Curie’s principle

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. 

Farina (19) has provided a number of examples of the application of Curie s principle in his review entitled High Symmetry Chiral Molecules, and Prelog (20) has extended the same principle to his simplex in relation to his studies of generalized pseudoasymmetry. 

According to Eq. (7.1) P is zero for the two cases of uniform director fields and pure twist. Hence both cases can serve as a zero state as far as flexoelectric excitations are concerned. It is important to note that a twist is not associated with a polarization (i.e. C2 is identically zero, cf. Fig. 7.2). An imstrained nematic has a centre of symmetry (centre of inversion). On the other hand, none of the elementary deformations - splay, twist or bend have a centre of symmetry. According to Curie s principle they could then be associated with the separation of charges analogous to the piezoeffect in solids. This is true for splay and bend but not for twist because of an additional symmetry in that case if we twist the adjacent directors in a nematic on either side of a reference point, there is always a two-fold symmetry axis along the director of the reference point. In fact, any axis perpendicular to the twist axis is such an axis. Due to this symmetry no vectorial property can exist perpendicular to the director. In other words, a twist does not lead to the separation of charges. This is the reason why twist states appear naturally in liquid crystals and are extremely common. It also means that an electric field cannot induce a twist just by itself in the bulk of a nematic. If anything it reduces the twist. A twist can only be induced in a situation where a field turns the director out of a direction that has previously been fixed by boundary conditions (which, for instance, happens in the pixels of an IPS display). 

An influential interpretation of Curie s principle is provided by Chalmers (1970). See also Barman (2004). 

Brush SG (1999) Dynamics of theory change in chemistry part 2. Benzene and molecular orbitals, 1945-1980. Stud Hist Philos Sci 30(2) 263-302 Chalmers A (1970) Curie s principle. Br J Philos Sci 21(2) 133-148... 

Houk KN, Gonzalez J, Li Y (1995) Pericyclic reaction transition states passions and punctilios, 1935-1995. Acc Chem Res 28 81-91 Ismael J (1997) Curie s principle. Synthese 110 167-190... 

As soon as we put together an Sc with chiral molecules, the centre of inversion and plane of symmetry are removed, leaving only the axis C2. Imagining a macroscopic electric polarisation in this direction, no symmetry operation can cancel it out. In these conditions, Curie s principle tells us that this polarisation must exist in a generic manner. This observation, made by R.B. Meyer in... 

It follows from the earlier discussion that the definition of SSB in atomic systems is directly related to their possible observation physical phenomena exist if they can be obsetyed. The problem of symmetry with respect to observation descends from Pierre Curie s principle formulated in 1894 [15]. It states ... 

Curie s principle is used directly as a postulate in quantum field theory assuming that the observable part of the Universe (the vacuum) is in a state of broken symmetry (Section Vni). In application to atomic systems Curie s principle implies that the symmetry of the observable phenomena (the effect) may not be... 

The properties of systems with multiminimum APES are significantly different from those of stable single-minima ones, and under certain external perturbations the multiminimum nature of the systan may be exposed showing its properties in the distorted SSB configurations. The presence of SSB in such systems can be observed by a variety of methods including external electric and magnetic fields, and especially in a variety of spectroscopies [6,7, 34-37]. The Curie s principle limitations of external perturbations that allow for observation of these SSB can be formulated here in simple (practical) terms. Denote the lifetime of the SSB system in one of the equivalent minima by t and the characteristic time of measurement of the external (e.g., spectroscopic) perturbation by t. Then the SSB will be seen in the measurement if... 

Condition (3) is a particular case of Curie s principle leading to the rule of relativity of the observed SSB to the means of observation [6], when the broken symmetry is seen in measurements with methods for which t other experiments when t >t. Significantly different t values are inherent in spectroscopies such as EXAFS, EPR, NMR, optical, vibrational, Mossbauer, and ultrasound. Examples of this relativity to the means of observation of SSB are given in Section Vll. 

Now apply the stress in the most general way, i.e., such that its three axes of two-fold symmetry and its three planes of reflection symmetry do not have the same directions as the axes and planes in the unstrained medium. Curie s principle then says that the stressed medium will retain none of its symmetry axes and planes. 

We recall from the discussion in Sec. 2.3.5 that a necessary condition for the appearance of a polarization was that the medium lacks a center of symmetry. The reason for this was that since at equilibrium the stress as well as the strain will be centro-symmetric, the piece of matter cannot develop charges of opposite sign at opposite ends of a line through its center if it has a center of symmetry, in accordance with Curie s principle. On the other hand, inspection of Fig. 31 immediately reveals that none of the three strains splay, twist, and bend has a center of synametry. Hence Curie s principle allows a local polarization to appear as a result of such local deformations in the director field, even if the medium itself has a center of symmetry. We see from Fig. 31 that the splay deformation violates the ->- invariance, whereas twist and bend... 

When a fluid is subject to a simultaneous Couette shear field and a concentration or temperature gradient, the shear field, being a tensorial quantity, cannot affect vector quantities such as mass or heat currents (Curie s principle). However, when... 

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