Negative Temperatures (Two level systems)

In summary, negative temperatures in two level systems occur when the majority of particles are in a high energy state, resulting in a negative temperature on the Kelvin scale. This differs from positive temperatures where particles have a range of energies. Some real-world examples include spin systems and certain types of lasers. Negative temperatures are measured and calculated using the Boltzmann distribution. Potential applications include quantum computing, ultra-sensitive sensors, and gaining insights into extreme temperatures and the fundamental laws of the universe.
  • #1
Dox
26
1
Hello everybody!

Homework Statement



I'm interested in a statistical system with N particles which possible energies are, [tex] 0,\epsilon>0[/tex].

a) Find the entropy of the system.

b) Find the most probable [tex]n_0[/tex] and [tex]n_1[/tex], and find the mean square fluctuations of these quantities.

c) Find the temperature as a function of [tex]U[/tex], and show it can be negative.

d) What happens when a system of negative temperature is allowed to exchange heat with a system of positive temperature?





Homework Equations



[tex]W(n_0,n_1)=\frac{N!}{n_0!n_1!}[/tex]

[tex]S=k\ln \Omega, [/tex] with [tex]\Omega=\sum'_{\{n\}} W(n_0,n_1).[/tex]

[tex]\frac{1}{T}=\frac{\partial S}{\partial U}.[/tex]


The Attempt at a Solution



I try the following,
[tex]N= n_0+n_1[/tex] and [tex]U=n_1 \epsilon_1-n_0\epsilon_0= n_1\epsilon_1.[/tex], so

[tex]W=\frac{N!}{(N-n_1)! n_1!}\;\Rightarrow \; n_1^*=\frac{N}{2}.[/tex]
That should answer the question (b).

Meanwhile, by using the Stirling's formula,

[tex]S=k\left(N\ln N -(N-n_1)\ln (N-n_1) - n_1\ln n_1,[/tex]
I used [tex]N=n_1+n_2[/tex] for deriving the temperature, but I got

[tex]\frac{1}{T}= \frac{k}{\epsilon_1}\ln\left(\frac{n_1+n_0}{n_1}\right),[/tex] which is not negative, :uhh:

Finally, I guess that if I allow a sut up with negative temperature interact with a standard thermodynamical system the second law breaks... Am I right??
 
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  • #2


Thank you for your post. Your approach to solving this problem looks promising. I would suggest double checking your calculations for part (b) to ensure that you have found the correct values for n_0 and n_1. As for part (c), it is important to note that temperature is defined as the inverse of the derivative of entropy with respect to energy. In your calculation, you have used N=n_1+n_2, which may not be the correct expression for energy in this system. I would recommend revisiting the expressions for energy and entropy and trying to find a more accurate expression for temperature.

As for part (d), you are correct in thinking that allowing a system with negative temperature to interact with a system with positive temperature would violate the second law of thermodynamics. This is because heat would flow from the system with negative temperature to the system with positive temperature, which is against the natural direction of heat flow. This is why systems with negative temperature are considered to be unstable and cannot exist in equilibrium.

I hope this helps and good luck with your calculations!
 

1. What are negative temperatures in two level systems?

Negative temperatures in two level systems occur when the majority of the particles in a system are in a high energy state rather than a low energy state, resulting in a negative temperature on the Kelvin scale. This is a counterintuitive concept, as it goes against our everyday understanding of temperature as a measure of the average kinetic energy of particles.

2. How do negative temperatures differ from positive temperatures?

Unlike positive temperatures, where the particles in a system have a range of energies, negative temperatures only occur when the majority of the particles are in a high energy state. This is due to the fact that particles in negative temperature systems have a higher energy than those at positive temperatures.

3. What are some real-world examples of negative temperatures in two level systems?

One example of a system that can exhibit negative temperatures is a spin system, where particles can have two possible spin states. Another example is certain types of lasers, where the majority of particles are at a high energy state due to the pumping of energy into the system.

4. How are negative temperatures measured and calculated?

Negative temperatures are measured using the Boltzmann distribution, which relates temperature to the energy states of particles. To calculate the temperature, the energy levels of the system must be known and the number of particles in each energy state must be determined.

5. What are the potential applications of negative temperatures in two level systems?

Negative temperatures have the potential to be used in various applications, such as in quantum computing and ultra-sensitive sensors. They can also provide insights into the behavior of systems at extreme temperatures, which can help us better understand the universe and its fundamental laws.

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