Effective Theories
I have been building models for years. Models and effective theories... As an example assume that you would like to describe something as complex as a magnet. The first step should be building a simple model keeping only a few key physical ingredients. So for a magnet, consider that all its atoms live on a regular grid and use the property that each atom can behave like a little magnet itself. Then assume that the atomic magnets are in only two states: pointing up or down. Focus on a 2d magnet first. This becomes the simplest model of a magnet: the Ising model. Onsager, in the 1940's, solved it. Solving here means describing its behaviour when changing the temperature. He proved that 2d magnets acquire a macroscopic magnetisation at sufficiently low temperature: all the atomic magnets point in the same direction.
For a high energy physicist like me, this description is not satisfactory. This is still far too complex and does not focus on the key features of the phenomenon. Does it really matter that the model is 2d? Does the result change if the grid is chosen to be triangular and not rectangular? Following ideas introduced in the 1950's by the Russian physicists Ginzburg and Landau, we can get rid of a lot of these details and use an effective description. The only important variable of the problem is the macroscopic magnetisation. It should be either vanishing at high temperature when the magnet is demagnetised and a constant at low temperature below a transition temperature. This behaviour is essentially captured by modeling the potential energy of the magnet as shown in the figure above. The potential is like the profile of a mountain and the blue dot represents the position of a ball. The ball wants to stay at the minimum of the potential. The position of the ball along the x axis gives the magnetisation. In the left figure, at high temperature, the balls sits at a unique minimum whose position corresponds to a vanishing magnetisation. This is the high temperature phase. When the temperature is lowered, the potential is modified and two minima appear. The ball rolls towards the right minimum where the magnetisation is not zero. This gives a perfect effective description of the phenomenon with only one relevant parameter: the temperature of the transition. In an effective description, it cannot be calculated and has to be measured. If one wants to have an order of magnitude estimate of this temperature, one has to go back to the microscopic description with atomic magnets. The temperature can then be estimated to be of the order of the energy of interaction between the atoms as this is the only energy which appears in the Ising model.
This type of construction used to work perfectly. Up until nearly twenty years ago when the discovery that the Universe is currently undergoing a phase of acceleration changed the picture radically. No effective theory can really tell us why the rate of this acceleration is so low. A new era dawned.