Onsager matrix

Note that q = 1 is equivalent to saying that the Onsager matrix L,y has a determinant equal to zero, implying that /] = J2 = 0 can be attained for nonzero values of the thermodynamic forces X and X2 [cf. Eq. (6)]. 

Figure 5. Cascade construction with a continuum of copies of the system with Onsager matrix Lij located in [y, y + dy], along with corresponding heat baths for the conversion of a heat flow Q y) into a power contribution dW(y).

As may easily be confirmed from (13.42), the Onsager matrix L is a positive-semidefinite matrix satisfying the eigenvalue equation... 

In this case, the system of Eqs. (14) is complete so that the matrix (15e) represents a symmetrical Onsager matrix the equality L12 = L2l with kg = kg then yields the condition of the reversible-thermodynamic equilibrium in the PDC-column... 

Onsager s approach, by definition, is valid in the vicinity of equilibrium, and deviations of Cj from c, eq are assumed to be small. The symmetry of the Onsager matrix, = Lji, follows from the principle of microreversibility (see the classical monograph by de Groot and Mazur, 1962). 

By a deft application of the transfer matrix teclmique, Onsager showed that the free energy is given by... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

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. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

If it can be shown that the prefactor is the identity matrix plus a matrix linear in x, then this is, in essence, Onsager s regression hypothesis [10] and the basis for linear transport theory. 

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... 

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

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... 

Krauss and Brus [15] measured the dielectric constant of CdSe dots, and found a value of 8. Hence, at room temperature we find r0ns — 70 A (angstrom) (however, note that the dielectric constant of the matrix is not identical to that of the dot). Since the length scale of the dots is of the order of a few nanometers, the Coulomb interaction seems to be an important part of the problem. This according to the theory in Ref. 16 is an indication of possible deviations from the universal 1 /2 power-law behavior. It is also an indication that an ejected electron is likely to return to the dot and not escape to the bulk (since the force is attractive). In contrast, if the Onsager radius is small, an ejected electron would most likely escape to the bulk, leaving the dot in state off forever (i.e., Polya... 

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. 

After applying the Onsager relations to the linear matrix solutions, we have... 

Equations (6.299) and (6.300) show that Onsager s reciprocal rules hold. The Js eq and Jweq have a microscopic definition represented by perturbation matrix elements and a macroscopic definition represented by the equilibrium exchange rate. As long as the criteria of linearization are satisfied, the statistical rate theory may be used to describe systems with temperature differences at an interface besides the driving forces of pressure and concentration differences. 

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... 

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). 

For a symmetric matrix vkl, both absolute equilibrium flows Js>eq and Jty eq must be identical and replaced by a universal constant Teq. However, if the matrix vM is not symmetric, which is usual, the equilibrium flows are related to each other so that the Onsager symmetry is achieved... 

Substituting Eq. (8.67) into Eq. (8.101) indicates that the matrix L is symmetric and the matrix elements Lu obey the Onsager reciprocal relations. 

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. 

---