Onsagers relations

Onsager relation implies that measurement of one of these effects is sufficient to detemiine the coupling for both. The coefficient L is proportional to the heat conductivity coefficient and is a single scalar quantity in... 

It follows from the second law of thermodynamics, the Onsager relation Ltj = Lji, the thermodynamic stability condition (G3)... 

The relationship between fluctuation and dissipation is reminiscent of the reciprocal Onsager relations that link affinity to flux. The two relationships become identical under Onsager s regression hypothesis which states that the decay of a spontaneous fluctuation in an equilibrium system is indistinguishable from the approach of an undisturbed non-equilibrium system to equilibrium. The conclusion important for statistics, is that the relaxation of macroscopic non-equilibrium disturbances is governed by the same (linear) laws as the regression of spontaneous microscopic fluctuations of an equilibrium system. In the specific example discussed above, the energy fluctuations of a system in contact with a heat bath at temperature T,... 

The coupling of two different electrokinetic ratios (Estr/p and V/I) through Equation (65) is an illustration of a very general law of reciprocity due to L. Onsager (Nobel Prize, 1968). The general theory of the Onsager relations, of which Equation (65) is an example, is an important topic in nonequilibrium thermodynamics. 

Exercise. Formulate the Onsager relations for nonlinear systems of diffusion type. Exercise. Sometimes Pe(x) contains an additional phase-space factor w(x), so that its dependence on 6 is displayed by... 

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. 

Irreversible thermodynamics thus accomplishes two things. Firstly, the entropy production rate EE t allows the appropriate thermodynamic forces X, to be deduced if we start with well defined fluxes (eg., T-VijifT) for the isobaric transport of species i, or (IZT)- VT for heat flow). Secondly, through the Onsager relations, the number of transport coefficients can be reduced in a system of n fluxes to l/2-( - 1 )-n. Finally, it should be pointed out that reacting solids are (due to the... 

The remaining fluxes and forces are independent and thus the Onsager relations Lki = 4 hold. The number of independent transport coefficients is l/2- -(n — 1), With the help of the above conditions, it is possible to verify the symmetry of both matrices L and L [M. Martin, et at. (1988)]... 

In Equation 1, R and V refer to the relaxed (low frequency) and unrelaxed (high frequency) dielectric constants, and AH is the measured activation energy for the y process. The latter was nearly independent of blend composition an average value of 8.7 kcal/mole was used. The integral in Equation 1 was found to be approximately independent of frequency in the range studied. The loss peak in absolute terms is rather weak, and values of eR — V were of the order of 10"2 and less. From these values, it was also possible to calculate the apparent dipolar density, Np2, using the Onsager relation (9) ... 

These jumps follow from the Onsager relations. Unlike the 2-slit geometry, the closed ABI requires many reflections of the electron waves from the forks connecting the ring with the leads. Each such reflection adds a term to the interference sum of amplitudes, and modifies the simple 2-slit formula. In fact, unitarity (conservation of current) and time reversal symmetry imply that = Q 4>) [11], and therefore (3 (as well as 7 etc.) must be equal to 0 or 7T. The additional reflections also explain the need for higher harmonics near resonances. Below we include these many reflections, and replace the 2-slit formula by a new one - which can be used to extract olqd from the closed interferometer data [12]. 

The Onsager relations, which require that T depends on only via cos , imply that the ratio K = ABtB/(ADtQD) = x[GD(ek) 1 + Eext(ek)], with the real coefEcient x = IiJn./ -lr-1 n [Pg.9]

We can specially show that the main principles of nonequilibrium thermodynamics (the Onsager relations, the Prigogine theorem, symmetry principle) and other theories of motion (for example, theory of dynamic systems, synergetics, thermodynamic analysis of chemical kinetics) are observed in the MEIS-based equilibrium modeling. In order to do that, we will derive these statements from the principles of equilibrium thermodynamics. 

From a satisfactory, to a certain extent, explanation based on the second law of the Prigogine theorem we can pass to an absolutely macroscopic explanation of the Onsager reciprocal relations by changing the order of proofs accepted in the nonequilibrium thermodynamics (in the nonequilibrium thermodynamics the Prigogine theorem is derived from the Onsager relations). 

It is obvious that using the properties of homogeneity and additivity of thermodynamic functions it is easy to obtain the Onsager relations in a general form... 

We have managed to interpret the theorem of minimum entropy generation and the Onsager relations on the basis of the second law therefore, we can additionally explain the Curie symmetry principle in terms of equilibrium. Let us suppose that far from the equilibrium between flows and forces there are nonlinear relationships... 

Fig. 23. Test plot of the Onsager relation (Eq. (4-2)) for the data of Figure 22 at different extents of conversion as determined by DSC measurements. (Reprinted from Ref.71 with permission of the Society of Plastics Engineers)...

As the existence of MChA can be deduced by very general symmetry arguments and the effect does not depend on the presence of a particular polarization, one may wonder if something like MChA can also exist outside optical phenomena, e.g. in electrical conduction or molecular diffusion. Time-reversal symmetry arguments cannot be applied directly to the case of diffusive transport, as diffusion inherently breaks this symmetry. Instead, one has to use the Onsager relation. (For a discussion see, e.g., Refs. 34 and 35.) For any generalized transport coefficient Gy (e.g., the electrical conductivity or molecular diffusion tensor) close to thermodynamic equilibrium, Onsager has shown that one can write... 

The form of the expressions for the rate of entropy production does not uniquely determine the thermodynamic forces or generalized flows. For an open system, for example, we may define the energy flow in various ways. We may also define the diffusion in several alternative ways depending on the choice of reference average velocity. Thus, we may describe the flows and the forces in various ways. If such forces and flows, which are related by the phenomenological coefficients obeying the Onsager relations, are subjected to a linear transformation, then the dissipation function is not affected by that transformation. 

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

There is no definite sign for Eq. (3.317). When the generalized flows are expressed by linear phenomenological equations with constant coefficients obeying to the Onsager relations... 

Reference to the preceding section, and to Eq. (6.4.4), shows that if steady state conditions are applied and if there are no constraints placed on the various forces all currents vanish thus, the first term drops out. This is also the condition for which the rate of entropy conduction is a minimum. Thus, considering 8 to be an implicit function of the fluxes, the left hand side must then vanish. The Onsager reciprocity condition is thereby established Lik - l i One should note that if the reciprocity condition were to fail the various XA would be interrelated, contrary to the assumption that they are independent. Furthermore, the assumption of the Onsager relations leads directly to the minimization of 8 under steady state conditions. 

THE INTERACTION BETWEEN THERMODYNAMIC PROCESSES AND LINEAR ONSAGER RELATIONS... 

Let us express the stationary rates via all channels of the concurrent stepwise reactions In the form that conforms to the modified Onsager relations ... 

In a similar way, substituting the series of certain transformations at their stationary modes by effective transformations will allow the exact expressions of the reciprocity coefficients Ay to be found for even very complex schemes of cocurrent stepwise transformations, provided that these are linear with respect to the intermediates. Unfortunately, for an arbitrary case of cocurrent stepwise transformations that are nonlinear in respect to their intermediates and proceed far from equilibrium, it is not possible to write general equations that are analogous to the modified Onsager relations. 

We saw in Chapter 2 that in the range of validity of linear nonequilibrium thermodynamics (i.e., in the scope of linear Onsager relations), a system approaching its stationary state is characterized by a monotonous decrease in the rate of entropy production (energy dissipation rate) resulting from the existence of internal irreversible processes dP < 0 and < 0. 

In the region of the linear Onsager relations validity, both summands in expression (3.1) also are identical, and derivative dP/dt represents the principle of the minimized rate of entropy production ... 

Far from equilibrium, the linear Onsager relations are not satisfied, so partial differentials dxP and djP are no longer total differentials. Hence, variations of parameter P in time depend on the transition route and are not applicable as an unambiguous criterion of the system evolution. How ever, Glansdorf and Prigogine demonstrated that far from equilibrium, any spontaneous evolution is characterized by a monotone decrease in the partial force differential dxP expressed as the inequahty... 

The Eloriuti-Boreskov-Onsager Relations for Parallel Catalytic... 

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