Entropy density

Here 0 is the Heaviside function. The projection operator formalism must be carried out in matrix from and in this connection it is useful to define the orthogonal set of variables, k,uk,5k > where the entropy density is sk = ek — CvTrik with Cv the specific heat. In terms of these variables the linearized hydrodynamic equations take the form... 

Zheludev, A., Papoular, V., Ressouche, E. and Schweizer, J. (1995) A non-uniform reference model for maximum-entropy density reconstruction from diffraction data, Acta Cryst., A51, 450-455. 

Molecular Distribution Functions for the Entropy Density in an Infinite System. 

The meson fields op, too and po are found by solving a set of equations self-consistently as shown in [11], Also expressions for the energy density, pressure and the entropy density can be found there. The empirical values of the binding energy of nuclear matter and nuclear matter density are reproduced using the above mentioned parameterization. The nuclear matter EOS can be found expressing the chemical potentials as functions of temperature, baryon density... 

A basic question concerns whether the configurational entropy should be defined as an entropy per unit mass (or molar entropy) or as an entropy per unit volume (a site entropy or an entropy density ) [22]. The majority of experimental studies [15, 49] concerning the validity of the AG theory use the molar entropy, while Binder and co-workers [55] have employed the site entropy in their computational studies of the applicability of AG theory. In fact, our calculations indicate [22] that the configurational entropy per unit volume... 

The site entropy is thus a sensible candidate for describing fluid relaxation outside the immediate vicinity of the glass transition. In a more precise language, is actually an entropy density, and the maximum in Sc T) derives from an interplay between changes in the entropy and fluid density as the temperature is varied. Explicit calculations demonstrate that the maximum in Sc T) disappears in the limit of an incompressible fluid, which is physically achieved in the limit of infinite pressure. The pressure dependence of Sc T) is described in Section X, where it is found that the maximum in Sc T) becomes progressively shallower and 7a becomes larger with increasing pressure. 

The configurational entropy of the AG model is identified by us with the entropy per lattice site ic of the LCT, since the use of the entropy density is the... 

The local rate of entropy-density creation is denoted by. The total rate of entropy creation in a volume V is fv dV. For an isolated system, dS/dt = fv dV. 

From Eq. 2.4, the time derivative of entropy density in a cell is... 

S, 5, a Entropy, entropy per mole (or particle), entropy density JK 1, JK mor1, JK-1m-3... 

As discussed in Section II, the five hydrodynamic variables are the number, energy, and momenta densities. Instead of considering the number and energy density, it turns out to be more convenient to consider the entropy density, which can be expressed as a combination of energy and number density. The second variable is now constructed so that it is orthogonal to the entropy density. To explicitly write the states we start with writing a, which is related to the entropy density, s0p>... 

In contrast, the barrier for cooperative rearrangements in the Adam-Gibbs model is inversely proportional to configurational entropy density [41]... 

Because ASPEN is to be used with coal conversion processes, its streams can be designated to carry an arbitrary number of solids or solid phases. This is done by specifying any number of substreams. In fact, the conventional vapor/liquid stream is normally assumed as a substream and solids can comprise other substreams. For the conventional vapor/liquid substream, process data is carried on component molar flows, total molar flow, temperature pressure, specific enthalpy, specific entropy, density, molar vapor fraction, molar liquid fraction, and molecular weight. For solid substreams, which are called "non-conventional substreams," the characterizing data is not as deterministic. The information associated with these substreams is called "attributes". Such attributes may be particle size distribution, ultimate and proximate analyses, or other material specific information. Another type of substream is an "informa-... 

In nonequilibrium systems, the intensive properties of temperature, pressure, and chemical potential are not uniform. However, they all are defined locally in an elemental volume with a sufficient number of molecules for the principles of thermodynamics to be applicable. For example, in a region A , we can define the densities of thermodynamic properties such as energy and entropy at local temperature. The energy density, the entropy density, and the amount of matter are expressed by uk(T, Nk), s T, Nk), and Nk, respectively. The total energy U, the total entropy S, and the total number of moles N of the system are determined by the following volume integrals ... 

Since the temperature is not uniform for the whole system, the total entropy is not a function of the other extensive properties of U, V, and N. However, with the local temperature, the entropy of a nonequilibrium system is defined in terms of an entropy density, sk. 

The reversibility of molecular behavior gives rise to a kind of symmetry in which the transport processes are coupled to each other. Although a thermodynamic system as a whole may not be in equilibrium, the local states may be in local thermodynamic equilibrium all intensive thermodynamic variables become functions of position and time. The definition of energy and entropy in nonequilibrium systems can be expressed in terms of energy and entropy densities u(T,Nk) and s(T,Nk), which are the functions of the temperature field T(x) and the mole number density Y(x) these densities can be measured. The total energy and entropy of the system is obtained by the following integrations... 

The total entropy of a system is related to the local entropy density sv... 

Equation (3.200) is the expression for a nonconservative change in local entropy density, and allows the determination of the entropy production from the total change in entropy and the evaluation of the dependence of on flows and forces. 

At stationary state, the local entropy density must remain constant because of the condition dsjdt = 0. However, the divergence of entropy flow does not vanish, and we obtain... 

If we consider the change of local entropy of a system at steady state ds/dt = 0, the local entropy density must remain constant because external and internal parameters do not change with time. However, the divergence of entropy flow does not vanish div J, = . Therefore, the entropy produced at any point of a system must be removed or transferred by a flow of entropy taking place at that point. A steady state cannot be maintained in an adiabatic system, since the entropy produced by irreversible processes cannot be removed because no entropy flow is exchanged with the environment. For an adiabatic system, equilibrium state is the only time-invariant state. 

Equation (A.26) is used to find an expression for the time variation of the local entropy density (Haase. 1990). 

The total entropy production as needed in Equation (A.22) is related to the entropy density as follows ... 

Consider a tall vertical cylinder of height h and constant cross-section A containing a fixed number of gas molecules of various chemical species. The new feature of this system is the nonuniform distribution of the gas in the cylinder under the action of the gravitational field. Accordingly, as shown in Fig. 5.1.1, we divide the cylinder into volume elements (subsystems) of constant cross-section A and vertical extension dz. The matter within each element is characterized by an energy density, u, an entropy density, s, and concentration cA... 

The remaining terms on the right of (6.2.13) must represent source terms if Eq. (6.2.13) is to be interpreted as an entropy balance equation d(p"5)/dt - - V JS + 9. Having thus identified - V Jg we can express 9 as the rate of entropy density generation locally as follows ... 

Note that we have succeeded in setting up a continuity equation for entropy density, the Second Law, in local form,... 

The form of (6.2.15) and (6.2.18) is highly significant. In each case the rate of local entropy density generation, due to irreversible processes occurring totally within a local volume element, may be written as a sum of terms of the general form i wherein the Ji represent either general... 

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