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American Journal of Physics

Negative temperatures and negative dissipation

Stefan Machlup

Published Nov 1, 1975• 11 citations
Publisher

Abstract

Negative absolute temperature, defined in terms of the Boltzmann factor, corresponds to an upside-down energy pyramid. The example of the laser shows population inversion to be accompanied by the switch from absorption to emission, that is, from positive to negative dissipation, from increasing disorder throughout a system to increasing order in a subsystem. Early treatments of the negative-temperature concept are by Onsager and by Ramsey. Ramsey’s ’’essential requirements’’ are shown to be too strong. A TS diagram for a Carnot cycle operating between two negative temperatures clarifies the origin of amplification. Some biological negative-resistance amplifiers (nerve, muscle), in particular those involving ordered arrays of water molecules, are conjectured to have negative-temperature subsystems.

DOI
10.1119/1.9938

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Negative absolute temperature, defined in terms of the Boltzmann factor, corresponds to an upside-down energy pyramid. The example of the laser shows population inversion to be accompanied by the switch from absorption to emission, that is, from positive to negative dissipation, from increasing disorder throughout a system to increasing order in a subsystem. Early treatments of the negative-temperature concept are by Onsager and by Ramsey. Ramsey’s ’’essential requirements’’ are shown to be too strong. A TS diagram for a Carnot cycle operating between two negative temperatures clarifies the origin of amplification. Some biological negative-resistance amplifiers (nerve, muscle), in particular those involving ordered arrays of water molecules, are conjectured to have negative-temperature subsystems.
Small systems consisting of a few particles are increasingly technologically relevant. In such systems, an intense debate in microcanonical statistical mechanics has been about the correctness of Boltzmann's surface entropy versus Gibbs' volume entropy. Both entropies have shortcomings─while Boltzmann entropy predicts unphysical negative/infinite absolute temperatures for small systems with an unbounded energy spectrum, Gibbs entropy entirely disallows negative absolute temperatures, in disagreement with experiments. We consider a relative energy window, motivated by the Heisenberg energy-time uncertainty principle and eigenstate thermalization in quantum mechanics. The proposed entropy ensures positive, finite temperatures for systems without a maximum limit on their energy and allows negative absolute temperatures in bounded energy spectrum systems, e.g., with population inversion. It also closely matches canonical ensemble predictions for prototypical systems, thus correctly describing the zero-point energy of an isolated quantum harmonic oscillator. Overall, we enable accurate thermodynamic models for isolated systems with few degrees of freedom.
Inverted quantum states provide a challenge to classical thermodynamics, since they appear to contradict the classical formulation of the second law of thermodynamics. Ramsey interpreted these states as stable equilibrium states of negative thermodynamic temperature, and added a provision to allow these states to the Kelvin-Planck statement of the second law [N. F. Ramsey, Phys. Rev. 103, 20 (1956)PHRVAO0031-899X10.1103/PhysRev.103.20]. Since then, Ramsey's interpretation has prevailed in the literature. Here, we present an alternative option to accommodate inverted states within thermodynamics, which strictly enforces the original Kelvin-Planck statement of the second law, and reconciles inverted states and the second law by interpreting the former as unstable states, for which no temperature-positive or negative-can be defined. Specifically, we recognize inverted quantum states as temperature-unstable states, for which all processes are in agreement with the original Kelvin-Planck statement of the second law, and positive thermodynamic temperatures in stable equilibrium states. These temperature-unstable states can only be created by work done to the system, which is stored as energy in the unstable states, and can be released as work again, just as in a battery or a spring.
Rate equation and population inversion In a material which has only two energy levels, χ ″ is always positive because ρ 11 > ρ 22 for E 1 < E 2 at thermal equilibrium. Prior to the invention of lasers, there was no known method to achieve ρ 22 > ρ 11 . However, we now know that a negative χ ″ in Eq. (5.41a) (i.e. ρ 22 > ρ 11 ) can be achieved by pumping processes that are available in materials that have multiple energy levels, as described in the following. Let there be many energy levels in the material under consideration, as shown in Fig. 6.1. Let there be a mechanism in which the populations at E 1 and E 2 , i.e. N 1 = Nρ 11 and N 2 = Nρ 22 , are increased by pumping from the ground state at pump rates R 1 and R 2 . In solid state lasers, the pumping action may be provided by an intense optical radiation causing stimulated transition between the ground state and other higher energy states, where the particles in the higher energy states relax preferentially into the E 2 state. In gas lasers, the molecules in the ground state may be excited into higher energy states within a plasma discharge; particles in those higher energy states then relax preferentially to the E 2 state. Alternatively, collisions with particles of other gases may be utilized to increase the number of particles in the E 2 state. Various schemes to pump different lasers are reviewed in ref. In order to obtain amplification, it is necessary to have R 2 ≫ R 1 .
During six decades of quantum electronics, a vast majority of new types of quantum generators have been developed. Although the principle of population inversion has united different ranges of electromagnetic spectra (and respective quantum generators), the existence of the title laser without the population inversion, makes that the exception had confirmed the rule, i. e. that this title deserves to be discussed further. Developing of formalisms describing the operation of quantum generators, by now have produced several approaches, which must have a quantum mechanics base. For the practical reasons, negative coefficient of absorption is acquired using classic electromagnetics as well, however for the population purposes, quantum representation must be entered. A few levels of formalisms will be set in this paper, linked to quantum generators accenting the optical portion of the spectra. The lowest level descriptions are based on lumped circuits. This could be expanded to equivalents of other physical problems, using program packages developed for the electrical engineering application purposes (Spice, etc.). Schematics are defined at the macro as well as micro equivalent levels (atomic – electronic levels). The kinetic equations with simpler approach will be considered as well as simplified laser equations based on quantum/ semi-quantum approach. The use of Fourier analysis or other appropriate transformations leads to formulating the main five laser equations which serve as the base for various working regimes of quantum generators and amplifiers (free generation regime, Q switch, synchronization, operation with filters, two modes regime, regime with losses, etc.). The Lyapunov stability theorem has to be included here, etc. For some of the chosen types of quantum generator, analytical modeling will be analyzed as well as the results of program packages developed for the lasers dynamics, regimes and parameters. The systems pumped with electronic beams (relativistic) will be considered and the nuclear physics statements discussed which must be included at the beginning, in order to consider further necessary parts of the condensed – solid state theory and laser techniques, after slowing down towards thermal energies. Existing program packages provide fast modeling and visualization of laser energy distribution, temperature, modes, etc. in active material with or without the resonator. A modeling will be performed for the specified geometries and a temperature distribution in active material will be captured during operation of a chosen laser system. Different pump geometries will be compared. Contemporary lasers with the shortest existing pulse durations demand new formalisms. Areas of nonlinear optics and quantum electrody-namics, Glauber states and similar, are areas that have to be included. Two main formalisms thermodynamical and quantum mechanical with transition probabilities using perturbation methods and secondary quantization naturally had to be complemented if the Brillouin, Raman, Compton, soliton, fiber and other lasers are included more generally.