Answer to Question #111834 in Classical Mechanics for Tanya Saini

Question

Consider an isolated system of N>>1distinct particles ,each of which can be in one of two states with energies 0 and E respectively.give the range of energies for which the system has negative temperature.The system is separated into two sub-systemsA and B,each having N/2 particles.Subsystem A has energy EA =NE/8 and B has energy EB=5 NE/8.Using the above result ,specify which of these subsystems has a positive temperature and negative temperature.The two subsystems are allowed to exchange energy.once equilibrium is attained,is the temperature of two systems positive or negative..?

Answer 1

For an isolated system of N>>1 distinct particles at temperature $T\to\infty$ both energy states for particles will be equally probable, so an infinitely large temperature will correspond to the system energy $E_S=N\frac{E}{2}$ [1]. If the system goes into a state with an inverted population of levels i.e. the number of particles at the upper energy level will be greater than at the lower one we can talk about a state with a negative temperature.

Using this result we can specify that the subsystem of $\frac {N}{2}$ particles have a negative temperature if $E_{halfN}>\frac{N}{2}\frac{E}{2}=NE/4$ . It should be pointed out that a subsystem B (or A) consisting of N/2 particles cannot have an energy greater than $E_B=E\cdot \frac{N}{2}$. At this energy all the particles are at the upper level of energy E. Thus it seems that an error has crept into the task condition and B has an energy $E_B=3 NE/8>NE/4$ . Thus subsystem A has positive temperature, and subsystem B a negative one.

When the two subsystems are allowed to exchange energy we get the overall energy $E_S=E_A+E_B=NE\frac{1}{8}+NE\frac{3}{8}=N\cdot\frac{ E}{2}$ and the system has a temperature formally corresponding to an infinite value $T\to\infty$.

Answer: The range of energies $E_S$ for which the system of N distinct particles has negative temperature is $NE>E_S>N\frac{E}{2}$ . The subsystem A has positive temperature, and system B a negative one. When the two subsystems are allowed to exchange energy the system will have a temperature formally corresponding to an infinite value.

[1] https://en.wikipedia.org/wiki/Negative_temperature

https://en.wikipedia.org/wiki/Negative_temperature#/media/File:Temperature_vs_E_two_state.svg