Research Papers: Energy Systems Analysis

On a Scale-Invariant Model of Statistical Mechanics and the Laws of Thermodynamics

[+] Author and Article Information
Siavash H. Sohrab

Department of Mechanical Engineering,
Northwestern University,
2145 Sheridan Road,
Evanston, IL 60208-3111
e-mail: s-sohrab@northwestern.edu

1Corresponding author.

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received November 9, 2015; final manuscript received November 10, 2015; published online January 13, 2016. Editor: Hameed Metghalchi.

J. Energy Resour. Technol 138(3), 032002 (Jan 13, 2016) (12 pages) Paper No: JERT-15-1432; doi: 10.1115/1.4032241 History: Received November 09, 2015; Revised November 10, 2015

A scale-invariant model of statistical mechanics is applied to describe modified forms of zeroth, first, second, and third laws of classical thermodynamics. Following Helmholtz, the total thermal energy of the thermodynamic system is decomposed into free heat U and latent heat pV suggesting the modified form of the first law of thermodynamics Q = H = U + pV. Following Boltzmann, entropy of ideal gas is expressed in terms of the number of Heisenberg–Kramers virtual oscillators as S = 4 Nk. Through introduction of stochastic definition of Planck and Boltzmann constants, Kelvin absolute temperature scale T (degree K) is identified as a length scale T (m) that is related to de Broglie wavelength of particle thermal oscillations. It is argued that rather than relating to the surface area of its horizon suggested by Bekenstein (1973, “Black Holes and Entropy,” Phys. Rev. D, 7(8), pp. 2333–2346), entropy of black hole should be related to its total thermal energy, namely, its enthalpy leading to S = 4Nk in exact agreement with the prediction of Major and Setter (2001, “Gravitational Statistical Mechanics: A Model,” Classical Quantum Gravity, 18, pp. 5125–5142).

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Grahic Jump Location
Fig. 1

Scale-invariant model of statistical mechanics. Equilibrium-β-dynamics on the left-hand side and nonequilibrium laminar-β-dynamics on the right-hand side for scales β = g, p, h, f, e, c, m, a, s, k, and t as defined in Sec. 2. Characteristic lengths of (system, element, and atom) are (Lββ,ℓβ) and λβ is the mean-free-path [50].

Grahic Jump Location
Fig. 3

(a) Transition of eddy eij from fluid element-j to fluid element-i leading to emission of cluster cij and (b) transition of subparticle (electron) sij from atom-j to atom-i leading to emission of photon kij [50]

Grahic Jump Location
Fig. 2

Maxwell–Boltzmann distributions for ECD, EMD, and EAD scales at 300 K [52]

Grahic Jump Location
Fig. 5

Hierarchy of absolute zero Tβ = 0β and “vacuum” T temperatures and associated entropy Sβ and Gibbs free energy Gβ [44]

Grahic Jump Location
Fig. 6

Asymptotic approach of entropy Sβ to zero with absolute temperature Tβ at scale β




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