transforms of the internal energy can also be regarded as transforms of H, A, or G. Since transforms of U, H, A, and G are useful, they can be referred to as the transformed internal energy U ′, transformed enthalpy H ′, transformed Helmholtz energy A ′, and transformed Gibbs energy G ′ in a context where it is clear what additional intensive natural variables have been introduced. The chemical potential μ i of a species is an especially important intensive property because its value is uniform throughout a multi-phase system at equilibrium even though the phases may be different states of matter or be at different pressures, gravitational potentials, or electric potentials. When the chemical potential of a species is held constant, a Legendre transform can be used to define a transformed Gibbs energy, which is minimized at equilibrium at a specified chemical potential of that species. For example, transformed chemical potentials are useful in biochemistry because it is convenient to use pH as an independent variable. Recommendations are made to clarify the use of transformed thermodynamic potentials of systems and transformed chemical potentials of species.

Use of Legendre transforms in chemical thermodynamics (IUPAC Technical Report)

Abstract The fundamental equation of thermodynamics for the internal energy U may include terms for various types of work and involves only differentials of extensive variables. The fundamental equation for U yields intensive variables as partial derivatives of the internal energy with respect to other extensive properties. In addition to the terms from the combined first and second laws for a system involving PV work, the fundamental equation for the internal energy may involve terms for chemical work, gravitational work, work of electric transport, elongation work, surface work, work of electric and magnetic polarization, and other kinds of work. Fundamental equations for other thermodynamic potentials can be obtained by use of Legendre transforms that define these other thermodynamic potentials in terms of U minus conjugate pairs of intensive and extensive variables involved in one or more work terms. The independent variables represented by differentials in a fundamental equation are referred to as natural variables. The natural variables of a thermodynamic potential are important because if a thermodynamic potential can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be obtained by taking partial derivatives of the thermodynamic potential with respect to the natural variables. The natural variables are also important because they are held constant in the criterion for spontaneous change and equilibrium based on that thermodynamic potential. By use of Legendre transforms any desired set of natural variables can be obtained. The enthalpy H, Helmholtz energy A, and Gibbs energy G are defined by Legendre transforms that introduce P, T, and P and T together as natural variables, respectively. Further Legendre transforms can be used to introduce the chemical potential of any species, the gravitational potential, the electric potentials of phases, surface tension, force of elongation, electric field strength, magnetic field strength, and other intensive variables as natural variables. The large number of transformed thermodynamic potentials that can be defined raises serious nomenclature problems. Some of the transforms of the internal energy can also be regarded as transforms of H, A, or G. Since transforms of U, H, A, and G are useful, they can be referred to as the transformed internal energy U', transformed enthalpy H', transformed Helmholtz energy A', and transformed Gibbs energy G' in a context where it is clear what additional intensive natural variables have been introduced. The chemical potential μ i of a species is an especially important intensive property because its value is uniform throughout a multiphase system at equilibrium even though the phases may be different states of matter or be at different pressures, gravitational potentials, or electric potentials. When the chemical potential of a species is held constant, a Legendre transform can be used to define a transformed Gibbs energy, which is minimized at equilibrium at a specified chemical potential of that species. For example, transformed chemical potentials are useful in biochemistry because it is convenient to use pH as an independent variable. Recommendations are made to clarify the use of transformed thermodynamic potentials of systems and transformed chemical potentials of species.

Hadronic modes in the quark plasma with an internal symmetry

We show that requiring the quark partition function to be color singlet of SU(3) color gauge group leads to reordering the thermodynamic potential in terms of the colorless multi-quark modes ($q{\bar q}, qqq, {\bar q}{\bar q}{\bar q}, ...$) at any given temperature. In accord with the "preconfinement" property of QCD, under a suitable confining mechanism, these could evolve into color singlet hadrons/baryons at low temperatures. At fairly high temperatures these multi-quark color singlet structures propagate in the plasma like hadronic modes, just as in the more familiar low temperature phase. This suggests that there exists a strong correlation in the plasma at all temperatures which is in conformity with lattice results.

Self-Consistent Approximations in Relativistic Plasmas: Quasiparticle Analysis of the Thermodynamic Properties

We generalize the concept of conserving, Phi-derivable, approximations to relativistic field theories. Treating the interaction field as a dynamical degree of freedom, we derive the thermodynamic potential in terms of fully dressed propagators, an approach which allows us to resolve the entropy of a relativistic plasma into contributions from its interacting elementary excitations. We illustrate the derivation for a hot relativistic system governed by electromagnetic interactions.

Thermodynamics of a Quark Plasma Beyond the Mean Field: A Generalized Beth-Uhlenbeck Approach

The thermodynamics of a quark plasma is studied within the two-flavour Nambu-Jona-Lasinio model. After reviewing the results which are obtained in the mean-field (Hartree) approximation, including chiral and quark number susceptibilities, we treat the contributions to the thermodynamic potential which arise to order 1/ N c , where N c is the number of colours. These contributions involve the π and σ mesons and the scattering properties of quarks. We express the thermodynamic potential in terms of the bound states and scattering phase shifts of a quark-antiquark pair. This result resembles the Beth-Uhlenbeck formula for the second virial coefficient for a gas of non-relativistic particles. Our results hold also for the relativistic case and include the influence of temperature and chemical potential on the S-matrix that arises as a consequence of these dependences in the collective excitations that are formed.

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.