DOCSLIB.ORG

Mesoscopy and Thermodynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mesoscopy and Thermodynamics EPJ manuscript No. (will be inserted by the editor) Mesoscopy and thermodynamics L. G. Moretto, J. B. Elliott, and L. Phair Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Received: date / Revised version: date Abstract. The interplay of thermodynamics and statistical mechanics is discussed with special attention to mesoscopic systems and phase transitions. PACS. 25.70.Pq – 05.70.Jk – 24.60.Ky – 64.60.Ak 1 Introduction course of theoretical speculation is wisely trimmed by the correctness of thermodynamical empiricism. Thermodynamics is an empirical science devoted to the The “recent” extension of experimental [1–3] and the- description of macroscopic systems. It follows formally oretical research to systems that are neither wholly mi- from its three principles and stands firmly independent croscopic nor macroscopic, thus aptly named mesoscopic, of any microscopic description. Statistical mechanics, on should be enlightened by the same wisdom. While names the other hand, is founded on the general principles of and functions (temperature, entropy, free energy...) were mechanics, and attempts to bridge the gap from the mi- borrowed from thermodynamics, the urge to label the field croscopic description of a system to its macroscopic ther- as “novel” led to rash statements such as the primacy of modynamical properties. This attempt has met with un- one ensemble (microcanonical) over the others, or even deniable success through the introduction of ensembles to the affirmation of a “new” thermodynamics or to the such as microcanonical, canonical, grand canonical, etc., “failure” of the “old” thermodynamics and its principles. whereby microstates, constrained to satisfy some specific These rash statements have often been accompanied macroscopic conditions are used to obtain average values by a cavalier disregard for the physical definitions and of macroscopic variables. Its success, however, must be identification of phases, claiming the presence of phase measured in terms of its agreement with thermodynam- transitions on the basis of possibly myopic interpretations ics, which is the final judge. Thus the potentially unbridled of anecdotal numerical results. 2 L. G. Moretto et al.: Mesoscopy and thermodynamics The resulting rush to calculate properties of a plethora any given energy. The result of these calculations is the of individual mesoscopic systems underscores the essential level density ρ(E) as a function of energy, or, equivalently, limitations of these efforts, which renounce the beautiful the entropy S(E) as a function of energy. Anomalies ap- generality of thermodynamics for results as limited and pear in the dependence of S(E) on E, like regions of con- specific as the systems under consideration become legion. stant slope, negative first derivatives and positive second This direction leads to as many individual “statistical me- derivatives (concex intruders). The elementary nature of chanics” as there are mesoscopic systems. We shall discuss this procedure seems to ward off any confusion. this point further and suggest a natural solution. Yet the transition from microcanonical to the more The study of numerical rather than analytical methods commonly used canonical language and variables becomes has given rise to two different problems. On one hand, the dangerous and misleading in correspondence to the very size of the system is finite, on the other, the choice of the anomalies for which one is searching. Thus the resulting algorithm becomes of paramount significance. negative temperature, negative specific heat, etc. tend to create misunderstanding and confusion. The first problem, which leads to the need of extrap- Our own approach uses the reductionistic device of de- olating correctly to the thermodynamic limit, is a nega- scribing the plethora of mesoscopy in terms of the “old” tive aspect of these numerical computations. On the other thermodynamics, which is augmented by the least num- hand, the ever increasing importance of microscopic sys- ber of new quantities necessary to describe the finiteness tems in experimental and applied science, makes the finite of the systems. This idea, of course, is not new. It is em- size of great interest in itself, and the attending differences bodied, for instance, in the “liquid drop” model, so much from the thermodynamic limit are drawing ever increasing successful in nuclear physics. It consists of expanding ther- attention. The second problem is related to the first. Some modynamic functions such as the free energy in terms of algorithms strive to compute the partition function versus a “finiteness” variable A−1/3 where A is the “size - mass” temperature (in the canonical approach), while others try of the finite system. So we write: to compute the entropy versus energy (the microcanoni- 2/3 1/3 cal approach). A lot of discussion has arisen concerning F (A, T ) = av(T )A + as(T )A + acA + ··· (1) the merits of the two approaches, in particular with re- where av(T ), as(T ), ac(T ) are the temperature T depen- gards to the observation of interesting anomalies and to dent volume, surface, curvature, etc. free energy coeffi- the extrapolation to the thermodynamic limit. cients. The leading term is the extensive volume term In the struggle between the two camps, the upper hand av(T )A, the others (which can be shape dependent, e.g. 0 seems to be taken by the microcanonical algorithms which as(T ) = as(T )f(s/s0) where s is the actual surface s of just generate and count configurations in phase space at the object and s0 is the surface of its equivalent sphere) L. G. Moretto et al.: Mesoscopy and thermodynamics 3 are the terms that account for finiteness and vanish for The lack of particle or energy exchange with the environ- A → ∞. The shape variables can either be fixed, or let ment makes the calculational aspect particularly transpar- free to assume the value demanded by the free energy ent. Literally one counts the number of configurations of minimization. The fact that in the nuclear case the ex- the system for a narrow energy interval around any given pansion can be safely stopped at the surface term while energy; that is the level density at that energy. Its loga- retaining accuracies of the order of one percent down to rithm is the entropy, now the “thermodynamic” potential sizes as small as A = 20 gives us the hope that just by in terms of V , N and E. That is all there is to do. However, adding a surface term we may obtain similarly adequate measurements typically deal with the canonical world of descriptions for many other mesoscopic systems. Further- temperature T and variables depending on T . more, for short range forces it happens that the surface Consequently one takes the derivative of entropy with coefficient is nearly equal and opposite to the volume co- respect to energy and calls that the inverse temperature efficient, i.e. a (T ) = −a (T ). This leads to a great deal ∂S 1 v s = β = . (2) ∂E T of economy and to an extreme generality. From this equation other quantities like the caloric curve It is important to appreciate that in any calculation ∂E E = E(T ) and specific heat CV = ∂T |V are determined. applied to mesoscopic systems, the presence of the sur- But it is here that problems arise. Equation (2) is a macro- face as a physical boundary, and boundary condition is of scopic thermodynamic relationship. But what is its mi- paramount importance for all calculations. Many of the croscopic origin? And does it apply all the time? In the claimed novel and general features resulting from finite following we consider this in detail. system calculations are either strongly dependent on such boundary conditions, or are artifacts of unwittingly intro- 2.1 The transformation from energy scale to duced boundary conditions which may not have necessar- temperature scale ily any specific physical meaning. We shall exemplify the above ideas by considering phase We are going to perform this transformation in two equiva- coexistence and phase transitions. lent ways: the first more physical, the second more formal. Let us consider a microcanonical system fully charac- terized by the relationship 2 Microcanonical and canonical worlds and ρ = ρ(E),S = S(E). (3) their languages Let us put a small system (thermometer) with a few de- The microcanonical world is conceptually the simplest grees of freedom (e.g. a classical one dimensional harmonic one. It deals with an ensemble of totally isolated systems. oscillator) into contact with the microcanonical system. 4 L. G. Moretto et al.: Mesoscopy and thermodynamics Any energy state ε of the thermometer is associated The independent variable β is the inverse of the canonical with a probability p (ε) equal to the degeneracy of the temperature T . In the canonical world at constant β (T ), microcanonical system at energy E − ε the microcanonical world at constant E is represented as a weighted sum, the weight being a Boltzmann factor. p(ε) ' ρ(E − ε)ρT (ε) (4) Equation (9) transforms the microcanonical world of where the level density of the one dimensional harmonic energy E and level density ρ(E) in to the canonical world oscillator ρT (ε) is constant. Given the “smallness” of the 1 of temperature T = β and partition function Z(β). This thermometer, ε is microscopically small compared to the transformation is a Laplace transform. “macroscopic” energy E. So the following Taylor expan- Notice that for any given T or β there is a distribution sion is in order of energies, and not a single value. The distribution in E 2 ∂ ln ρ 1 2 ∂ ln ρ ln p(ε) = ln ρ(E)− ε + ε 2 −· · · . (5) is the integrand of Eq. (9) ∂E ε=0 2 ∂E ε=0 But (E − E )2 p(E) = ρ(E) exp(−βE) = A exp − 0 .
Load more

Recommended publications
  • Temperature Dependence of Breakdown and Avalanche Multiplication in In0.53Ga0.47As Diodes and Heterojunction Bipolar Transistors
    This is a repository copy of Temperature dependence of breakdown and avalanche multiplication in In0.53Ga0.47As diodes and heterojunction bipolar transistors . White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/896/ Article: Yee, M., Ng, W.K., David, J.P.R. et al. (3 more authors) (2003) Temperature dependence of breakdown and avalanche multiplication in In0.53Ga0.47As diodes and heterojunction bipolar transistors. IEEE Transactions on Electron Devices, 50 (10). pp. 2021-2026. ISSN 0018-9383 https://doi.org/10.1109/TED.2003.816553 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 50, NO. 10, OCTOBER 2003 2021 Temperature Dependence of Breakdown and Avalanche Multiplication in InHXSQGaHXRUAs Diodes and Heterojunction Bipolar Transistors M.
    [Show full text]
  • PHYS 596 Team 1
    Braun, S., Ronzheimer, J., Schreiber, M., Hodgman, S., Rom, T., Bloch, I. and Schneider, U. (2013). Negative Absolute Temperature for Motional Degrees of Freedom. Science, 339(6115), pp.52-55. PHYS 596 Team 1 Shreya, Nina, Nathan, Faisal What is negative absolute temperature? • An ensemble of particles is said to have negative absolute temperature if higher energy states are more likely to be occupied than lower energy states. −퐸푖/푘푇 푃푖 ∝ 푒 • If high energy states are more populated than low energy states, entropy decreases with energy. 1 휕푆 ≡ 푇 휕퐸 Negative absolute temperature Previous work on negative temperature • The first experiment to measure negative temperature was performed at Harvard by Purcell and Pound in 1951. • By quickly reversing the magnetic field acting on a nuclear spin crystal, they produced a sample where the higher energy states were more occupied. • Since then negative temperature ensembles in spin systems have been produced in other ways. Oja and Lounasmaa (1997) gives a comprehensive review. Negative temperature for motional degrees of freedom • For the probability distribution of a negative temperature ensemble to be normalizable, we need an upper bound in energy. • Since localized spin systems have a finite number of energy states there is a natural upper bound in energy. • This is hard to achieve in systems with motional degrees of freedom since kinetic energy is usually not bounded from above. • Braun et al (2013) achieves exactly this with bosonic cold atoms in an optical lattice. What is the point? • At thermal equilibrium, negative temperature implies negative pressure. • This is relevant to models of dark energy and cosmology based on Bose-Einstein condensation.
    [Show full text]
  • Consistent Thermostatistics Forbids Negative Absolute Temperatures
    ARTICLES PUBLISHED ONLINE: 8 DECEMBER 2013 | DOI: 10.1038/NPHYS2815 Consistent thermostatistics forbids negative absolute temperatures Jörn Dunkel1* and Stefan Hilbert2 Over the past 60 years, a considerable number of theories and experiments have claimed the existence of negative absolute temperature in spin systems and ultracold quantum gases. This has led to speculation that ultracold gases may be dark-energy analogues and also suggests the feasibility of heat engines with efficiencies larger than one. Here, we prove that all previous negative temperature claims and their implications are invalid as they arise from the use of an entropy definition that is inconsistent both mathematically and thermodynamically. We show that the underlying conceptual deficiencies can be overcome if one adopts a microcanonical entropy functional originally derived by Gibbs. The resulting thermodynamic framework is self-consistent and implies that absolute temperature remains positive even for systems with a bounded spectrum. In addition, we propose a minimal quantum thermometer that can be implemented with available experimental techniques. ositivity of absolute temperature T, a key postulate of Gibbs formalism yields strictly non-negative absolute temperatures thermodynamics1, has repeatedly been challenged both even for quantum systems with a bounded spectrum, thereby Ptheoretically2–4 and experimentally5–7. If indeed realizable, invalidating all previous negative temperature claims. negative temperature systems promise profound practical and conceptual
    [Show full text]
  • Two Level System & Negative Temperature
    PHYS4006: Thermal and Statistical Physics Lecture Notes Part - 4 (Unit - IV) Two Level System & Negative Temperature Dr. Neelabh Srivastava (Assistant Professor) Department of Physics Programme: M.Sc. Physics Mahatma Gandhi Central University Semester: 2nd Motihari-845401, Bihar E-mail: neelabh@mgcub.ac.in Two level energy system • Consider a system having two non-degenerate microstates with energies ε1 and ε2. The energy difference between the levels is ε= ε - ε 2 1 ε2 Let us assume that the system is in ε= ε - ε thermal equilibrium at temperature T. 2 1 So, the partition function of the system is – ε1 Z exp 1 exp 2 ........(1) kT kT The probability of occupancy of these states is given by - exp 1 exp 1 kT kT 1 P 1 Z exp 1 exp 2 1 exp 2 1 kT kT kT 2 exp 2 exp 2 exp 2 1 kT kT kT P 2 Z exp 1 exp 2 1 exp 2 1 kT kT kT or, 1 and exp P1 ......(2) T P ........(3) 1 exp 2 T 1 exp T where 2 1 k From (2) and (3), we can see that P1 and P2 depends on T. Thus, if T=0, P1=1 and P2=0 (System is in ground state) When T<<θ, i.e. kT<<ε then P2 is negligible and it begins to increase when temperature is increased. 3 • Consider a paramagnetic substance having N magnetic atoms per unit volume placed in an external magnetic field B. Assume that each atoms has spin ½ and an intrinsic magnetic moment μB.
    [Show full text]
  • Atomic Gases at Negative Kinetic Temperature
    PHYSICAL REVIEW LETTERS week ending PRL 95, 040403 (2005) 22 JULY 2005 Atomic Gases at Negative Kinetic Temperature A. P. Mosk Department of Science & Technology, and MESA+ Research Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands and Laboratoire Kastler-Brossel, Ecole Normale Supe´rieure, 24 rue Lhomond, 75231 Paris CEDEX 05, France (Received 26 January 2005; published 21 July 2005) We show that thermalization of the motion of atoms at negative temperature is possible in an optical lattice, for conditions that are feasible in current experiments. We present a method for reversibly inverting the temperature of a trapped gas. Moreover, a negative-temperature ensemble can be cooled (reducing jTj) by evaporation of the lowest-energy particles. This enables the attainment of the Bose- Einstein condensation phase transition at negative temperature. DOI: 10.1103/PhysRevLett.95.040403 PACS numbers: 03.75.Hh, 03.75.Lm, 05.70.2a Amplification of a macroscopic degree of freedom, such ized in the vicinity of a band gap in a three-dimensional as a laser or maser field, or the motion of an object, is a (3D) optical lattice. The band gap is an energy range inside phenomenon of enormous technological and scientific im- which no single-particle eigenstates exist. If the atoms are portance. In classical thermodynamics, amplification is not constrained to energies just below the band gap the effec- possible in thermal equilibrium at positive temperature. In tive mass will be negative in all directions and the entropy contrast, due to the fluctuation-dissipation theorems [1,2], of the ensemble will decrease with energy.
    [Show full text]
  • Query on Negative Temperature, Internal Interactions and Decrease
    Query on Negative Temperature, Internal Interactions and Decrease of Entropy Yi-Fang Chang Department of Physics, Yunnan University, Kunming, 650091, China (e-mail: yifangchang1030@hotmail.com) Abstract: After negative temperature is restated, we find that it will derive necessarily decrease of entropy. Negative temperature is based on the Kelvin scale and the condition dU>0 and dS<0. Conversely, there is also negative temperature for dU<0 and dS>0. But, negative temperature is contradiction with usual meaning of temperature and with some basic concepts of physics and mathematics. It is a question in nonequilibrium thermodynamics. We proposed a possibility of decrease of entropy due to fluctuation magnified and internal interactions in some isolated systems. From this we discuss some possible examples and theories. Keywords: entropy; negative temperature; nonequilibrium. PACS: 05.70.-a, 05.20.-y 1.Restatement of Negative Temperature In thermodynamics negative temperature is a well-known idea proposed and expounded by Ramsey [1] and Landau [2], et al. Ramsey discussed the thermodynamics and statistical mechanics of negative absolute temperatures in a detailed and fundamental manner [1]. He proved that if the entropy of a thermodynamic system is not a monotonically increasing function of its internal energy, it possesses a negative temperature whenever ( S / U ) X is negative. Negative temperatures are hotter than positive temperature. He pointed out: from a thermodynamic point of view the only requirement for the existence of a negative temperature is that the entropy S should not be restricted to a monotonically increasing function of the internal energy U. In the thermodynamic equation relating TdS and dU, a temperature is 1 T ( S / U) X .
    [Show full text]
  • NIKOLAI G. BASOV the High Sensitivity of Quantum Amplifiers, and There Was Investigated the Possibility of the Creation of Various Types of Lasers
    N IKOLAI G . B A S O V Semiconductor lasers Nobel Lecture, December 11, 1964 In modern physics, and perhaps this was true earlier, there are two different trends. One group of physicists has the aim of investigating new regularities and solving existing contradictions. They believe the result of their work to be a theory; in particular, the creation of the mathematical apparatus of modern physics. As a by-product there appear new principles for constructing devices, physical devices. The other group, on the contrary, seeks to create physical devices using new physical principles. They try to avoid the inevitable difficulties and con- tradictions on the way to achieving that purpose. This group considers various hypotheses and theories to be the by-product of their activity. Both groups have made outstanding achievements. Each group creates a nutrient medium for the other and therefore they are unable to exist without one another; although, their attitude towards each other is often rather critical. The first group calls the second « inventors », while the second group accuses the first of abstractness or sometimes of aimlessness. One may think at first sight that we are speaking about experimentors and theoreticians. However, this is not so, because both groups include these two kinds of physicists. At present this division into two groups has become so pronounced that one may easily attribute whole branches of science to the first or to the second group, although there are some fields of physics where both groups work together. Included in the first group are most research workers in such fields as quan- tum electrodynamics, the theory of elementary particles, many branches of nuclear physics, gravitation, cosmology, and solid-state physics.
    [Show full text]
  • Light and Thermodynamics: the Three-Level Laser As an Endoreversible Heat Engine
    Light and thermodynamics: The three-level laser as an endoreversible heat engine Peter Muys Laser Power Optics, Ter Rivierenlaan 1 P.O. box 4, 2100 Antwerpen Belgium peter.muys@gmail.com +32 477 451916 Abstract: In the past, a number of heat engine models have been devised to apply the principles of thermodynamics to a laser. The best one known is the model using a negative temperature to describe population inversion. In this paper, we present a new temperature scale not based on reservoir temperatures. This is realized by revealing a formal mathematical similarity between the expressions for the optimum power generated by an endoreversible heat engine and the optimum output-coupled power from a three-level laser resonator. As a consequence, the theory of endoreversibility can be applied to enable a new efficiency analysis of the cooling of high- power lasers. We extend the endoreversibility concept also to the four-level laser model. 1. Introduction It was found very early in the history of the laser that a three-level optical amplifier could be considered as a heat engine [1] . The justification was based on a quantum model of the lasing element, and hence was named a “quantum heat engine”. The engine is working between two reservoirs, the hot and the cold one, at respective (absolute) temperatures TH and TC. In the ideal case of reversible operation, this engine follows the Carnot law: its efficiency with which heat (pump power) is converted to work (laser power) is limited by TC =1 − (1) TH A number of conceptual difficulties were pointed out later [2] , such as the debatable assignment of negative temperatures to the different atomic transitions, since these are out of equilibrium, and hence not suited to the concept of temperature which presupposes thermal equilibrium.
    [Show full text]
  • Statistical Physics
    Part II | Statistical Physics Based on lectures by H. S. Reall Notes taken by Dexter Chua Lent 2017 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Part IB Quantum Mechanics and \Multiparticle Systems" from Part II Principles of Quantum Mechanics are essential Fundamentals of statistical mechanics Microcanonical ensemble. Entropy, temperature and pressure. Laws of thermody- namics. Example of paramagnetism. Boltzmann distribution and canonical ensemble. Partition function. Free energy. Specific heats. Chemical Potential. Grand Canonical Ensemble. [5] Classical gases Density of states and the classical limit. Ideal gas. Maxwell distribution. Equipartition of energy. Diatomic gas. Interacting gases. Virial expansion. Van der Waal's equation of state. Basic kinetic theory. [3] Quantum gases Density of states. Planck distribution and black body radiation. Debye model of phonons in solids. Bose{Einstein distribution. Ideal Bose gas and Bose{Einstein condensation. Fermi-Dirac distribution. Ideal Fermi gas. Pauli paramagnetism. [8] Thermodynamics Thermodynamic temperature scale. Heat and work. Carnot cycle. Applications of laws of thermodynamics. Thermodynamic potentials. Maxwell relations. [4] Phase transitions Liquid-gas transitions. Critical point and critical exponents. Ising model. Mean field theory. First and second order phase transitions. Symmetries and order parameters. [4] 1 Contents II Statistical Physics Contents 0 Introduction 3 1 Fundamentals of statistical mechanics 4 1.1 Microcanonical ensemble . .4 1.2 Pressure, volume and the first law of thermodynamics . 12 1.3 The canonical ensemble . 15 1.4 Helmholtz free energy .
    [Show full text]
  • 3.6 MC-CE Negative Temperature II
    Negative temperature II Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: November 29, 2018) Fig. (a) Subsystem 1 has an upper limit to energy it can have, whereas subsystem 2 is a normal thermodynamic system, which has no upper limit to the energy it can have and is always T at a positive absolute temperature ( 2 0 ). In the initial state, with the adiabatic partition T in place, subsystem 1 starts at a negative absolute temperature ( 1 0 ). (b) The adiabatic partition is replaced by a diathermic partition. In the approach to thermal equilibrium, heat is transferred, on average, from subsystem 1, which is initially at a negative absolute temperature, to subsystem 2 which is always at a positive absolute temperature. 1 S 1 S 1 , 2 T U T U 1 1 V N 2 2 V N 1, 1 2, 2 As the closed composite system as shown in the above figure, approaches thermal equilibrium on average, S S dS dS1 dU 2 dU 0 12U 1 U 2 1VN 2 VN 11, 22 , U U Since 1 2 constant, we get dU dU 2 1 Hence we have S S dSdS[1 2 ] dU 1 2 U U 2 1VN 2 VN 11, 22 , 1 1 ( ) dU T T 2 2 1 T T 1 2 dU 0 TT 2 1 2 where 1 S 1 S 1 , 2 T U T U 1 1 V N 2 2 V N 1, 1 2, 2 T T T T dU Suppose that 1 and 2 are positive temperature.
    [Show full text]
  • The Physics of Negative Absolute Temperatures
    The Physics of Negative Absolute Temperatures Eitan Abraham∗ Institute of Biological Chemistry, Biophysics, and Bioengineering, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK Oliver Penrosey Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Colin Maclaurin Building, Heriot-Watt University, Edinburgh EH14 4AS, UK (Dated: November 25, 2016) Negative absolute temperatures were introduced into experimental physics by Purcell and Pound, who successfully applied this concept to nuclear spins; nevertheless the concept has proved con- troversial : a recent article aroused considerable interest by its claim, based on a classical entropy formula (the `volume entropy') due to Gibbs, that negative temperatures violated basic princi- ples of statistical thermodynamics. Here we give a thermodynamic analysis which confirms the negative-temperature interpretation of the Purcell-Pound experiments. We also examine the princi- pal arguments that have been advanced against the negative temperature concept; we find that these arguments are not logically compelling and moreover that the underlying `volume' entropy formula leads to predictions inconsistent with existing experimental results on nuclear spins. We conclude that, despite the counter-arguments, negative absolute temperatures make good theoretical sense and did occur in the experiments designed to produce them. PACS numbers: 05.70.-a, 05.30.d Keywords: Negative absolute temperatures, nuclear spin systems, entropy definitions I. INTRODUCTION amine, in section IV, the statistical mechanics argument which has been held to rule out negative temperatures; The concept of negative absolute temperature, first we find that this argument is not logically compelling mooted by Onsager[1, 2] in the context of two- and moreover (subsection IV D) that it leads to predic- dimensional turbulence, fits consistently into the descrip- tions inconsistent with known experimental results.
    [Show full text]
  • Population Inversion, Negative Temperature, and Quantum
    Population Inversion, Negative Temperature, and Quantum Degeneracies Zotin K.-H. Chu WIPM, 30, West Xiao-Hong Shan, Wuhan 430071, PR China Abstract We revisit the basic principle for lasing : Population inversion which is nevertheless closely linked to the negative temperature state in non-equilibrium thermodynamics. With the introduction of quantum degeneracies, we also illustrate their relationship with the lasing via the tuning of population inversion. Introduction Atomic and molecular physics arc certainly the oldest subfields of quantum physics and ev- erybody knows the major role that they played in the early 1920s when the first principles of quantum mechanics were elaborated. After this golden era, many physicists considered research in these subfields essentially complete and in fact, for several decades, activity in atomic and molecular physics decreased steadily in comparison with the large effort directed towards nu- clear and high-energy physics. The advent of lasers (LASER means : Light Amplification by Stimulated Emission of Radiation), and more precisely of tunable lasers, changed this situation and since the early 1970s, atomic and molecular physics, gradually intermingling with laser physics, has become an area where more and more activity is contributing substantially to the understanding of many phenomena. There has been a gradual change in the use of lasers in the teaching as well as research labora- tory. Increasingly, the laser is not being used for spectroscopy, but as a tool [1]. However, the art of making a laser operate is to work out how to get population inversion for the relevant arXiv:0902.0421v2 [physics.gen-ph] 9 Apr 2009 transition.
    [Show full text]