The n vector model

We now describe the system of atomic moments or spins 5j in more detail on a specific model, the n vector model. We assume that the magnetic atoms are located on a periodic lattice. Each magnetic atom (i) carries a spin S( this is a vector, with n components 5(1, Sa... 5( . In our considerations, we ignore all quantum effects the components Sia are just numbers. There is one constraint—i.e., the total len 5 of each spin is fixed. We choose the following normalization  

Neighboring spins are coupled, and their energy is minimized when they are parallel. The coupling energy, or Hamiltonian has the form 

The constant is positive, Ky = K, for nearest-neighbor pairs (if) and vanishes for all other choices of i and j. In eq. (X.12) we have also incorporated terms (— H-S() describing the effect of an external field H. The partition function of the spin system is 

It has been a traditional temptation to expand the partition function in powers of the coupling energy Ky for each pain 

Usually when such expansions are inserted into eq. (X.IS) forZ, they result in a complicated structure. There is one case, however, when they 

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We will discuss if the size exponents (i/ or ve) are affected by the lattice (configurational) fluctuations if r (p) is different from i/ or if i/ is different from i/J for p < 1. This question arises naturally from the application of the Harris criterion [8] to the n-vector model in the n —> 0 limit [2], when the partition function graphs are all SAWs. A naive application of the criterion to the SAWs suggested [6] a possible disorder induced crossover in the critical behaviour of SAW statistics for any amount of disorder (p < 1). A modified... 

Le Guillou JC, Zinn-Justin J (1977) Critical exponents for the n-vector model in three dimensions from field theory. Phys Rev Lett 39 95-98 Madras N, Slade G (1993) The self-avoiding walk. Biikhaeuser, Basel Mandelbrot BB (1983) The fractal geometry of nature. Freeman W.H, New York Manning GS (1969) Limiting laws and counterion ctmdcmsation in polyelectrolyte solutions I. 

R. Guida, J. Zinn-Justin, Critical exponents of the N-vector model, J. Phys. A Math. Gen. 31 (1998) 8103-8121. 

Apart from this n-vector model allowing for a -component order parameter, there is also the need to consider order parameters of tensorial character. This happens, for example, when we consider the adsorption of molecules such as N2 on grafoil. For describing the orientational ordering of these dumbbell-shaped molecules, the relevant molecular degree of freedom which matters is their electric quadrupole moment tensor,... 

A relation between polymer statistics and phase transition was established by de Gennes (1979) and des Cloiseaux (1974) showing a correspondence between the polymer chain modeled by SAWs and the n-vector spin model of magnetization in the limit n 0. Similarities between correlation length (0 and the end-to-end distance (Re) can be noticed by compar-ing O Eqs. 8.12 and O 8.13. Correspondence between 1/N and -jr- is viewed as 0 and... 

D. B. Murray and B. G. NicKel, Revised estimates for critical exponents for the continuum n-vector model in 3 dimensions, preprint (1991). 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

We examined the role of vector percolation in the fracture of model nets at constant strain and subjected to random bond scission, as shown in Fig. 11 [1,2]. In this experiment, a metal net of modulus Eo containing No = 10" bonds was stressed and held at constant strain (ca. 2%) on a tensile tester. A computer randomly selected a bond, which was manually cut, and the relaxation of the net modulus was measured. The initial relaxation process as a function of the number of bonds cut N, could be well described by the effective medium theory (EMT) via... 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

A pictorial representation of the Tg-S mixing process follows from Fig. 6. Just as in normal n.m.r. or e.s.r. spectroscopy, precession can be represented by a vector model. When placed in an external magnetic field the two unpaired electrons of the radical pair 1 and 2 will precess... 

A single experiment consists of the measurement of each of the m response variables for a given set of values of the n independent variables. For each experiment, the measured output vector which can be viewed as a random variable is comprised of the deterministic part calculated by the model (Equation 2.1) and the stochastic part represented by the error term, i.e.,... 

Experimental data are available as measurements of the output vector as a function of time, i.e., [yj, t ], i=l,...,N where withyj we denote the measurement of the output vector at time t,. These are to be matched to the values calculated by the model at the same time, y(t,), in some optimal fashion. Based on the statistical properties of the experimental error involved in the measurement of the output vector, we determine the weighting matrices Qj (i=l,...,N) that should be used in the objective function to be minimized as mentioned earlier in Chapter 2. The objective function is of the form,... 

Having the smoothed values of the state variables at each sampling point and having estimated analytically the time derivatives, n we have transformed the problem to a usual nonlinear regression problem for algebraic models. The parameter vector is obtained by minimizing the following LS objective function... 

The reader will recognize these terms as having of the same form as the correction terms in the two-environment model discussed earlier. With N — 1, 6 j = 0 and the model reduces to the laminar-chemistry approximation. With N —2, additional information is obtained concerning the second-order moments of the composition vector. Likewise, by using a larger N, the Mh-order moments are controlled by the DQMOM correction terms found from Eq. (89). 

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