Local equilibrium defined

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is... 

The above estimates suggest that, with the scaling (1.7) valid, it takes about time td for an ionic system to restore local electro-neutrality on the macroscopic length scale and to reach local equilibrium. By local equilibrium we merely imply a state with the normalized ionic fluxes ji, defined by... 

It has long been known that defect thermodynamics provides correct answers if the (local) equilibrium conditions between SE and chemical components of the crystal are correctly formulated, that is, if in addition to the conservation of chemical species the balances of sites and charges are properly taken into account. The correct use of these balances, however, is equivalent to the introduction of so-called building elements ( Bauelemente ) [W. Schottky (1958)]. These are properly defined in the next section and are the main content of it. It will be shown that these building units possess real thermodynamic potentials since they can be added to or removed from the crystal without violating structural and electroneutrality constraints, that is, without violating the site or charge balance of the crystal [see, for example, M. Martin et al. (1988)]. 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

Note that we have assumed the vacancies to be ideally diluted. We can then introduce a perturbation of the planar boundary, z = A +0(x,y,t), and define °(x,y) = Cartesian coordinates perpendicular to z. In this way, the morphological stability becomes a two-dimensional problem. Since we also assume that local equilibrium prevails at both interfaces (surfaces), the boundary conditions are... 

In order to find the eigenvalues of L, let us define a projection operator P. This operator projects onto the local equilibrium states and is written as... 

Now we define Q, which is the projection operator that rejects the local equilibrium states ... 

The distribution function/(v) is Maxwellian at local equilibrium, and is defined by... 

Dissipative structures can sustain long-range correlations. The temperature and chemical potential are well defined with the assumption of local equilibrium, and the stationary probability distribution is obtained in the eikonal approximation so the fluctuation-dissipation relation for a chemical system with one variable is... 

Linear nonequilibrium thermodynamics has some fundamental limitations (i) it does not incorporate mechanisms into its formulation, nor does it provide values for the phenomenological coefficients, and (ii) it is based on the local equilibrium hypothesis, and therefore it is confined to systems in the vicinity of equilibrium. Also, properties not needed or defined in equilibrium may influence the thermodynamic relations in nonequilibrium situations. For example, the density may depend on the shearing rate in addition to temperature and pressure. The local equilibrium hypothesis holds only for linear phenomenological relations, low frequencies, and long wavelengths, which makes the application of the linear nonequilibrium thermodynamics theory limited for chemical reactions. In the following sections, some of the attempts that have been made to overcome these limitations are summarized. 

Ql is the submatrix of corresponding to the n - n c fastest variables, and so the slow manifold is defined by assuming that motion in the direction of the Schur vectors associated with the fast variables is zero. This means that the system is in local equilibrium with respect to its fastest time-scales. 

Non-equilibrium thermodynamics was founded by Onsager. The theory was further elaborated by de Groot and Mazur and Prigogine. The theory is based on the hypothesis of local equilibrium a volume element in a non-equilibrium system is in local equilibrium when the normal thermodynamic relations apply to the element. Evidence is emerging that show that many systems of interest in the process industry are in local equilibrium by this criterion. " Onsager prescribed that each flux be connected to its conjugate force via the extensive variable that defines the flux. - ... 

Let the n 1 three-dimensional vectors zSi (i = 1,1) be the mass-weighted Jacobi vectors for a reference molecular configuration. The reference configuration is usually set to be a local equilibrium structure of the molecule oriented to a certain orientation. The Eckart subspace is defined as a (3n — 6)-dimensional subspace in the (3n — 3)-dimensional translation-reduced configuration space, which is parameterized by Jacobi vectors pf (/ = 1,..., m - 1) with three additional constraint conditions called the Eckart conditions,... 

We define two states A, the state of the reactants, and B, the state of the product. Both states are assumed to be in local equilibrium the transition time between the two states is considerably longer than relaxation times to local equilibrium within any of the states. The reaction progress is described by the conditional probability that the system will be at state B after time t, given that it was at the reactant state, A, at time t = 0... 

The purpose of the study of irreversible thermodynamics is to extend classical thermodynamics to include systems in which irreversible processes (e.g., diffusion and heat transfer) are taking place. Such an extension is made possible by assuming that for systems not too far from equilibrium the postulate of local equilibrium applies Departures from local equilibrium are sufficiently small that all thermodynamic state quantities may be defined locally by the same relations as for systems at equilibrium. ... 

Assuming that these relations are valid for a system at local equilibrium, the differential d may be substituted by the V-operator. Hence, from (2.460) and (2.463) we may define the following short notations ... 

The idea of an associated equilibrium state is thus quite general it is a very powerful concept that can be used more accurately and more confidently than the concept of local equilibrium. Confidence comes fi om two sources first we can see that the real nonequilibrium state and the imagined equilibrium state are closely similar, and second, we can bring the exact differences to light—even, in some circumstances, express them quantitatively and demonstrate their smallness. It is the second aspect that brings confidence, and allows us to use associated equilibrium states to define potentials and predict material responses, even in situations where the exact magnitude of the small differences cannot actually be worked out in numbers. 

Since Eq, (8.50) holds only under equilibrium conditions, its use for nonequilibrium problems may be justified only under the assumption of local equilibrium. As was done with the other properties, a total emissivity (depending on the body temperature) can be defined by integrating over the entire wavelength range. 

The ideas described previously for understanding film morphology in terms of the local equilibrium and in terms of the surface tension are useful, but film growth occurs far from equilibrium (ex vi termini). Thus, kinetic processes control the details of film growth and the final film morphology. According to the paper by Burton, Cabrera, and Frank [13], the kinetic rates and processes are controlled by the thermodynamic driving force Ap, defined as the positive difference between the chemical potential of a molecule in the vapor phase and that in the crystal phase. 

It is assumed that local equilibrium prevails at the raterface betwean phases, where the compositions are % and X%. If transfer of A takes place from the lighter liquid to the heavier liquid, the individual coefficients kT and kx for the light and heavy phases, respectively, are defined as follows ... 

To draw a clear picture of the physical mechanism of the electrostatic retardation effect any existence of additional non-electrostatic barriers, such as discussed in Section 4.4, or other mechanisms influencing the transfer of surfactant molecules between the subsurface and the interface, are avoided. Thus, the electrostatic retardation effect is considered under the assumption of local equilibrium between the subsurface, defined as the bulk layer adjacent to the interface (cf. Chapter 2), and the interface. 

Local equilibrium between fluid and rock. A common assumption of stable isotope transport is that of local equilibrium (Baumgartner and Rumble 1988, Bowman et al. 1994 Gerdes et al. 1995 a, b). Equilibrium is assumed for a small volume of the transport system. Within this volume, all phases are in equilibrium at all times (Thompson 1959). A molar equilibrium constant, Kd can be defined to describe the thermodynamic stable isotope equilibrium between two phases or species. The stable isotope concentration of a mineral (s) can be related to that of a fluid species through the equilibrium constant, Kq ... 

Even though nonequilibrium in this larger sense, the solid surface can come to a local equilibrium with molecules of an adjacent phase, in terms of chemical and van der Waals interactions, exchange of thermal energy, etc. The difference y yo - y can thus be viewed as arising from the microscopic or local interactions between the molecules in the solid and vapor and the solid and liquid phases. On this basis, it is only indirectly and may be only incidentally related to the surface free energy of the pure solid. Indeed, it is different to define operationally the surface free energy of a nonequilibrium solid surface. 

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