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

Question

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

Answer 1

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.

Answer 2

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.