Adapted from P. Coles, 1999, The Routledge Critical Dictionary of the New Cosmology, Routledge Inc., New York. Reprinted with the author's permission. To order this book click here:

In particular many-particle systems, phase transitions take place when processes occur that move the system between some disordered phase, characterised by a certain degree of symmetry, and an ordered phase with a smaller degree of symmetry. In this type of order-disorder transition, some macroscopic quantity called the order parameter (here denoted by the symbol Phi) grows from its original value of zero in the disordered phase. The simplest physical examples of materials exhibiting these transitions are ferromagnetic substances and crystalline matter. At temperatures above a critical temperature TC (the Curie temperature), the stable state of a ferromagnet is disordered and has net magnetisation M = 0; the quantity M in this case plays the role of the order parameter. At temperatures below TC a nonzero magnetisation appears in different domains (called Weiss domains) and its direction in each domain breaks the rotational symmetry possessed by the original disordered phase at T > TC. In the crystalline phase of solids the order parameter represents the deviation of the spatial distribution of ions from the homogeneous distribution they have at their critical temperature, which is simply the freezing point of the crystal, Tf. At temperatures T < Tf the ions are arranged on a regular lattice, which possesses a different symmetry to the original liquid phase.

The lowering of the degree of symmetry of the system takes place even though the laws of physics that govern its evolution maintain the same degree of symmetry, even after the phase transition. For example, the macroscopic equations of the theory of ferromagnetism and the equations in solid-state physics do not favour any particular spatial position or direction. The ordered states that emerge from such phase transitions do, however, have a degree of symmetry which is less than that governing the system. We can say that the solutions corresponding to the ordered state form a degenerate set of solutions with the same symmetry as that possessed by the laws of physics. Returning to the above examples, the magnetisation M can in theory assume any direction. Likewise, the ions in the crystalline lattice can be positioned in an infinite number of different ways. Taking into account all these possibilities, we again obtain a homogeneous and isotropic state. Any small fluctuation, in the magnetic field of the domain for a ferromagnet or in the local electric field for a crystal, will have the effect of selecting one preferred solution from this degenerate set, and the system will end up in the state corresponding to that fluctuation. Repeating the phase transition with random fluctuations will eventually produce randomly aligned final states.

This is a little like the case of a free particle with velocity v and position r, described in Newtonian mechanics by the requirement that dv/dt = 0. This (very simple) law has both translational and rotational symmetries. The solutions of the equation are of the form r(t) = r0 + v0t, with some arbitrary initial choice of r = r0 and v = v0 at t = 0. These solutions form a set which respects the symmetry of the original equation. But the initial conditions r0 and v0 select, for one particular time, a solution from this set which does not have the same degree of symmetry as that of the equations of motion.

A symmetry-breaking transition, during which the order parameter Phi grows significantly, can be caused by external influences of sufficient intensity. For example, a strong magnetic field can magnetise a ferromagnet, even above the Curie temperature. Such phenomena are called induced symmetry-breaking processes, in order to distinguish them from spontaneous symmetry-breaking. The spontaneous breaking of a symmetry comes from a gradual change of the parameters of a system, such as its temperature.

It is useful to consider the free energy F of a system, which is defined in thermodynamics to be F = E - TS, where E is the internal energy, T is the temperature and S is the entropy; the condition for the system to have a stable equilibrium state is that F must have a minimum. The free energy coincides with the internal energy only at T = 0. At higher temperatures, whatever the form of E, an increase in the entropy (i.e. disorder) generally leads to a decrease in the free energy F, and it is therefore favourable. for systems in which there is a phase transition, F is a function of the order parameter Phi. Under some circumstances the free energy F must have a minimum at Phi = 0 (corresponding to the disordered state), while in others it must have a minimum for some Phi neq 0 (corresponding to the ordered state).

Let us now consider the simplest possible example. Suppose that the situation we are dealing with respects a symmetry between Phi and -Phi. The free energy function F must also respect this symmetry, so that we can expand F(Phi) in a power series containing only even powers of Phi:

F(Phi) approx F0 + alpha(T) Phi2 + beta(T) Phi4

where the coefficients alpha and beta of this expansion depend explicitly on temperature T. If alpha > 0 and beta > 0, the curve has one minimum at the origin, so the equilibrium state lies at Phi = 0. If the coefficients change in such a way that at lower temperatures alpha < 0 and beta > 0, the original minimum becomes a maximum and two new minima appear at Phi = ± sqrt(alpha / 2beta); in this case the disordered state is unstable, while the minima correspond to degenerate ordered states with the same probability. Any small external perturbation which renders one of the two minima slightly deeper or nudges the system towards one of them can make the system evolve towards one rather than the other. This is how spontaneous symmetry-breaking occurs. If there is only one parameter describing the system, say the temperature, and the coefficient alpha is written as alpha = a(T - TC) with alpha > 0, this transition is called a second-order phase transition, and it proceeds by a process known as spinodal decomposition: the order parameter appears or disappears gradually, and the difference DeltaF in free energy between the two states is infinitesimally small when T is close to TC.

There are also first-order phase transitions, in which at T = TC the order parameter appears or disappears rapidly, and the difference DeltaF between the old and new minima is finite. This difference is called the latent heat of the phase transition. In first-order phase transitions, when T decreases the phenomenon known as supercooling can occur: the system remains in the disordered state characterised by Phi = 0 even when T has fallen below the critical temperature. This is because the original minimum represents a metastable equilibrium. A local minimum of the free energy still persists there, protected by a potential barrier that prevents the system from moving immediately into the global minimum. As T decreases further, or the system is perturbed by either internal or external fluctuations, the system finally evolves into the global minimum that represents the true energetically stable state. This liberates the latent heat that was stored in the metastable state. The system, still in the ordered state, is heated up again to a temperature of around TC by the release of this latent heat, a phenomenon called reheating.

To apply these notions to cosmology we need also to take into account quantum and thermal fluctuations and other effects of non-equilibrium thermodynamics. Different symmetries in the past also involved different fields interacting according to different potentials. Nevertheless, the basic general properties of phase transitions lie behind many models of the inflationary Universe.


Kolb, E.W. and Turner, M.S., The Early Universe (Addison-Wesley, Redwood City, CA, 1990).