Onsager principle

Comparison of Eq. 2.50 with Eqs. 2.49 and 2.51 shows that Lij = Lji and therefore demonstrates the role of microscopic reversibility in the symmetry of the Onsager coefficients. More demonstrations of the Onsager principle are described in Lifshitz and Pitaerskii [6] and in Yourgrau et al. [8]. 

Here one can take advantage of the Onsager principle, that is equate factors at cross members. 

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... 

We notice that the correlation function defined by Eq. (147) is stationary. Thus, it fits the Onsager principle [101], which establishes that the regression to equilibrium of an infinitely aged system is described by the unperturbed correlation function. The authors of Ref. 102 have successfully addressed this issue, using the following arguments. According to an earlier work [96] the GME of infinite age has the same time convoluted structure as Eq. (59), with the memory kernel T(t) replaced by (1>,XJ (f). They proved that the Laplace transform of Too is... 

We do not go through the details of the demonstration done in Ref. 96 to get this result. We limit ourselves to noticing that if (u) is replaced by the brand new property /(it), then the memory kernel of Eq. (243) becomes identical to the memory kernel of Eq. (66). Then, there exists an even more compelling reason to accept the result of Ref. 96. This is done through the realization of the Onsager principle via the GME of infinite age. This GME reads... 

This GME, being of infinite age, fits the Onsager principle. Therefore, we can identify the stationary correlation function with the out of equilibrium distribution, namely,... 

The following properties are characterized by symmetric tensors of rank 2 magnetic susceptibility (negative eigenvalues for diamagnetic materials) electrical and thermal conductivities (these tensors are symmetrical according to the Onsager principle) thermal expansion. 

Linear response theory is an example of a microscopic approach to the foundations of non-equilibrium thennodynamics. It requires knowledge of tire Hamiltonian for the underlying microscopic description. In principle, it produces explicit fomuilae for the relaxation parameters that make up the Onsager coefficients. In reality, these expressions are extremely difficult to evaluate and approximation methods are necessary. Nevertheless, they provide a deeper insight into the physics. 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

Continuum models of solvation treat the solute microscopically, and the surrounding solvent macroscopically, according to the above principles. The simplest treatment is the Onsager (1936) model, where aspirin in solution would be modelled according to Figure 15.4. The solute is embedded in a spherical cavity, whose radius can be estimated by calculating the molecular volume. A dipole in the solute molecule induces polarization in the solvent continuum, which in turn interacts with the solute dipole, leading to stabilization. 

On a related point, there have been other variational principles enunciated as a basis for nonequilibrium thermodynamics. Hashitsume [47], Gyarmati [48, 49], and Bochkov and Kuzovlev [50] all assert that in the steady state the rate of first entropy production is an extremum, and all invoke a function identical to that underlying the Onsager-Machlup functional [32]. As mentioned earlier, Prigogine [11] (and workers in the broader sciences) [13-18] variously asserts that the rate of first entropy production is a maximum or a minimum and invokes the same two functions for the optimum rate of first entropy production that were used by Onsager and Machlup [32] (see Section HE). 

This uses the fact that dr = dT. For macrostates all of even parity, this says that for an isolated system the forward transition x > x will be observed as frequently as the reverse x —> x. This is what Onsager meant by the principle of dynamical reversibility, which he stated as in the end every type of motion is just as likely to occur as its reverse [10, p. 412]. Note that for velocity-type variables, the sign is reversed for the reverse transition. 

The basic principle of heat-flow calorimetry is certainly to be found in the linear equations of Onsager which relate the temperature or potential gradients across the thermoelements to the resulting flux of heat or electricity (16). Experimental verifications have been made (89-41) and they have shown that the Calvet microcalorimeter, for instance, behaves, within 0.2%, as a linear system at 25°C (41)-A. heat-flow calorimeter may be therefore considered as a transducer which produces the linear transformation of any function of time f(t), the input, i.e., the thermal phenomenon under investigation]] into another function of time ig(t), the response, i.e., the thermogram]. The problem is evidently to define the corresponding linear operator. 

THL.3. 1. Prigogine, Remarque sur le principe de reciprocite d Onsager et le couplage des reactions chimiques (Remarks on Onsager s reciprocity principle and the coupling of chemical reactions). Bull. Cl. Set Acad. Roy. Belg. 32, 30-35 (1946). 

The allowance for polarization in the DH model obviates the need for separation of long-range and short-range attractive forces and for inclusion of additional repulsive interactions. Belief in the necessity to include some kind of covolume term stems from the confused analysis of Onsager (13), and is compounded by a misunderstanding of the standard state concept. Reference to a solvated standard state in which there are no interionic effects can in principle be made at any arbitrary concentration, and the only repulsive or exclusion term required is that described by the DH theory which puts limits on the ionic atmosphere size and hence on the lowering of electrical free energy. The present work therefore supports the view of Stokes (34) that the covolume term should not be included in the comparison of statistical-mechanical results with experimental ones. 

The importance of Onsager s earlier work on coupled diffusional processes was also increasingly appreciated. Onsager had based his proof of force-flux reciprocal relations on the physical principle of detailed balance (as suggested by G. N. Lewis), which rules... 

Onsager s principle supplements these postulates and follows from the statistical theory of reversible fluctuations [5]. Onsager s principle states that when the forces and fluxes are chosen so that they are conjugate, the coupling coefficients are... 

The statistical-mechanics derivation of Onsager s symmetry principle is based on microscopic reversibility for systems near equilibrium. That is, the time average of a correlation between a driving force of type a and the fluctuations of quantity P is identical with respect to switching a and / [6]. 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

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