Degree of Freedom Selection State Variables, Order Parameters and Configurational Coordinates

1 Degree of Freedom Selection State Variables, Order Parameters and Configurational Coordinates 

State Variables. The construction of effective theories is predicated in part on the ability to discern some trimmed down description of the disposition of the system of interest. From the perspective of equilibrium thermodynamics, the ability to characterize a system in terms of a series of state variables exemplifies the type of degree of freedom thinning we have in mind here. A particular virtue of the 

The order parameter is essentially a kinematic measure, describing the state of order within a system without any intrinsic reference to what factors drove the system to the state of interest. For example, in thinking about the transition between the ordered and disordered states of an alloy, it is useful to define an order parameter that measures the occupation probabilities on different sublattices. Above the order-disorder temperature, the sublattice occupations are random, while below the critical temperature, there is an enhanced probability of finding a particular species on a particular sublattice. The conventional example of this thinking is that provided by brass which is a mixture of Cu and Zn atoms in equal concentrations on a bcc lattice. The structure can be interpreted as two interpenetrating simple cubic lattices where it is understood that at high temperatures we are as likely to find a Cu atom on one sublattice as the other. A useful choice for the order parameter, which we denote by r], is 

Configurational Coordinates and Defects. A second broad class of ideas that have proven quite useful in the construction of effective theories of material response are those in which all attention is focused on the defects populating the system. That is, all reference to either atomic coordinates or continuum fields is surrendered and instead a fheory is built around the dynamical evolution of the defects themselves. In this setting, kinematics is reduced to a question of 

The basic ideas in this section reflect an extension of the kinematic ideas of continuum mechanics in the sense that rather than enumerating the properties of a system on a discrete atom-by-atom basis, the state of the system is subsumed into either a set of continuous fields or a collection of coordinates which identify the positions of the defects that populate the system. We have seen that the class 

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