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Is there a negative absolute
temperature?
Jian-Sheng Wang
Department of Physics,
National University of Singapore
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Abstract
In 1956, Ramsey, based on experimental evidence of nuclear spin, developed a theory of negative temperature. The concept is challenged recently by Dunkel and Hilbert [Nature Physics 10, 67 (2014)] and others. In this talk, we review what thermodynamics is and present our support that negative temperature is a valid concept in thermodynamics and statistical mechanics.
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References
• R.H. Swendsen and J.-S. Wang, arXiv:1410.4619• And other unpublished notes
• J. Dunkel and S. Hilbert, Nature Physics 10, 67 (2014); S. Hilbert, P. Hänggi, and J. Dunkel, arXiv:1408.5382.
• S. Braun, et al, Science 339, 52 (2013); D. Frenkel and P.B. Warren, arXiv:1403.4299; J.M.G. Vilar and J.M. Rubi, J. Chem. Phys. 140, 201101 (2014).
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Outline
• Empirical temperatures and the Kelvin absolute temperature scale
• Negative T ?• Thermodynamics
• Classic: Traditional• Modern: Callen formulation• Post-modern: Lieb and Yngvason axiomatic foundations
• Volume or ‘Gibbs’ entropy – evidence of violations of thermodynamics
• Conclusion
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thermometersle
ngth
Ideal gas equation of state
pV = NkBT
p: pressure, fixed at 1 atmV: volume, V = length cross section areaN: number of moleculeskB: Boltzmann constantT: absolute temperature
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“Ising thermometer”
Spin up, = +1
Spin down, = -1
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Fundamental thermodynamic equation
Entropy S
Energy E
SG
SB
E: (internal) energy, Q: heat, T: temperatureμ: chemical potential
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S. Braun et al 39K atoms on optical lattice experiment
The system is described by the Bose-Hubbard model , A: entropy and temperature scale. B: energy bound of the three terms in . C: measured momentum distributions. From S. Braun, et al, Science 339, 52 (2013).
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Thermodynamics: traditional
Sadi Carnot (1796 -1832) Rodulf Clausius (1822-1888)
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The idea (see, e.g., A. B. Pippard, “the elements of …”)• Define empirical thermometer, based on 0th law of
thermodynamics• Build Carnot cycle with two isothermal curves and
two adiabatic curves• Compute the efficiency of cycle and find the
relation of empirical temperature and the Kelvin scale
• Define entropy according to Clausius
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Applying the procedure to Ising paramagnet, • The relation between empirical and Kelvin scale is • Equation of state is • Carnot cycle lead to • One find
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Carnot cycle in the paramagnet
Magnetic field h
Magnetization MHeat absorbed by the
system
Work done to the system
𝜃𝐿
𝜃𝐻
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Zeroth Law of thermodynamics
Max Planck: “If a body A is in thermal equilibrium with two other bodies B and C, then B and C are in thermal equilibrium with one another.”
Two bodies in thermal equilibrium means: if the two bodies are to be brought into thermal contact, there would be no net flow of energy between them.
Basis for thermometer and definition of isotherms
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Callen postulates (see also R H Swendsen, “introduction to ..”)1. Existence of state functions. (Equilibrium) States
are characterized by a small number of macroscopically measurable quantities. For simple system it is energy E, volume V, and particle number N.
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Callen postulates (see also R H Swendsen, “introduction to ..”)2. There exists a state function called “entropy”, for
whichthe values assumed by the extensive
parameters of an isolated composite system in the absence of an internal constraint are those that maximize the entropy over the set of all constrained macroscopic states.
The above statement is a form of Second Law of thermodynamics.
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Callen postulates (see also R H Swendsen, “introduction to ..”)3. Additivity: The entropy of a composite system
consisting of 1 and 2 is simply.
4. Monotonicity of entropy: entropy S is an increasing function of energy E. Can we remove this?
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Second law according to Callen
Combined and allow to exchange energy
𝐸10
𝐸20
? 𝐸2=𝐸10+𝐸2
0−𝐸1
Total entropy
𝐸1
𝐸1max
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Second law according to Callen
Combined and allow to exchange energy
𝐸10
𝐸20
= 𝐸2=𝐸10+𝐸2
0−𝐸1
Total entropy
𝐸1
𝐸1max
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E.H. Lieb & J. Yngvason, Phys Rep 310, 1 (1999)• Build the foundation of thermodynamics and the
second law on the concept of “adiabatic accessibility.”
• Starting with a set of more elementary axioms and prove the Callen postulates as theorems.
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Adiabatic Accessibility, X ≺ Y
“A State Y is adiabatically accessible from a state X, in symbols X ≺ Y, if it is possible to change the state from X to Y by means of an interaction with some device and a weight, in such a way that the device returns to its initial state at the end of the process whereas the weight may have changed its position in a gravitational field.”
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Order relation ≺1. Reflexivity, X ≺ X2. Transitivity, X ≺ Y & Y ≺ Z implies X ≺ Z3. Consistency, X≺X’ & Y≺Y’ implies (X,Y) ≺ (X’,Y’)4. Scaling invariance, if X ≺ Y, then t X ≺ t Y for all t
> 05. Splitting and recombination, for all 0 < t < 1, X ≺
(tX, (1-t)X), and (tX, (1-t)X) ≺ X
6. Stability, (X, Z0) ≺ (Y, Z1) (for any small enough > 0) implies X ≺ Y
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Comparison Hypothesis (CH)
• Definition: We say the comparison hypothesis holds for a state space if any two states X and Y in the space are comparable, i.e., X ≺ Y or Y ≺ X.
• Compare to Carathéodory: In the neighborhood of any equilibrium state of a system there are states which are inaccessible by an adiabatic process.
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Entropy Principle
• There is a real-valued function on all states of all systems (including compound systems), called “entropy” S such that
• Monotonicity: When X and Y are comparable then X ≺ Y if and only if S(X) S(Y)
• Additivity: S((X,Y)) = S(X) + S(Y)• Extensivity: for t > 0, S(tX) = t S(X)
• The above is proved with axiom 1-6 and CH, i.e. 1-6 plus CH and entropy principle are equivalent. Callen’s maxima entropy postulate is proved as a theorem 4.3 on page 57.
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Our definition of entropy• Work with composite system, determine the
(unnormalized probability) weight that the system is in a state ; we have
• Define (in equilibrium W obtains max value consistent with the constraints)
• For a classical gas, density of states is
• Additivity is built in (neglecting subsystem interactions)
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Volume (or Gibbs) entropy SG
• Total density of states up to energy E,
• Volume or Gibbs entropy is defined by
• Note that
and
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Adiabatic invariance, see, e.g. S.-K. Ma, Chap.23• We change the model parameters such that
• If then we say is an adiabatic invariant
• Volume entropy is an adiabatic invariant for any number of particles
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Why volume entropy is wrong
• It violates Zeroth Law
• It violates Second Law
• It violates Third Law (when applied to a simple quantum oscillator)
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Temperatures of three bodies according to TG
A B C
A
A
B B C A C
B C
TA TB TC
TAB TBC TAC
TABC
Starting with three systems A, B, C, such that there is no energy transfer when making contact, then according to SG, all seven cases will have different temperatures of TG.
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Two-level system, ,
0
Boltzmann distribution
,
T can be positive or negative in the above formula, can be derived in Boltzmann way as in Frenkel & Warren.
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Temperature TG increases if you combine two loafs of bread into one
0
0
0
T1,G = 25 T2,G = 28
T1+2,G = T1,GT2,G=213
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Heat flows from cold to hot according to TG
Energy of the two-level system vs time. Squares: NA = 5, NB=1, temperature of the oscillator T = 64. Dots: NA = 1000, NB=1000, T = .
0
Quantum harmonic oscillator energy level
Two-level system
ħ =
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Violation of Callen’s second postulate
Total entropy
𝐸1
𝐸1max
N1 E1max for SB E1
max for SG
5 4 4
10 8 9
50 40 43
100 80 87
500 400 433
1000 800 867
Two identical two-level systems 1 and 2 with N2 = 2N1 and total energy E1+2=(4/5)(N1+N2). SG gives wrong results for by about 8%.
𝐸1eq
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Entropy and thermodynamic limit Entropy of
(distinguishable) quantum harmonic oscillators computed according to SG for the number of oscillators N = 1, 2, 5, 20, 80, and (from bottom to top) or SB with one particle larger, i.e., N = 2, 3, 6, etc.
Temperature for N=1 cannot be properly defined.
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Opposing view
• Ensembles are not equivalent, especially so for the case when energy distributions are inverted
• Thermodynamics applies to any number of particles, N = 1, 2, 3, …
• Heat flows from hot to cold is “naïve”, T is not a state function
• People have been using the wrong definition of entropy of Boltzmann for the last 60 years without realizing it
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Conclusion
• The volume entropy SG fails to satisfy the postulates of thermodynamics – the zeroth law and the second law. It lacks additivity important for the validity of thermodynamics
• For classical systems, SG satisfies an exact adiabatic invariance (due to Hertz) while Boltzmann entropy does not. However, the violations are of order 1/N and go away for large systems
• Thermodynamics is a macroscopic theory which applies to large systems only