Casimir-Onsager

In the presence of magnetic fields B and or in systems undergoing angular rotations at rates (o the above expression is replaced by the Casimir-Onsager reciprocity relation... 

If odd variables, the b, are also included, then a generalization by Casimir [8] results in the Onsager-Casimir relations... 

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

This follows because, in the grouped representation, Q+ contains nonzero blocks only on the diagonal and is symmetric, and Q contairisnonzero blocks only off the diagonal and is asymmetric. These symmetry rules are called the Onsager-Casimir reciprocal relations [10, 24], They show that the magnitude of the coupling coefficient between a flux and a force is equal to that between the force and the flux. 

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

One approach has been suggested by WALDMANN [2.89] and WALDMANN and VESTNER [2.90]. This invoives the use of a truncated version of Maxwell s moment equations. Boundary conditions for Waldmann s "generalized hydrodynamics are established by the methods of nonequilibrium thermodynamics with restrictions arising from the second law and Onsager-Casimir symmetries. This gives, of course, phenomenological coefficients for the boundary laws. These coefficients are implicitly dependent on the accommodation coefficients, but the nature of this dependence cannot be determined solely from the phenomenological theory. 

Here the first equation is the usual Fourier law, the second relates the viscous pressure tensor to the internal variable W, and the last is the evolution of the internal variable. The matrix of the transport coefficients Ly is positive definite with L q = —Lq due to Onsager-Casimir reciprocal rules. 

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. ... 

The above results have been obtained by straightforward calculations within the method of statistical mechanics. However, Onsager and Casimir used a new hypothesis. They assumed that the decay of a fluctuation follows, on the average, the ordinary phenomenological macroscopic laws. The latter equations express linear relations between the flows, which are expressed as the time derivatives of macroscopic variables a s, and the forces, which are also functions of a,. s. This can be written as... 

A more detailed discussion of the Onsager-Casimir symmetry is given in Section 7.3.2.1 of Ref. [167]. 

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