Phase Transitions

A short discussion about phase transitions is given. A first-order phase transition is characterized by a discontinuity in a first derivative of the free energy, F, such as the energy, E, the entropy, S and the magnetization, M. A first-order transition occurs in a nematic liquid crystal, where at low temperature, T the elongated molecules are ordered in some direction in space, while a random molecular arrangement occurs at high T. At the critical temperature, T _c, the system co-exists in two states (phases), ordered (1) and disordered (2), of equal free energies, F _ordered = F _disordered, which leads to a finite energy difference, (E _1,2 (called latent heat) and an entropy difference, (S _1,2. Therefore, the function f_T (E) = n(E)exp[−E/k _B T] (n(E) is the density of states) has two peaks at T _c, around E _ordered and E _disordered, which are very sharp in a macroscopic system becoming smeared in a small system.

In a second-order phase transition the first derivatives of the free energy are continuous, while at T _c, a divergence to infinity (controlled by critical exponents) occurs in second derivatives of F, such as the specific heat, C_V , and the magnetic susceptibility, ( _T . Thus, C_V /N ~ T _c/(T − T _c)^α, χ _T /N ~ T _c/(T − T _c)^γ, and ξ ~ T _c/(T − T _c)^ν, where ξ is the correlation length. Such a transition occurs in the Ising model of spins, σ _i  = +1 or σ _i  = −1, on a square lattice with nearest neighbor attractions. The divergence stems from increasing fluctuations, which is a manifestation of the creation of comparable-size “droplets” of same spin. Another measure of the droplet effect is the spin-spin correlations as a function of distance. Thus, two spins on the same droplet will have the same sign, therefore they will be correlated. As T → T _c, the droplet size increases together with the fluctuations and the correlation length, ξ.