Negative temperature

In quantum thermodynamics, certain systems can achieve negative temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. 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 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 as 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. Classical thermodynamic systems cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to 'saturate' in entropy. This is only possible if the number of high energy states is limited. In classical Boltzmann statistics, the number of high energy states is unlimited (particle speeds can in principle be increased indefinitely). Systems bounded by a maximum amount of energy are generally forbidden in classical mechanics, and the phenomenon of negative temperature is strictly a quantum mechanical 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. The definition of thermodynamic temperature T as a function of the change in the system's entropy S under reversible heat transfer Qrev: 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: Equivalently, thermodynamic beta, or 'coldness', is defined as where k is the Boltzmann constant.