A phase transition is a qualitative change in equilibrium behavior as a control variable such as temperature, pressure or magnetic field is varied. Different phases are often distinguished by an order parameter: a macroscopic quantity that is zero in one phase and nonzero in another.

For a ferromagnet, magnetization $M$ is an order parameter. Above the critical temperature $T_c$, thermal disorder gives $M=0$ in zero applied field. Below $T_c$, the system can develop spontaneous magnetization with either sign.

A simple Landau model writes the free energy near the transition as $$F(M)=F_0+a(T-T_c)M^2+bM^4,$$ with $a,b>0$. Equilibrium values minimize $F$. For $T>T_c$, the quadratic coefficient is positive and the minimum is at $M=0$. For $T<T_c$, it becomes negative and two minima appear at $$M=\pm\sqrt{\frac{a(T_c-T)}{2b}}.$$ The order parameter therefore turns on continuously as the critical point is crossed.

Near a continuous transition, fluctuations can remain correlated over increasingly large distances. The correlation length is the characteristic distance over which local fluctuations remain statistically related; it can become very large near a critical point, accompanied by strong response and approximately scale-free behavior.

A first-order transition instead has coexistence of distinct phases and discontinuous changes in quantities such as density or entropy, often with latent heat in thermal transitions.

The microscopic physics differs between magnets, fluids and other systems, but order parameters, symmetry breaking, free-energy minima, fluctuations and critical scaling provide a reusable language for collective behavior.