A Temperature Below Absolute Zero: Atoms at Negative Absolute Temperature Are the Hottest Systems

[tama]

5. Thermodynamic beta

Wiki explains well:

In physics, certain systems can achieve negative temperature; that is, their thermodynamic temperature can be expressed as a negative quantity on the Kelvin scale. In colloquial usage, "negative temperature" may refer to temperatures that are expressed as negative numbers on the more familiar degrees Celsius or Fahrenheit scales, with values that are colder than the zero points of those scales but still warmer than absolute zero. By contrast, a system with a truly negative temperature in absolute terms 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.
That a system at negative temperature is hotter than any system at positive temperature is paradoxical if absolute temperature is interpreted as an average internal energy of the system. The paradox is resolved by understanding temperature through its more rigorous definition as the tradeoff between energy and entropy, with the reciprocal of the temperature, thermodynamic beta, as the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system. Systems with a negative temperature will decrease in entropy as one adds energy to the system.
Most familiar systems cannot achieve negative temperatures, because adding energy always increases their entropy. The possibility of decreasing in entropy with increasing energy requires the system to "saturate" in entropy, with the number of high energy states being small. These kinds of systems, bounded by a maximum amount of energy, are generally forbidden classically. Thus, negative temperature is a strictly quantum phenomenon. Some systems, however (see the examples below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.

http://en.wikipedia.org/wiki/Negative_temperature

Thermodynamic beta gives exactly same results as classical definition, but is more general notion, just like quantum physics is more general than classical mechanics.