Spherical Model - Critical Behaviour

Critical Behaviour

For the critical temperature occurs at absolute zero, resulting in no phase transition for the spherical model. For d greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite Curie temperature where ferromagnetism ceases. The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension d may be a real non-integer dimension.

The critical exponents and in the zero-field case which dictate the behaviour of the system close to were derived to be

$\alpha = \begin{cases} - \frac{4-d}{d-2} & \ \mathrm{if} \ 2<d<4 \\ 0 & \ if \ d > 4 \end{cases}$

$\gamma = \begin{cases} \frac{2}{d-2} & \ if \ 2 <d<4 \\ 1 & if \ d > 4 \end{cases}$

$\delta = \begin{cases} \frac{d+2}{d-2} & \ if \ 2 < d < 4 \\ 3 & if \ d > 4 \end{cases}$

which are independent of the dimension of d when it is greater than four, the dimension being able to take any real value.

Read more about this topic: Spherical Model

Famous quotes containing the words critical and/or behaviour:

“Probably more than youngsters at any age, early adolescents expect the adults they care about to demonstrate the virtues they want demonstrated. They also tend to expect adults they admire to be absolutely perfect. When adults disappoint them, they can be critical and intolerant.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.4 (1985)

“I look on it as no trifling effort of female strength to withstand the artful and ardent solicitations of a man that is thoroughly master of our hearts. Should we in the conflict come off victorious, it hardly pays us for the pain we suffer from the experiment ... and I still persist in it that such a behaviour in any man I love would rob me of that most pleasing thought, namely, the obligation I have to him for not making such a trial.”
—Sarah Fielding (1710–1768)