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:
“We noticed several other sandy tracts in our voyage; and the course of the Merrimack can be traced from the nearest mountain by its yellow sand-banks, though the river itself is for the most part invisible. Lawsuits, as we hear, have in some cases grown out of these causes. Railroads have been made through certain irritable districts, breaking their sod, and so have set the sand to blowing, till it has converted fertile farms into deserts, and the company has had to pay the damages.”
—Henry David Thoreau (1817–1862)
“Let children know you are human. It’s important for children to see that parents are human and make mistakes. When you’re sorry about something you’ve said or done, apologize! But don’t sound guilt ridden. It is best when parents apologize in a manner that is straightforward and sincere.”
—Saf Lerman (20th century)
“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)