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Consider a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +ɛ and -E. Assume that N is large. (a) Working in the microcanonical ensemble, find the entropy of the system S(E, N) as a function of fixed total energy E and number N (Hint: it is useful to consider the numbers N. and N. of up and down objects). (b) Using your result from (a) find the temperature T as a function of energy E and number N. Show that T will be negative if E>0. (c) What happens if such a system (1) with T1<0 comes into thermal contact with another such system (2) with T2>0? Does T, increase or decrease? Does T2 increase or decrease? In which direction does the heat flow?

Consider a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +ɛ and -E. Assume that N is large. (a) Working in the microcanonical ensemble, find the entropy of the system S(E, N) as a function of fixed total energy E and number N (Hint: it is useful to consider the numbers N. and N. of up and down objects). (b) Using your result from (a) find the temperature T as a function of energy E and number N. Show that T will be negative if E>0. (c) What happens if such a system (1) with T1<0 comes into thermal contact with another such system (2) with T2>0? Does T, increase or decrease? Does T2 increase or decrease? In which direction does the heat flow?

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Consider a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +ɛ and -E. Assume that N is large. (a) Working in the microcanonical ensemble, find the entropy of the system S(E, N) as a function of fixed total energy E and number N (Hint: it is useful to consider the numbers N. and N. of up and down objects). (b) Using your result from (a) find the temperature T as a function of energy E and number N. Show that T will be negative if E>0. (c) What happens if such a system (1) with T1<0 comes into thermal contact with another such system (2) with T2>0? Does T, increase or decrease? Does T2 increase or decrease? In which direction does the heat flow?
Transcribed Image Text:Consider a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +ɛ and -E. Assume that N is large. (a) Working in the microcanonical ensemble, find the entropy of the system S(E, N) as a function of fixed total energy E and number N (Hint: it is useful to consider the numbers N. and N. of up and down objects). (b) Using your result from (a) find the temperature T as a function of energy E and number N. Show that T will be negative if E>0. (c) What happens if such a system (1) with T1<0 comes into thermal contact with another such system (2) with T2>0? Does T, increase or decrease? Does T2 increase or decrease? In which direction does the heat flow?
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