N-vector model

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,... 

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. 

Key words Flory theory - Edwards model - 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 ... 

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

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

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... 

Panizzay, P., Cristobal, G., and Curely, J. "Phase separation in solution of worm-like micelles a dilute n—>0 spin-vector model". ]. Phys. Condens. Matter 10,11659-11678 (1998). 

In this volume, principal consideration is given to the lighter elements, so that the Russell-Saunders (549) vector model of the atom is used. In this model a multielectron atom is assumed to have the quantum numbers n, L = lif Ml, 8 = siy (or n, L, J = L + S, Mj). This implies stronger and Si-Sj coupling than U-Si coupling. It follows from Pauli s principle that for a closed shell =... 

A model y = /( ) is fitted to the mean-data n-vector y = (yi,..., y ) while estimating the best-fit parameter values 0. The latter notation ( ) is used to distinguish it from individual estimates, denoted... 

With this TVAR vector model, the predicting and filtering steps for the PM lo concentrations can be performed with the Kalman filter. The essential steps of the Kalman filter are to predict and filter the measured PMio concentration alternately. When the measured PMio concentrations up to the (n - 1 )th day are available, the predicting procedure is applied to give the one-step-ahead prediction of the PMio concentration. By using Equations (2.204) and (2.205), the predicted state vector on the nth day can be estimated from the filtered state vector on the (n - l)th day ... 

---