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:
“Those who talk about individuality the most are the ones who most object to deviation, and in a few years it may be the other way around. Some day everybody will just think what they want to think, and then everybody will probably be thinking alike; that seems to be what is happening.”
—Andy Warhol (1928–1987)
“Japanese mothers credit “effort” as the key determinant of a child’s achievement in school, while American mothers name “ability” as the more important factor.”
—Perry Garfinkel (20th century)
“It is critical vision alone which can mitigate the unimpeded operation of the automatic.”
—Marshall McLuhan (1911–1980)