Onsager coefficient matrix

Comparison with Eq. 3.2.5 shows that the Fick matrix [ >] is related to the Onsager coefficient matrix [//] by... 

T is the free energy fiinctional, for which one can use equation (A3.3.52). The summation above corresponds to both the sum over the semi-macroscopic variables and an integration over the spatial variableThe mobility matrix consists of a synnnetric dissipative part and an antisyimnetric non-dissipative part. The syimnetric part corresponds to a set of generalized Onsager coefficients. 

The Onsager coefficients (L),y are here evaluated in terms of the real symmetric matrix... 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

The elements of the matrix L are the Onsager coefficients. We introduce a matrix of decay rates F,... 

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

According to the Onsager assumption, the square matrix of the phenomenological coefficients... 

Equation (4.11) expresses the central Onsager theorem. It states the symmetry of the phenomenological coefficients (the L matrix) in the absence of magnetic fields. The foundation of this theorem is discussed elsewhere [J. H. Kreuzer (1981) S. R. de Groot, P. Mazur (1962)]. 

In a linear theory, the kinetic coefficients Ly are independent of the forces. They are, however, functions of the thermodynamic variables. In view of the Onsager relations, not only is the L matrix of the transport coefficients symmetric, but the transformed matrix is symmetric as well if the new fluxes are linearly related to the original ones. This also means that the new Ly (i st j) contain diagonal components of the original set. 

The matrix coefficients in (8.21) depend on the thermodynamic variables, which, in the case under discussion, are pressure p or density p (we can chose any of them, so as there exist an equation of state, connecting these variables), temperature T and internal variables a. The coefficients can be expanded into series near equilibrium values of internal variables. Zero-order terms of expansions of the components of the matrices in a series of powers of the internal variables are connected due to the Onsager principle (Landau and Lifshitz 1969) by some relations... 

Onsager s reciprocal relations state that, provided a proper choice is made for the flows and forces, the matrix of phenomenological coefficients is symmetrical. These relations are proved to be an implication of the property of microscopic reversibility , which is the symmetry of all mechanical equations of motion of individual particles with respect to time t. The Onsager reciprocal relations are the results of the global gauge symmetries of the Lagrangian, which is related to the entropy of the system considered. This means that the results in general are valid for an arbitrary process. 

By the Onsager reciprocal relations, the matrix of phenomenological coefficients is symmetric, LXq = LqX. Since the dissipation function is positive, the phenomenological coefficients must satisfy the inequalities... 

The matrix of the phenomenological coefficients must be positive definite for example, for a two-flow system, we have L0 > 0, Ip >0, and Z/.p Z,pZpo > 0.1,0 shows the influence of substrate availability on oxygen consumption (flow), and Ip is the feedback of the phosphate potential on ATP production (flow). The cross-coupling coefficient Iop shows the phosphate influence on oxygen flow, while Zpo shows the substrate dependency of ATP production. Experiments show that Onsagers s reciprocal relations hold for oxidative phosphorylation, and we have Iop = Zpo. 

In other words, the matrix giving the phenomenological coefficients for the system is symmetric. Another important part of this theory is that one can estimate the entropy generated in an irreversible process. Onsager showed that the local rate of entropy production per unit volume is... 

The Onsager reciprocal relation for the 2 x 2 matrix of phenomenological coupling coefficients is... 

In this framework, the intensity of an entropy source is represented by a quadratic form of thermodynamic forces. The corresponding phenomenological coefficients form a matrix with remarkable properties. These properties, formulated as the Onsager reciprocity theorem, allow to reduce the number of independent quantities and to find relations between various physical effects. 

The matrix in Eq. 10 is symmetrical according to the Onsager reciprocity law [11], which means Ls = Lp. The linear coefficients can be identified with known material properties... 

Starting from the entropy produced at the liquid surface the transport matrix for the heat and mass transfer is derived. Onsager symmetry and the role of the evaporation coefficient, the condensation coefficient and the energy accommodation coefficient are discussed. 

The matrix of phenomenological coefficients is symmetrical as the Onsager s reciprocal relations hold... 

They phenomenological coefficients in the matrix [L] obey Onsager s reciprocal rules, and hence there would be six instead of nine phenomenological coefficients to be determined. 

Here the first equation is the usual Fourier law, the second relates the viscous pressure tensor to the internal variable W, and the last is the evolution of the internal variable. The matrix of the transport coefficients Ly is positive definite with L q = —Lq due to Onsager-Casimir reciprocal rules. 

Then a set of linear rate equations may be written in terms of a matrix of phenomenological coefficients which satisfy the Onsager relation (Onsager, 1931) ... 

The previous results justify the form of equation [4.11] as coupled phenomena have a specific role and their entity depends on the value of the mixed coefficient However, a large number of coefficients is inconvenient and much experimental work would be necessary to estimate them independently. This may be avoided by using the symmetry law of Onsager The entire set of phenomenological coefficients forms a matrix where reciprocity relations are valid . In mathematical terms it can be expressed thus ... 

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