Supradegeneracy and the Second Law of Thermodynamics

Daniel P. Sheehan

doi:10.1515/jnet-2019-0051

Published by De Gruyter January 24, 2020

Supradegeneracy and the Second Law of Thermodynamics

Daniel P. Sheehan

From the journal Journal of Non-Equilibrium Thermodynamics

https://doi.org/10.1515/jnet-2019-0051

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Abstract

Canonical statistical mechanics hinges on two quantities, i. e., state degeneracy and the Boltzmann factor, the latter of which usually dominates thermodynamic behaviors. A recently identified phenomenon (supradegeneracy) reverses this order of dominance and predicts effects for equilibrium that are normally associated with non-equilibrium, including population inversion and steady-state particle and energy currents. This study examines two thermodynamic paradoxes that arise from supradegeneracy and proposes laboratory experiments by which they might be resolved.

Keywords: supradegeneracy; second law of thermodynamics; non-equilibrium; statistical mechanics; population inversion; laser

Funding statement: Financial support was provided by the Laney Thornton Foundation.

Acknowledgment

The author is grateful for discussions with L. S. Schulman, M. W. Anderson, M. Weber, T. Herrinton, G. Levy, J. Denur, D. Keogh, and P. Layton. Crystal Ibarra is thanked for artistic support. He also thanks the two anonymous reviewers for significant improvements to the manuscript.

Appendix

The following extends the introduction to supradegeneracy found in Section 2.

Picking up the example pertaining to eq. (2), the partition function for the discrete ladder with evenly spaced energy levels (capped at level N) is

(4)Zdisc(N)=eγ(N+1)−1eγ−1=eγN/2sinh(γ(N+1)2)sinh(γ/2),

where γ≡ln(p)−ϵβ. The partition function for a continuous ladder (capped at Emax) is

(5)Zcont(Emax)=1δ(eδEmax−1)=2δsinhδEmax2,

where δ≡ln(p)ϵ−β.

From these Z’s, all the standard thermodynamic functions and potentials (e. g., entropy, pressure, internal energy, enthalpy, and Helmholtz and Gibbs free energies) can be obtained using standard methods [1], [2], [3]. For example, the average level number for the discrete ladder is

(6)⟨n⟩=∂∂γ(ln(Zdisc))=(N+1)eγ(N+1)eγ(N+1)−1−γeγ−1.

In the limit (eγ≫1), one has ⟨n⟩→N. Its average energy is

(7)⟨E⟩≡−∂∂βlnZdisc=∂γ∂β∂∂γlnZ=⟨n⟩ϵ.

Similar results follow for the continuous ladder.

The density of states function for a capped continuous ladder (cutoff at Emax and go=1) is^[22]

(8)gcont(E)=expln(p)ϵE,(0≤E≤Emax).

Its energy distribution, assuming Boltzmann occupancy, is

(9)fB≡gcont(E)nB(E)=eμβe(ln(p)ϵ−β)E,

where μ is chemical potential. For Fermi–Dirac (FD) and Bose–Einstein (BE) occupancies, the energy distribution functions are

(10)fBE/FD=eln(p)Eϵe(E−μ)β±1,

where ± apply to FD(+) and BE(−) statistics. In the high-energy limit (e(E−μ)β≫1) the FD and BE distributions converge to the Boltzmann distribution, which is dominated by the supradegeneracy term; thus, all distributions exponentially increase up to their abrupt cutoff at Emax. The BE distribution is bimodal, retaining its low-energy (E=μ)-asymptotic peak originating with its underlying BE occupancy nBE, while its high-energy second peak is due to supradegeneracy.

In principle, particles in an uncapped supradegenerate ladder can ascend with no upper bound to their energy. This is suggested by Schulman’s observation concerning occupation probability [31] in Section 4. These supradegenerate runaways are reminiscent of electron runaways in plasmas [55]. Ultimately, of course, their energies will be truncated by physical constraints, as they are for plasma runaways.^[23]

Others have investigated the thermodynamics of related systems, in particular, heat baths possessing highly degenerate energy spectra [56], [57]. Although these bear some resemblance to the present supradegenerate systems, they are distinct in that they (i) are limiting cases of infinite heat baths rather than finite physical systems; (ii) are uncapped rather than capped in energy space; and (iii) do not appear to be physically realizable.

Particle ascent in energy ladders can be understood in terms of entropic forces [58], [59]. Formally, the entropic force is written F∇S≡T∇ηS, where T is the system temperature and ∇η is a generalized gradient of S with respect to the coordinate η, usually in configuration space. The ladder’s entropy is given as S(η)=kln(Ω(η)), where Ω is the Boltzmann multiplicity of states [1], [2], [3]. For example, consider the discrete configuration space ladder in Fig. 3, whose degeneracy increases with y. Here nN=EnEN=yH, and Ω(y)=exp[NHln(p)y]. The entropic force is

(11)F∇S=T∂S∂y=NHln(p)β.

Most of the counterintuitive effects associated with supradegeneracy can be understood with the aid of F∇S.

In traditional thermodynamics, the entropic force is overwhelmingly written in terms of configuration space gradients, as in (11). The supradegenerate laser (as well as the photovoltaic and thermophotovoltaic systems [5]), however, is subject to entropic forces in energy space (F∇S≡T(∂S∂E)).

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Received: 2019-07-10

Revised: 2019-11-22

Accepted: 2020-01-07

Published Online: 2020-01-24

Published in Print: 2020-04-26

Supradegeneracy and the Second Law of Thermodynamics

Abstract

Acknowledgment

Appendix

References

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