Onsagers Symmetry Principle

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... 

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. 

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... 

ONSAGER RECIPROCAL RELATIONS AND THE SYMMETRY PRINCIPLE in which... 

Unlike other branches of physics, thermodynamics in its standard postulation approach [272] does not provide direct numerical predictions. For example, it does not evaluate the specific heat or compressibility of a system, instead, it predicts that apparently unrelated quantities are equal, such as (1 A"XdQ/dP)T = - (dV/dT)P or that two coupled irreversible processes satisfy the Onsager reciprocity theorem (L 2 L2O under a linear optimization [153]. Recent development in both the many-body and field theories towards the interpretation of phase transitions and the general theory of symmetry can provide another plausible attitude applicable to a new conceptual basis of thermodynamics, in the middle of Seventies Cullen suggested that thermodynamics is the study of those properties of macroscopic matter that follows from the symmetry properties of physical laws, mediated through the statistics of large systems [273], It is an expedient happenstance that a conventional simple systems , often exemplified in elementary thermodynamics, have one prototype of each of the three characteristic classes of thermodynamic coordinates, i.e., (i) coordinates conserved by the continuous space-time symmetries (internal energy, U), (ii) coordinates conserved by other symmetry principles (mole number, N) and (iii) non-conserved (so called broken ) symmetry coordinates (volume, V). 

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]. 

Even if the linear law given by Eq. (436) applies, the irreversible thermodynamics of Onsager et al. still has nothing to contribute to chemical kinetics. Its central theme and source of usefulness is the statement (Onsager relations) that the matrix L is symmetric. This symmetry is a consequence of the principle of detailed balancing—a principle well known and used in chemical kinetics independently of irreversible thermodynamics. 

A typical proposal of how the Onsager relations provides useful information about chemical reactions proceeds as follows [DeGroot (39)] The true kinetics of a chemical reaction are very often not known and the principle of detailed balancing cannot be readily applied. In such cases the symmetry of the matrix L can be used to obtain some useful information about the process. One proceeds by noting that the conjugant forces (x) and fluxes (j) in Eq. (436) are related by... 

It is possible for more than two forces to couple. There exists a criterion which allows one to deduce a priori the number of effective couplings. This is Curie s principle of symmetry. The principle states that a macroscopic phenomenon in the system never has more elements of symmetry than the cause that produces it. For example, the chemical affinity (which is a scalar quantity) can never cause a vectorial heat flux and the corresponding coupling coefficient disappears. A coupling is possible only between phenomenon which have the same tensor symmetry. Thus Onsager reciprocity relation is not valid for a situation when the fluxes have different tensorial character. 

The presence of / > in Eq. (3) gives rise to oscillatory, nondissipative dynamics for A(/), while F gives rise to decay and dissipation. Since iClyX and ia> are composed entirely of equal-time correlation functions, these quantities are, in principle, available from equilibrium statistical mechanics. On the other hand, irreversible thermodynamics gives no expression for L, aside from certain symmetry requirements, the Onsager relations. 

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). 

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