Negative Temperatures Are Hotter Than Positive Ones: Where Adding Energy Decreases Entropy
Before discussing negative temperatures, we need a clear definition:
Temperature is a property of a system that determines whether or not it is in thermal equilibrium with other systems. Systems in thermal equilibrium have the same temperature T. [1]
This definition, however, requires understanding what an equilibrium state is: it’s a state where all the macroscopic physical properties of a system are uniform and unchanging in time.
Mathematically, temperature is defined as:
where S is entropy and U is internal energy.
This formulation, expressed in SI units, defines absolute zero as 0 K—the coldest possible temperature—where there is no thermal energy in the system, only zero-point energy (a lower bound that always exists). However, true absolute zero is experimentally unreachable, as stated by the third law of thermodynamics. Achieving it would require an infinite sequence of thermodynamic processes.
Negative temperatures emerge when the energy of a system has an upper bound. One such system consists of N fermions (spin ½) with a magnetic moment μ in an external magnetic field H.
The energy of this system is given by:
Since this includes a kinetic energy term and the series converges, there is an upper bound on the energy.
To calculate the entropy, we work within the canonical ensemble and derive the relevant expressions [2].
If we examine the temperature definition graphically, at the entropy curve’s peak, the slope becomes zero. To the left, the slope is positive (yielding positive temperatures), and to the right, the slope becomes negative, resulting in negative temperatures. At the peak, the temperature approaches +∞ from the left and –∞ from the right.
Thus, the temperature order from coldest to hottest is:
[+0 K, …, 300 K, …, +∞ K, …, –∞ K, …, –300 K, …, –0 K].
This non-intuitive interval stems from our mathematical definition of temperature. In systems with negative temperatures, adding energy actually *reduces* entropy, making the system more ordered—the opposite of what occurs at positive temperatures. [3]
If a system with negative temperature is brought into contact with one at positive temperature, heat flows from the negative-temperature system to the positive one, until equilibrium is reached. The final equilibrium temperature is a positive value higher than the original one. This counterintuitive behavior means negative-temperature systems only exist under carefully controlled conditions, with minimized interactions.
Allowing for negative temperatures in the Carnot cycle leads to a conclusion that the Kelvin–Planck formulation of the second law of thermodynamics requires revision (Ramsey) [2].
We have seen that if we extend the scope of the range of temperatures that we are interested in we get some un-intuitive results. The values of hotter systems may feel counterintuitive as negative systems are hotter than hypothetical systems at + infinity Kelvin as well as the fact that for negative systems when we add energy to the system it becomes more ordered rather than less ordered.
References:
1. Finn, C. B. P. (1993). *Thermal Physics* (2nd ed., Chapter 1). London: Chapman & Hall.
2. Pathria, R. K., & Beale, P. D. (2011). *Statistical Mechanics* (3rd ed., p. 80). Amsterdam: Elsevier.
3. Ramsey, N. F. (1956). Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures. *Physical Review*, 103(1), 20–28. https://doi.org/10.1103/PhysRev.103.20
External Resources:
– Finn’s Thermal Physics (3rd Ed): https://ia801503.us.archive.org/15/items/dzofferz_gmail_My/Finn%27s%20Thermal%20Physics%20%283rd%20Ed%29.pdf
– Pathria & Beale: http://linux0.unsl.edu.ar/~froma/MecanicaEstadistica/Bibliografia/PathriaBeale.pdf
– Ramsey (1956): https://physics.umd.edu/courses/Phys404/Anlage_Spring11/Ramsey-1956-Thermodynamics%20and%20S.pdf
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