Casimir

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

Casimir, in his memoirs, mentions that the first Philips representative to visit Brookhaven gave the design team a set of guaranteed magnetic specifications which were too conservative, and Brookhaven was pleased to find that the material was better than they had been led to expect. Casimir reckons that Brookhaven must have concluded that Philips was dumb but honest . 

Casimir, H.B.G. (1983) Haphazard Reality Half a Century of Seienee, Chapter 8 (Harper and Row, New York) p. 224. 

M. Krech. The Casimir Effect in Critical Systems. Singapore World Scientific, 1994, Chap. 3. 

B. Mosaic Stiffening and Temperature Evolution of the Boson Peak The Negative Griineisen Parameter An Elastic Casimir Effect ... 

VI. THE NEGATIVE GRUNEISEN PARAMETER AN ELASTIC CASIMIR EFFECT ... 

The temperature independence of this contribution to the Griineisen constant is the main difference between Eq. (84) and the original calculation by Phillips. The numerical value of the expression should be nearly the same for all substances and is about 8. This suggests that the direct coupling to phonons is a potential contributor to the elastic Casimir effect at temperatures around 1 K. Remember, however, the sign of the expression in Eq. (84) is unknown and its numerical value of lO only provides an estimate from the above. 

Erom the qualitative analysis in this section, we tentatively conclude that there are several contributions of comparable magnitude to the thermal expansion at low temperatures. Higher order effects may also be present. In this case, it may be more straightforward to estimate the interaction between ripplons as extended membranes without using a multipole expansion, as indeed is done when computing the regular Casimir force between extended plates. 

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. 

The asymmetric part of the transport matrix gives zero contribution to the scalar product and so does not contribute to the steady-state rate of first entropy production [7]. This was also observed by Casimir [24] and by Grabert et al. [25], Eq. (17). 

As stressed at the end of the preceding section, there is no proof that the asymmetric part of the transport matrix vanishes. Casimir [24], no doubt motivated by his observation about the rate of entropy production, on p. 348 asserted that the antisymmetric component of the transport matrix had no observable physical consequence and could be set to zero. However, the present results show that the function makes an important and generally nonnegligible contribution to the dynamics of the steady state even if it does not contribute to the rate of first entropy production. 

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. 

P,T-odd interaction operator, 252-253 basis functions, 259—261 Casimir s generalization, nonlinear... 

The term molecular crystal refers to crystals consisting of neutral atomic particles. Thus they include the rare gases He, Ne, Ar, Kr, Xe, and Rn. However, most of them consist of molecules with up to about 100 atoms bound internally by covalent bonds. The dipole interactions that bond them is discussed briefly in Chapter 3, and at length in books such as Parsegian (2006). This book also discusses the Lifshitz-Casimir effect which causes macroscopic solids to attract one another weakly as a result of fluctuating atomic dipoles. Since dipole-dipole forces are almost always positive (unlike monopole forces) they add up to create measurable attractions between macroscopic bodies. However, they decrease rapidly as any two molecules are separated. A detailed history of intermolecular forces is given by Rowlinson (2002). 

Casimir and Polder29 later showed that the dispersion interaction constant Dab can be expressed in the surprising form... 

Casimir and Polder also showed that, at very long range (i.e., separations greater than a characteristic distance R of a few hundred angstrom units), the dispersion interaction takes the modified asymptotic form... 

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