Max Entropy & Negative Temperature: Exploring the Limits

In summary: This is because in the case of negative temperatures, the entropy decreases as energy increases, which is the opposite of what happens with positive temperatures. This is also reflected in the fact that negative temperatures have a higher value for 1/T (the partial derivative of entropy with respect to energy) compared to positive temperatures. In summary, negative temperatures are defined in certain unstable systems where the entropy decreases as energy increases, and they can have a higher value for 1/T compared to positive temperatures. However, they cannot exist in macroscopic objects and only occur in a few particles. This definition of temperature may not always be applicable in all cases and there may be other
  • #1
turin
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Is there a theoretical maximum entorpy? I just read on Wikipedia that there are certain conditions in which the absolute temperature can be negative, based on the definition: T-1 = dS/dE. If the entropy has nowhere to go but down (from Smax), can temperature still be defined?
 
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  • #2
Temperature has not a top limit... so entropy has not a top limit as well ...

Nernst's enunciate of the 3rd principle says that the limit as T goes to 0 of entropy is 0...

Lets take the first principle: [tex]dE = TdS + y_{i} dx_{i}[/tex]

T is defined as:

[tex] T = \frac{ \partial E}{\partial S} [/tex] and the variation of energy with entropy is usually positive.

If "E" is the ocupation of energy levels, and "S" is disorder...

[tex] \frac{1}{T} = \frac{ \partial S}{\partial E} > 0 [/tex]

If we have a system with a finite number of energy levels, if E increases the order increases as well and entropy decreases, so that partial is < 0 and T < 0 !

That can occur for example in a population inversion:

I can draw a little scheme:

E2:
E1: oooooo

T = 0, S = min

E2: o
E1: ooooo

T > 0, S > 0

E2: ooo
E1: ooo

T = ± infinity
S = max

E2: ooooo
E1: o

E increases
S decreases
so T < 0

Negative temperatures are over infinite temperature.

Bye

-

I forgot to say that this can only occur in unstable systems, and a macroscopic object can not be at a negative temperature, only can occur in a few particles...
 
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  • #3
This puzzled me much time ago. I have read a lot of books talking about what migui has explained about inverting population.

My question is: Are negative temperatures "hotter" than positives ones?. I mean, does it have smaller levels of thermal energy?. I was wondering if it would be employable in classical engines or something like that. All of you know what happens if we substitute for negative temperatures in the Carnot efficiency or in Kelvin relation. Engines would generate work without any energy added, and freezers would produce mechanical work. The classical thermodynamics fall down automatically with negative temperatures. So what happens with this contradiction?.
 
  • #4
How can T = (+/-)infinity at that midpoint in in the 6 particle example? Shouldn't T = (+/-)infinity correspond to a change in entropy requiring an infinite change in energy (which it doesn't seem to in the 6 particle example)? Do I have to extend your example to an infinite number of particles (or a continuum) so that the inflection point has perfectly zero slope?

I find that Smax = S3,3 = kB ln(6!/(3!)2) = 3.00 kB

but

S2,4 = S4,2 = kB ln(6!/2!4!) = 2.71 kB

(It's been a while, so please correct me if I'm disrespecting the entropy calculation)

So, the way I'm seeing it (in this discrete example, at least), the temperature is not infinite, it is just not defined, because it should equal a positive and negative finite number according to the definition (depending on the direction of ΔE). In that sense, I would expect the limit, rather than approaching some infinite value, to simply be undefined (even mathematically).

I guess I am wondering if this is really the correct (sufficient) way to define temperature:

T = (∂S/∂E)-1

since it doesn't seem to be definable in this manner in all cases. Are there other parts of the definition besides this mathematical statement that I have left out?
 
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  • #5
"Are negative temperatures "hotter" than positives ones?".

Yes
 

What is maximum entropy?

Maximum entropy is a concept in statistical mechanics that refers to the maximum level of disorder or randomness in a system. It is often used to describe the state of a system in equilibrium, where the system has reached a state of maximum disorder and has the highest possible entropy.

How is maximum entropy related to negative temperature?

Negative temperature is a concept that describes a state of a system where the particles have higher energy levels than the particles in a system with positive temperature. In this state, the system has reached a state of maximum entropy, as the particles are in their most disordered state.

What are the limits of maximum entropy?

The concept of maximum entropy does not have any specific limits, as it is a theoretical concept used to describe the state of a system in equilibrium. However, in practical terms, the maximum entropy of a system is limited by the number of particles and the energy levels they can occupy.

How is maximum entropy measured?

Maximum entropy is measured using the concept of entropy, which is a measure of the disorder or randomness in a system. In statistical mechanics, entropy is calculated using the Boltzmann equation, which takes into account the number of particles and their energy levels.

What are the applications of maximum entropy and negative temperature?

Maximum entropy and negative temperature have applications in various fields, including physics, chemistry, and information theory. They are used to describe the state of systems in equilibrium and have been applied to understand the behavior of complex systems, such as black holes and quantum systems.

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