Entropy density local

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. 

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 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... 

The local entropy density, j(r, t), was used frequently in Chapter 10. The total entropy S of the fluid contained in a region of volume V is then... 

This is the local equation for the entropy density. It is of fundamental importance in what follows. The quantity a, the local entropy production, is the entropy produced irreversibly per unit time per unit volume and is analogous to the property a a defined prior to Eq. (10.3.4). The quantity Js is the entropy flux and V Js represents the rate of change of the local entropy due to an inflow of entropy from neighboring regions of the fluid. 

One of the important postulates of irreversible thermodynamics is the postulate of local equilibrium discussed in Chapter 10. Accordingly, the local rate of change of the entropy density is... 

If the fluctuation in the local volume fraction of component 2 in a binary mixture is denoted by dcp the fluctuation in entropy density is... 

In continuous systems, the local increase in entropy can be defined by using the entropy density s(x,t), which is the entropy per unit volume. The total entropy change is ds = dgS + diS and results from the flow of entropy due to exchanges with surroundings (dgs) and from the changes inside the system (dis). Therefore, the local entropy production can be defined by... 

From the internal energy density u and entropy density s, we obtain the local variables of... 

Fluctuations of such extent involve collective motion of a great number of molecules and therefore can be described by the laws of macroscopic physics, namely, thermodynamics and hydrodynamics. Thus, small parts of the system where fluctuations of the macroscopic values manifest themselves in the properties of scattered light (the Fourier transform) contain rather many molecules that enables one to speak of local values of such macroscopic terms as entropy, enthalpy, and pressure. Every point f corresponding to a small space element in liquid at an instant i can be ascribed some values of entropy density a(r,i), of molecule number density p(r,l), of energy e(f,l), of pressure P(r,i), and of the dielectric constant e(f,t). 

The local increase of entropy in continuous systems can be defined by using the entropy density s x, t). As was the case for the total entropy, ds = d s + df,s, with diS > 0. We define local entropy production as... 

The above definitions can be extended to other local thermodynamic functions, such as the pressure, entropy density, specific heats, etc. Thus all intensive thermodynamic variables are, by definition, to be functionally related to the mean energy and number densities in the same way as at equilibrimn. It then follows that the various thermodynamic identities derived by equilibrium arguments are still valid for the non-equilibrium state. (It is worth emphasizing that this procedure is quite general, and is not restricted to linear processes.)... 

We take -W and K in one form of the theory, or -W and K in the other. If m is a constant, then the surface excesses of matter and entropy are given correctly by (3.26), and Ae local densities of matter and entropy may be identified simply as the functions p(z) and t)(z) that minimize the variational integral in (3.18). But Ae local energy clensity is not then correctly identified as the 0(z) that minimizes Ae variational inte al in (3.22) it is rather (z)+ mp (z), because of the term -8X/fl(-l/T0 = nip (z) in the integrand in (3.28). This circumstance, in which Acre is no density-gradient term in Ae local entropy density while there is one in the local energy density, is typical though if m were... 

Bongiorno et al. and Abraham " have also proposed gradient expansions that invoke explicitly the original suggestion of van der Waals that the entropy density of an inhomogeneous fluid is a function only of the local density (cf. 8 3.4). This leads them (if we ignore any dependence of g on p, on which they disagree) to a further approximation for the coefficient, namely... 

This simple idea lies behind van der Waals s equation of state, whidi was extended to the treatment of the surface of a liquid by Hill. He recognized the value of dividing u(r) into a repulsive part Uo(r) and an attractive part Ui(r), but used once again the simplifiration of van der Waals of assuming that the entropy density of the reference (or repulsive) system at i is determined solely the local den p(t). 

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