Certain thermodynamic systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers on non-thermodynamic Celsius or Fahrenheit scales, which are nevertheless higher than absolute zero. Temperature is related to how particles are distributed among the available energy states of a system. At ordinary positive temperatures, lower-energy states are more likely to be occupied than higher-energy states. At negative temperatures, this distribution is reversed, so higher-energy states become more populated than lower-energy ones. For this reason, a system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. A standard example of such a system is population inversion in laser physics. Temperature is also related to how a system's entropy changes as energy is added. Thermodynamic systems with no upper limit on their energy cannot achieve negative temperatures. This is the case for ordinary particles, such as atoms or dust, whose kinetic energy can increase without bound. In such systems, adding energy always increases the number of accessible states and therefore the entropy. Negative temperatures are only possible in systems that have a maximum energy they can hold. As such systems approach this maximum energy, the number of accessible states begins to decrease with further increases in energy, causing the entropy to fall and making the temperature negative. Examples of such systems include two-level systems and nuclear spin ensembles in an external magnetic field.
History The possibility of negative temperatures was first predicted by Lars Onsager in 1949. Onsager was investigating 2D vortices confined within a finite area, and realized that since their positions are not independent degrees of freedom from their momenta, the resulting phase space must also be bounded by the finite area. Bounded phase space is the essential property that allows for negative temperatures, and can occur in both classical and quantum systems. As shown by Onsager, a system with bounded phase space necessarily has a peak in the entropy as energy is increased. For energies exceeding the value where the peak occurs, the entropy decreases as energy increases, and high-energy states necessarily have negative Boltzmann temperature. The limited range of states accessible to a system with negative temperature means that negative temperature is associated with emergent ordering of the system at high energies. For example in Onsager's point-vortex analysis negative temperature is associated with the emergence of large-scale clusters of vortices. This spontaneous ordering in equilibrium statistical mechanics goes against common physical intuition that increased energy leads to increased disorder. It seems negative temperatures were first found experimentally in 1951, when Purcell and Pound observed evidence for them in the nuclear spins of a lithium fluoride crystal placed in a magnetic field, and then removed from this field. They wrote:
A system in a negative temperature state is not cold, but very hot, giving up energy to any system at positive temperature put into contact with it. It decays to a normal state through infinite temperature.
Definition of temperature The absolute temperature (Kelvin) scale can be loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of thermodynamic temperature in terms of Boltzmann's entropy formula. This reveals the tradeoff between internal energy and entropy contained in the system, with "coldness", the reciprocal of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system. The definition of thermodynamic temperature T is a function of the change in the system's entropy S under reversible heat transfer Qrev:
T = d Q r e v d S . {\displaystyle T={\frac {dQ_{\mathrm {rev} }}{dS}}.}
Entropy being a state function, the integral of dS over any cyclical process is zero. For a system in which the entropy is purely a function of the system's energy E, the temperature can be defined as:
T = ( d S d E ) − 1 . {\displaystyle T=\left({\frac {dS}{dE}}\right)^{-1}.}
Equivalently, thermodynamic beta, or "coldness", is defined as
β = 1 k T = 1 k d S d E , {\displaystyle \beta ={\frac {1}{kT}}={\frac {1}{k}}{\frac {dS}{dE}},}
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