Midterm Solution

Problem 1

Describe the difference between a first order and a second order phase transition. Give examples of each type. (10 points)

When determining the type of phase transition you must look at the change in the physical quantities of the material. If the change is sudden, a discontinuity in the first derivative of the free energy, then you have a first order phase transition. Examples of this would be water going from a liquid state to a gaseous state. A second order phase transition is continuous in the first derivative of the free energy and displays discontinuity only in the second derivative. With this fact, second order phase transitions will experience spontaneous symmetry breaking. Examples of a second order phase transition includes ferromagnetism, antiferromagnetism, and superconductivity.

Problem 2

Give some examples of spontaneous symmetry breaking. How does it depend on the system size? Explain how it can happen, in apparent violation of the ergodicity hypothesis of Boltzmann. According to Boltzmann, what determines the probability of some microscopic configuration (10 points)

Examples of spontaneous symmetry breaking are the same as those in second order phase transitions ferromagnetism, antiferromagnetism, and superconductivity. In order to have spontaneous symmetry breaking you must have an infinitely large system. If we have a finite system, spontaneous symmetry breaking could not occur. An example of spontaneous symmetry breaking as previously stated would be ferromagnetism. If we have a finite number of spins in our system then you will not be able to find a phase transition to the ferromagnetic state. With an infinite number of spins you will be in the thermodynamic limit and a ferromagnetic state will be able to be found.