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:
“Gold and silver are but merchandise, as well as cloth or linen; and that nation that buys the least, and sells the most, must always have the most money.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“The important question is not, what will yield to man a few scattered pleasures, but what will render his life happy on the whole amount.”
—Joseph Addison (1672–1719)
“An audience is never wrong. An individual member of it may be an imbecile, but a thousand imbeciles together in the dark—that is critical genius.”
—Billy Wilder (b. 1906)