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. Boltzmann's ergodicity hypothesis states, in equilibrium each possible state of the system has the same energy and the same probability of existing. This means each possible state is equally populated. Viewing this hypothesis it would look like the system would transition over time gradually visiting each possible state. The system does not necessarily violate the ergodicity hypothesis. The problem is the transition to each possible state can sometimes take much longer than our lifetimes or maybe even longer than the existence of the universe. We can determine this time as it is proportional to the probability. Looking at the probability of a microscopic configuration we get:

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