text: Free energies in the presence of small amplitude fluctuations, e.g. in concentration, can be evaluated using an approximation introduced by Ginzburg and Landau to describe magnetic field gradients in superconductors. This approach allows one to approximate the free energy as an expansion in terms of the concentration gradient , a vector. Since free energy is a scalar and we are probing near its minima, the term proportional to  is negligible. The lowest order term is the quadratic expression , a scalar. Here  is a parameter that controls the free energy cost of variations in concentration .
  The Cahn–Hilliard free energy is then
  where  is the bulk free energy per unit volume of the homogeneous solution, and the integral is over the volume of the system.
  We now want to study the stability of the system with respect to small fluctuations in the concentration , for example a sine wave of amplitude  and wavevector , for  the wavelength of the concentration wave. To be thermodynamically stable, the free energy change  due to any small amplitude concentration fluctuation , must be positive.
  We may expand  about the average composition c as follows:
  :and for the perturbation  the free energy change is
  :when this is integrated over the volume , the  gives zero, while . So, then
  As , thermodynamic stability requires that the term in brackets be positive. The  is always positive but tends to zero at small wavevectors, large wavelengths. Since we are interested in macroscopic fluctuations, , stability requires that the second derivative of the free energy be positive. When it is, there is no spinodal decomposition, but when it is negative, spinodal decomposition will occur. Then fluctuations with wavevectors  become spontaneously unstable, where the critical wave number  is given by:
  :which corresponds to a fluctuations above a critical wavelength