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2 1 The two distinguishable particles can each be in one of the two boxes. The energy of the system depends on which box the particles occupy. Consider the system illustrated. The system consists of two distinguishable particles, each of which can be in either of two boxes. The system is in thermal equilibrium with a heat bath at temperature T. Assume that the energy of a particle is zero if it is in the left box and r if it is in the right box. There is also a correlation energy term that increases the energy by A if the two particles are in the same box. (a) Enumerate the 22 = 4 microstates and their corresponding energy. (b) Calculate the partition function Z for arbitrary values of r and A and use your result to find the mean energy. (c) What is the probability that the system is in a particular microstate? 1 and A = 15. Sketch the qualitative behavior of the (d) Suppose that r = heat capacity C as a function of T.

2 1 The two distinguishable particles can each be in one of the two boxes. The energy of the system depends on which box the particles occupy. Consider the system illustrated. The system consists of two distinguishable particles, each of which can be in either of two boxes. The system is in thermal equilibrium with a heat bath at temperature T. Assume that the energy of a particle is zero if it is in the left box and r if it is in the right box. There is also a correlation energy term that increases the energy by A if the two particles are in the same box. (a) Enumerate the 22 = 4 microstates and their corresponding energy. (b) Calculate the partition function Z for arbitrary values of r and A and use your result to find the mean energy. (c) What is the probability that the system is in a particular microstate? 1 and A = 15. Sketch the qualitative behavior of the (d) Suppose that r = heat capacity C as a function of T.

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2 1 The two distinguishable particles can each be in one of the two boxes. The energy of the system depends on which box the particles occupy. Consider the system illustrated. The system consists of two distinguishable particles, each of which can be in either of two boxes. equilibrium with a heat bath at temperature T. The system is in the Assume that the energy of a particle is zero if it is in the left box and r if it is in the right box. There is also a correlation energy term that increases the energy by A if the two particles are in the same box. (a) Enumerate the 22 = 4 microstates and their corresponding energy. (b) Calculate the partition function Z for arbitrary values of r and A and use your result to find the mean energy. (c) What is the probability that the system is in a particular microstate? (d) Suppose that r = 1 and A = 15. Sketch the qualitative behavior of the heat capacity C as a function of T.
Transcribed Image Text:2 1 The two distinguishable particles can each be in one of the two boxes. The energy of the system depends on which box the particles occupy. Consider the system illustrated. The system consists of two distinguishable particles, each of which can be in either of two boxes. equilibrium with a heat bath at temperature T. The system is in the Assume that the energy of a particle is zero if it is in the left box and r if it is in the right box. There is also a correlation energy term that increases the energy by A if the two particles are in the same box. (a) Enumerate the 22 = 4 microstates and their corresponding energy. (b) Calculate the partition function Z for arbitrary values of r and A and use your result to find the mean energy. (c) What is the probability that the system is in a particular microstate? (d) Suppose that r = 1 and A = 15. Sketch the qualitative behavior of the heat capacity C as a function of T.
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