Negative Absolute Temperatures: Explanation & Examples

It’s a famous result in thermodynamics that you cannot reach absolute zero no matter how hard you try. Given the definition of absolute zero, it is also meaningless to speak of “colder” temperatures beyond that point. So what are negative absolute temperatures? Below I will explain what they mean, when they can occur, and how they are realized experimentally.

Concept of temperature

First, it’s helpful to review the concept of temperature. In thermodynamics, temperature is the quantity that is equal for two systems in thermal equilibrium — in other words, two systems that, when brought into thermal contact, exchange no net heat. This thermodynamic definition is macroscopic and does not depend on microscopic structure.

For our discussion we need a microscopic/statistical definition. A useful statistical definition of temperature is

1/T ≡ dS/dE,

where S is the system entropy and E is its internal energy. It is also helpful to remember the intuitive association between temperature and the average energy per degree of freedom in a system.

The Boltzmann factor and energy spectra

In statistical mechanics, the relative occupation of two energy levels follows the Boltzmann factor. For levels ε1 and ε2,

n2/n1 = exp[-(ε2 - ε1)/(kT)],

where n1 and n2 are the average occupancies and k is Boltzmann’s constant.

For systems with an infinite ladder of states bounded only from below (a ground state, then energy increasing without bound), adding energy always allows particles to occupy ever-higher states. In such systems the equilibrium occupancies decrease monotonically with energy, and the Boltzmann factor implies the temperature must be positive.

A simple two-level temperature model

Now consider a microscopic system with a finite set of energy levels. The simplest nontrivial example is a two-level system with energies ε1 and ε2 and a fixed total number of particles n. Let

n = n1 + n2 and E = n1 ε1 + n2 ε2.

Using the combinatorial entropy S = k ln Ω with Ω = n!/(n1! n2!) and Stirling’s approximation, one can write the entropy in terms of the total energy E (after eliminating n1 and n2 in favor of E and n). The important qualitative feature is that this entropy has a maximum when

E = (n/2)(ε + ε2).

That maximum means occupancies equalize (n1 = n2) at that energy. Taking the derivative dS/dE to obtain 1/T shows that the temperature is positive for E < (n/2)(ε1 + ε2), infinite at the entropy maximum, and negative for E > (n/2)(ε1 + ε2). In other words, states with more population in the higher-energy level correspond to formally negative absolute temperature.

Key formula (two-level)

One convenient way to express the sign change is to solve for the ratio n2/n1 from the energy and substitute into 1/T = dS/dE. The result (after algebra) yields a temperature that changes sign when the higher level becomes more populated than the lower one. The algebraic form depends on the chosen elimination scheme; the qualitative conclusion above is the important point.

Interpreting negative absolute temperatures

Negative-temperature states are not “colder” than zero; they are hotter than any positive-temperature state. They have higher internal energy than any positive-temperature configuration for the same bounded spectrum, and the Boltzmann factor shows that higher-energy states are more populated than lower ones.

As particles fill the available levels from below, once the highest accessible energy level is reached further energy input increases the relative population of high-energy states. When populations invert so that higher-energy states are more occupied than lower ones, the formal temperature is negative. Passing through an infinite-temperature state (equal populations) takes you from positive to negative temperature. Ordered on an axis from coldest to hottest (in terms of energy content), the sequence runs:

0+ , (positive temperatures increasing) ... , +∞ = -∞ , (negative temperatures decreasing toward 0-)

Why use β = -1/(kT)?

It is often convenient to define β ≡ -1/(kT) because this parameter maps +∞ and -∞ to the same point (β = 0) and orders states by increasing energy. With β, the two zero-temperature limits appear at opposite ends of the axis, which is sometimes a clearer way to think about accessibility and extrema.

Experimental realization of negative temperatures

Although the total energy of a generic system may be unbounded above, particular degrees of freedom can have spectra that are bounded both below and above. A canonical example is a spin system in a magnetic field: each spin has a finite set of energy states.

Experimentalists create effectively isolated, bounded subsystems by making the energy exchange between the bounded degrees of freedom and the rest of the system much slower than the internal dynamics of the bounded subsystem. Under those conditions the bounded subsystem can reach a quasi-equilibrium with an inverted population and a formally negative temperature while the rest of the system is effectively inert on the same time scale.

Because negative-temperature states correspond to population inversion, they are related to the same physics used in lasers and other devices where high-energy levels become more populated than low-energy ones.

Thermodynamic consequences and caution

One might wonder how negative temperatures affect the laws of thermodynamics (for example, the second law). The literature treats these issues carefully: negative-temperature ensembles are well-defined for systems with bounded spectra, but naïve extensions of thermodynamic statements (such as placing a negative-temperature system in contact with a conventional positive-temperature reservoir) require caution. Practical heat-engine constructions that combine negative- and positive-temperature systems remain a subject of research and careful qualification.

For a related theoretical perspective, see the following discussion of duality in the Ising model:

High Temperature and Low Temperature Duality for the Ising Model on an Infinite Regular Tree