# If more heat means more energy and negative temperature is hotter than any temperature,has it more energy than any positive temperature's energy?

If heat means energy and negative temperature is the hottest thing possible (it is hotter than infinite positive temperature), then, has it infinite energy?

• How do you define negative temperature? – Sanya Nov 25 '16 at 23:18
• @Sanya from $\frac{1}{T} = \frac{\partial S}{\partial U}$ when $T<0$ – hyportnex Nov 26 '16 at 1:19

then, has it infinite energy?

This does not follow:

Take a pair of dice as our toy model, with macrostates characterized by the energy of the system given by the sum $n$ of dots.

The number $\Omega = \Omega(n)$ of compatible microstates for a given macrostate is the number of ways you can roll a particular sum, ie \begin{align} Ω(2) &= Ω(12) = 1 \\ Ω(3) &= Ω(11) = 2 \\ Ω(4) &= Ω(10) = 3 \\ Ω(5) &= Ω(9) = 4 \\ Ω(6) &= Ω(8) = 5 \\ Ω(7) &= 6 \end{align} or $$Ω(2≤n≤7) = n - 1 \\ Ω(7≤n≤12) = 13 - n$$

For convenience, we also set $$Ω(n<2) = Ω(n>12) = 0$$

In statistical mechanics, temperature is not defined as an average energy, but rather (in suitable units) as the reciprocal of thermodynamic $β$ and thus $$T = Ω/Ω'$$

For our discrete system, we replace the derivative with the mean of the forward and backward difference quotient of step size 1, ie \begin{align} Ω'(n) &= 1/2 \left( \frac{Ω(n+1) - Ω(n)}1 + \frac{Ω(n) - Ω(n-1)}1 \right) \\&= 1/2 \left( Ω(n+1) - Ω(n-1) \right) \end{align}

It follows \begin{align} Ω'(2≤n<7) &= 1 \\ Ω'(n=7) &= 0 \\ Ω'(7<n≤12) &= -1 \end{align} and thus \begin{align} T(2≤n<7) &= Ω(n) = n - 1 > 0 \\ T(n=7) &= ∞ \\ T(7<n≤12) &= -Ω(n) = n - 13 < 0 \end{align}

Even though our model has no notion of kinetic energy, there's still a nice relationship between total energy and temperature.

This is particularly instructive if we offset our energy scale:

For $E = n - 7$ we have \begin{align} E(T>0) &= T - 6 ∈ [-5,0[ \\ E(T=∞) &= 0 \\ E(T<0) &= T + 6 ∈ ]0,5] \end{align} As we see here, negative temperatures correspond to higher energies, and it only takes a finite amount of energy to go from positive to infinite and from infinite to negative temperature states.

• But correct me if i'm wrong: Negative temperature is hotter than any positive temperature right? So if we put a system with a positive temperature next to a negative one, then that "more than infinite heat" from the negative temperature system will pass to the positive one...so if it is (more than) infinitely hot then that amount of heat will be transmitted giving it infinite energy (because heat is energy) isn't it? – Noduagg Nov 26 '16 at 9:48

Actually, there is no such thing as equilibrium negative temperature. Roughly speaking, you can only get negative temperature in systems where the energy is bounded from above. Kinetic energy is not bounded from above, so you cannot get negative temperature in the degrees of freedom related to kinetic energy, and these degrees of freedom are present in all real systems. Therefore, you can only get negative temperatures for some, but not all degrees of freedom, such as degrees of freedom related to spin. Therefore, negative energy cannot have infinite (specific) energy, as it is only possible in systems with bounded energy.

• But correct me if i'm wrong: Negative temperature is hotter than any positive temperature right? So if we put a system with a positive temperature next to a negative one, then that "more than infinite heat" from the negative temperature system will pass to the positive one...so if it is (more than) infinitely hot then that amount of heat will be transmitted giving it infinite energy (because heat is energy) isn't it? – Noduagg Nov 26 '16 at 9:50
• @Noduagg: In my answer, I tried to explain that negative temperature systems cannot have infinite energy. – akhmeteli Nov 26 '16 at 15:46
• so, even if it has infinite heat (it is infinitely hot) and heat means energy (it should have infinite energy then) doesn't have it? why? if there's infinite heat, it should exist infinite energy too – Noduagg Nov 26 '16 at 23:25
• @Noduagg: Again, a negative temperature system cannot have infinite energy – akhmeteli Nov 26 '16 at 23:31
• @Noduagg: Heat is not the same as energy. Unlike energy, heat is not a function of the state of a system. – akhmeteli Nov 27 '16 at 0:03