Entropy production positive definiteness

Nonequilibrium thermodynamics estimates the rate of entropy production for a process. This estimation is based on the positive and definite entropy due to irreversible processes and of Gibbs relation... 

It is well-known from thermodynamics that the entropy production or better the related quantity, the dissipation, II = T5S/5t is a positive-definite function being related with the rates and forces via182,183... 

The positive definiteness requirement of allows us to derive certain restrictions on the values of the If we substitute the GMS relation for into Eq. 2.3.15 we obtain a very neat and compact expression for the rate of entropy production due to diffusion... 

The entropy production s, being a positive definite, equation (5) gives the condition of a minimum. It is a mathematical form of Prigogine s principle of minimum entropy production according to which at the steady state, all the flows corresponding to the unrestricted forces vanish. 

The results of calculating entropy production from head values ranging between elevations of 40 meters to about sea level are shown on Figure 4. Note that the high point on the potentiometric surface is designated as having a zero entropy level. This is the input boundary of the system, and by our definition, the entropy of the water attributed to position is zero at this point. Therefore in order to depict the entropy increase attributed to downgradient flow, the equation was modified to... 

The degree of completeness of the analogies between Eqs. (331) and (49) is quite remarkable oll and Jfoll are each symmetric, positive-definite forms, as are their direct submatrices too. The positive-definiteness of o stems from the positivity of the rate of irreversible entropy production. In contrast to the proof of the symmetry of the hydrodynamic resistance matrix (B22), the corresponding proof of the symmetry of the diffusion matrix is trivial. The latter may be taken to be symmetric by definition since its antisymmetric part gives rise to no observable macroscopic physical consequence. 

Since entropy production a is positive definite, it can be easily shown that... 

Resistive nodes In this first category of power continuous nodes the power continuity is hidden, as the power entering the resistive ports is converted into thermal power and not explicitly represented by a thermal port, such that energy seems to be dissipated, but careful use of concepts shows that only free energy can be dissipated and that the use of power as a flow of free energy corresponds to an implicit assumption, viz., that the temperature at the thermal port is constant or its fluctuations are slow with respect to the fluctuations of interest, such that the temperature can be considered constant. For a resistive node a semi-positive definite scalar potential function ( entropy production function or dissipation function ) of the independent variables exists that generates its constitutive relations. A resistive node has at least one port. Its node label is R. A modulated resistive node has node label MR. A resistive node or resistor is sometimes called a dissipative node or dissipator. 

Show with short calculus, that in the chapter of 2.2.2. the first variation of the global entropy production is zero, the second one is positive definite. 

We show that in the case of small perturbations this stability criteria is identical with the stability criteria deduced from the positive definite property of the second variation of the global entropy production. 

A dissipative system could be energy converter, when not all the terms of the entropy production are positive, their sum is positive definite only. The problem will be studied in a dissifjative system having two thermodynamic forces and currents and in case when the dissipation potentials are homogeneous Euler s function of degree k. In this case the currents and forces are connected with the following constitutive equations (see Appendix 7.6.)... 

In general, the diagonal elements of a positive definite matrix must be positive. In addition, a necessary and sufficient condition for a matrix Lg to be positive definite is that its determinant and all the determinants of lower dimension obtained by deleting one or more rows and columns must be positive. Thus, according to the Second Law, the proper coefficients L k should be positive the cross coefficients, (i 7 k), can have either sign. Furthermore, as we shall see in the next section, the elements Ljk also obey the Onsager reciprocal relations Ljk = Lkj. The positivity of entropy production and the Onsager relations form the foundation for linear nonequilibrium thermodynamics. 

If the transport-kinetic model were compatible with thermodynamics, the entropy production would be a suitable positive definite function. In this case, however, it would be a very tedious task to establish bounds of the type (2.8.11) —(2.8.13). In the case of the simplified transport model, Eqs.(2.4.1) and (2.4.2), a positive definite quantity a satisfying Eqs.(2.8.21) to (2.8.22) cannot be constructed for an arbitrary kinetic model. In many special cases, the construction is possible although a need not have thermodynamic significance. This will be illustrated in the following examples [17]. 

The noise with information entropy H( Y X) is the integral part of the definition of the transfer information process. It is not generated by a positive production of the heat AQo f > 0 in the working medium CP... 

---