Critical Exponent - The Most Important Critical Exponents

The Most Important Critical Exponents

Above and below the system has two different phases characterized by an order parameter, which vanishes at and above .

Let us consider the disordered phase ( > 0), ordered phase ( < 0 ) and critical temperature ( = 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. It's also another standard convention to use the super/subscript +(-) for the disordered(ordered) state. We have spontaneous symmetry breaking in the ordered phase. So, we will arbitrarily take any solution in the phase.

Keys
order parameter (e.g. for the liquid-gas critical point, magnetization for the Curie point,etc.)
specific free energy
specific heat;
source field (e.g. where P is the pressure and Pc the critical pressure for the liquid-gas critical point, reduced chemical potential, the magnetic field H for the Curie point )
the susceptibility/compressibility/etc.;
correlation length
the number of spatial dimensions
the correlation function
spatial distance

The following entries are evaluated at (except for the entry)

Critical exponents for > 0 (disordered phase)
Greek letter relation
Critical exponents for < 0 (ordered phase)
Greek letter relation
Critical exponents for = 0

The critical exponents can be derived from the specific free energy as a function of the source and temperature. The correlation length can be derived from the functional .

These relations are accurate close to the critical point in two- and three-dimensional systems. In four dimensions, however, the power laws are modified by logarithmic factors. This problem does not appear in 3.99 dimensions, though.

Read more about this topic:  Critical Exponent

Famous quotes containing the words the most, important and/or critical:

    Can we love our children when they are homely, awkward, unkempt, flaunting the styles and friendships we don’t approve of, when they fail to be the best, the brightest, the most accomplished at school or even at home? Can we be there when their world has fallen apart and only we can restore their faith and confidence in life?
    Neil Kurshan (20th century)

    Nothing is so important to man as his own state; nothing is so formidable to him as eternity. And thus it is unnatural that there should be men indifferent to the loss of their existence and to the perils of everlasting suffering.
    Blaise Pascal (1623–1662)

    An art whose medium is language will always show a high degree of critical creativeness, for speech is itself a critique of life: it names, it characterizes, it passes judgment, in that it creates.
    Thomas Mann (1875–1955)