Onsager’s theorem

Onsager s theorem deals with reciprocal relations in irreversible resistive processes, in the absence of magnetic fields [114], The resistive qualifier signifies that the fluxes at a given instant depend only on the instantaneous values of the affinities and local intensive parameters at that instant. For systems of this kind two independent transport processes may be described in terms of the relations... 

Onsager s theorem consists of proving that a reciprocal relationship of the type Lap = Lpa between the affinities and fluxes of coupled irreversible processes is universally valid in the absence of magnetic fields. 

Some of the well-known effects that have been successfully analyzed in terms of Onsager s theorem include ... 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical (i.e., xy = X/,-). This is known as a reciprocal relation. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Thus, the tensor above can be rewritten as... 

According to Onsager s theorem from nonequilibrium thermodyn lmics, the flux is proportional to the gradient of the chemical potential of this component (see Equation 2.4-33)... 

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. 

The usual emphasis on equilibrium thermodynamics is somewhat inappropriate in view of the fact that all chemical and biological processes are rate-dependent and far from equilibrium. The theory of non-equilibrium or irreversible processes is based on Onsager s reciprocity theorem. Formulation of the theory requires the introduction of concepts and parameters related to dynamically variable systems. In particular, parameters that describe a mechanism that drives the process and another parameter that follows the response of the systems. The driving parameter will be referred to as an affinity and the response as a flux. Such quantities may be defined on the premise that all action ceases once equilibrium is established. 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

We describe the linearly damped harmonic quantum oscillator in Heisenberg s interpretation by Onsager s thermodynamic equations. Ehrenfest s theorem is also discussed in this framework. We have also shown that the quantum mechanics of the dissipative processes exponentially decay to classical statistical theory. 

Note that this formula can be easily obtained from Gauss s theorem of the induced charges, which, in this case, will be equal to — q — s)/( w s)- The coefficient 1/2 comes from the linear-response hypothesis. Another useful formula is the solvation energy of the point dipole (/j.) in the center of a spherical cavity [29] (Bell formula, sometimes also called the Onsager formula) ... 

According to Onsager s reciprocity theorem Cl], the matrix of the phenomenological equations (7) is symmetric and we have... 

One can imagine limiting circumstances for which the latter equations are decoupled. Explicitly, for an almost ion-free (almost nonconductive) solution, one can see that 1 = 0, while Eq. (lb) reproduces Darcy s law. In the other extreme case of a solid (infinitely viscous) electrolyte, one has U = 0, and Eq. (la) reduces to the familar Ohm s law. In general, one can show that a = p, by virtue of the Onsager thermodynamic theorem [3] p denotes the transpose of the tensor p. 

Limit our investigation on the generalized Onsager s constitutive theory, containing the linear Onsager s theory as well. In this case the thermodynamic fluxes have no non dissipative parts and the thermodynamic forces could be deduced from the flux dissipation potential by the Edelens decomp>osition theorem. Assuming a dissipation potential... 

The above parts show the minimum principle for vector processes in the frame of the generalized Onsager constitutive theory by the directions of Onsager s last dissip>ation of energy principle. We had seen above that in case of source-free balances, this principle is equivalent with the principle of minimal entropy production. The equivalence of the two theorems in the frame of the linear constitutive theory was proven by Gyarmati [2] first. Furthermore, we showed that in case when the principle of minimal entropy production is used for the determination of the possible forms of constitutive equations, the results are similar to the linear theory in the frame of the Onsager s constitutive theory, where the dissipation potentials are homogeneous Euler s functions. 

Some of the variational principles which are to be described in the present article, are very closely connected with Onsager s reciprocity relationsAlthough there have been various methods of derivation of this theorem, we shall follow the traditional method of derivation by Onsager and Casimir. This is based on the consideration of fluctuations in an aged system, and this method is also connected with the derivation of Onsager s principle of least dissipation of energy. ... 

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)]. 

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