One Dimension
The thermodynamic limit exists as soon as the interaction decay is with .
- In the case of ferromagnetic interaction with Dyson proved, by comparison with the hierarchical case, that there is phase transition at small enough temperature.
- In the case of ferromagnetic interaction, Fröhlich and Spencer proved that there is phase transition at small enough temperature (in contrast with the hierarchical case).
- In the case of interaction with (that includes the case of finite range interactions) there is no phase transition at any positive temperature (i.e. finite ) since the free energy is analytic in the thermodynamic parameters.
- In the case of nearest neighbor interactions, E. Ising provided an exact solution of the model. At any positive temperature (i.e. finite ) the free energy is analytic in the thermodynamics parameters and the truncated two-point spin correlation decays exponentially fast. At zero temperature, (i.e. infinite ), there is a second order phase transition: the free energy is infinite and the truncated two point spin correlation does not decay (remains constant). Therefore is the critical temperature of this case. Scaling formulas are satisfied.
Read more about this topic: Ising Model
Famous quotes containing the word dimension:
“Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
—Tom Wolfe (b. 1931)
“God cannot be seen: he is too bright for sight; nor grasped: he is too pure for touch; nor measured: for he is beyond all sense, infinite, measureless, his dimension known to himself alone.”
—Marcus Minucius Felix (2nd or 3rd cen. A.D.)