# Negative Temperature

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## Negative temperature

The property of a thermodynamical system which satisfies certain conditions and whose thermodynamically defined absolute temperature is negative. The essential requirements for a thermodynamical system to be capable of negative temperature are: (1) the elements of the thermodynamical system must be in thermodynamical equilibrium among themselves in order for the system to be described by a temperature at all; (2) there must be an upper limit to the possible energy of the allowed states of the system; and (3) the system must be thermally isolated from all systems which do not satisfy both requirements (1) and (2); that is, the internal thermal equilibrium time among the elements of the system must be short compared to the time during which appreciable energy is lost to or gained from other systems.

The second condition must be satisfied if negative temperatures are to be achieved with a finite energy. Most systems do not satisfy this condition; for example, there is no upper limit to the possible kinetic energy of a gas molecule. Systems of interacting nuclear spins, however, have the characteristic that under suitable circumstances they can satisfy all three of the conditions, in which case the nuclear spin system can be at negative absolute temperature. *See* Kinetic theory of matter, Statistical mechanics

The transition between positive and negative temperatures is through infinite temperature, not absolute zero; negative absolute temperatures should therefore not be thought of as colder than absolute zero, but as hotter than infinite temperature. *See* Absolute zero, Temperature

## Negative Temperature

an effective quantity introduced for the description of nonequilibrium states of quantum systems. A system is in a negative temperature state when high energy levels are more populated than low energy levels. The probability for a system in an equilibrium state to be on the level ℰ_{n} is determined by the equation.

Here, £_{i}, are the energy levels of the system, *k* is the Boltzmann constant, and *T* is the absolute temperature, which characterizes the mean energy of the equilibrium system *U = Σ W _{n}£_{n}* Equation (1) indicates that the low energy levels are more populated by particles than the high energy levels. If external influences cause the system to enter a nonequilibrium state characterized by a greater population of the higher levels than the lower levels, it is formally possible to make use of equation (1) by setting

*T*< 0. The concept of negtive temperature is applicable only to quantum systems that contain a finite number of levels.

The absolute temperature *T* is defined in thermodynamics through its reciprocal *l/T*, which is equal to the derivative of the entropy *S* with respect to the mean energy of the system, the remaining parameters *x* being kept constant:

Equation (2) indicates that the negative temperature implies a decrease of entropy with increasing mean energy. Negative temperature, however, is introduced for the description of nonequi librium states, to which the laws of equilibrium thermodynamics are applicable only conditionally.

An example of a system with negative temperature is the system of nuclear spins in a crystal located in a magnetic field when the spins interact very weakly with the thermal lattice vibrations—that is, a system practically isolated from thermal motion. The time required for the attainment of thermal equilibrium of the spins with the lattice is of the order of several tens of minutes. During this time, the system of nuclear spins may be in a state with negative temperature, a state it entered under external influences.

In a narrower sense, negative temperature is a characteristic of the degree of inversion of the populations of two selected energy levels in a quantum system. In the case of thermodynamic equilibrium, the relation between the populations *N*_{1} and *N*_{2} of the levels £_{1} and £_{2} (£_{1} < £_{2}), that is, the mean numbers of particles in these states, is given by the Boltzmann equation:

where *T* is the absolute temperature of the substance. Equation (3) implies that *N*_{2} < *N*_{1}. Let us suppose the equilibrium of the system is disturbed, for example, by irradiation of the system with monochromatic electromagnetic radiation, whose frequency is close to the frequency of the transition between the levels ω_{21} = (£_{2}— £_{1}) ℏ and is different from the frequencies of other transitions. It is then possible to attain a state in which the population of the upper level is greater than that of the lower level: *N*_{2} > *N*_{1}. If the Boltzmann equation is conditionally applied to such a nonequilibrium state, negative temperature can be introduced with respect to the pair of energy levels £_{1} and £_{2} in accordance with the equation

Despite its formal character, this definition is convenient in a number of cases. For example, it permits fluctuations in equilibrium and nonequilibrium systems with negative temperature to be described by analogous equations. The concept of negative temperature is used in quantum electronics for a convenient description of the processes of amplification and generation in mediums with population inversion.

D. N. ZUBAREV