Onsager symmetry relation

Here, a is an arbitrary constant. Now, we see that the dissipative part of the kinetic matrix satisfies the Onsager symmetry relation and the positivity of the damping constant c trivially. The constant a can be evaluated as follows. In the case of zero for the dissipative part of the kinetic matrix, these equations must be transformed into Hamilton equations of a simple harmonic oscillator. From this fact, it follows that a is a universal constant and a=1. Also, the final form of the Onsagerian constitutive (kinetic) equation the of damped oscillator is... 

Onsager symmetry relation. For results on the rheological properties in the isotropic and in the nematic phases with stationary flow alignment, following from (2) and (5) see... 

Z can be deconstructed into a symmetrical tensor and a antisymmetrical tensor Z Z = Z +Z. The tensor Z is associated with the axial vector Z. The Onsager symmetry relations are written thus, in the presence of a magnetic field Z (H) = Z, (-H). From this, we can deduce that Z(H) and... 

An electrical current along one axis creates an electrical field along the other two axes. Similarly, a field in one direction produces a current in the other two directions. The Onsager symmetry relations give us Rxy = =... 

Because of the Onsager symmetry relation, r j = r, , only three coefficients are needed in the Equations [18.4] and [18.5], and six in the set Equations [18.9]-[18.12]. The nine coefficients needed were already determined by Inzoli et al (Inzoli et al, 2008,2009), and the values of the transport coefficients are shown in Fig. 18.5 (transport inside the membrane) and Fig. 18.6 (transport at the membrane surface). Altogether, the six equations and nine coefficients are sufficient to compute the profiles of the temperature and chemical potential across the heterogeneous system fully, for any set of inlet conditions on the /-side, or feed side. 

These symmetry relations are called the Onsager reciprocal relations. Their meaning is best illustrated by reference to the following problem. Suppose the interaction Hamiltonian is... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

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. 

Thus, the Maxwell-Stefan diffusion coefficients satisfy simple symmetry relations. Onsager s reciprocal relations reduce the number of coefficients to be determined in a phenomenological approach. Satisfying all the inequalities in Eq. (6.12) leads to the dissipation function to be positive definite. For binary mixtures, the Maxwell-Stefan dififusivity has to be positive, but for multicomponent system, negative diffusivities are possible (for example, in electrolyte solutions). From Eq. (6.12), the Maxwell-Stefan diffusivities in an -component system satisfy the following inequality... 

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... 

Linear phenomenological equations obey the Onsager reciprocal relations. For the nonlinear region, from the symmetry of the Jacobian of forces versus flows, we have... 

Equation 2.3.22 expresses the Onsager reciprocal relations (ORR) discussed briefly in Section 3.3. For ideal gases, this symmetry relation can be obtained from the kinetic theory of gases (HCB, 1964 Muckenfuss, 1973). 

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. 

Equation (11.5.15) forms the basis of the Onsager reciprocal relations. These symmetry relations can be expressed in matrix form as... 

The operator, exp (k/), is symmetric in the entropic scalar product. This enables the formulation of symmetry relations between observables and initial data, which can be validated without differentiation of empirical curves and are, in that sense, more robust and closer to direct measurements than the classical Onsager relations. In chemical kinetics, there is an elegant form of symmetry between A produced from B and B produced from A their ratio is equal to the equilibrium coefficient of the reaction A B and does not change in time. The symmetry relations between observables and initial data have a rich variety of realizations, which makes direct experimental verification possible. This symmetry also provides the possibility of extracting additional experimental information about the detailed reaction mechanism through dual experiments. The symmetry relations are applicable to all systems with microreversibility. 

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

The respective tensor of transport coefficients is determined by the symmetry Daoh of nematic and the Onsager reciprocal relations. Thermal conductivity tensor has two independent components km and k, and... 

Due to the symmetry property of crP and Ay, the four-rank viscosity tensor must be Viju = Vjiu = Vijik, and the Onsager reciprocal relations require that Vyu = Vkuy For Harvard notation (7.30) we have... 

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. 

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