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

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Siavash H. Sohrab
Keyword(s):  
Statistical Mechanics ◽  
Thermal Energy ◽  
Thermodynamic System ◽  
Ideal Gas ◽  
Absolute Temperature ◽  
Scale Invariant ◽  
Invariant Model ◽  
Definition Of ◽  
Classical Thermodynamics ◽  
Total Thermal Energy

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).

scholarly journals Some Implications of a Scale Invariant Model of Statistical Mechanics to Boltzmann versus Shannon Entropy in Thermodynamics and Information Theory

2021 ◽  
Vol 20 ◽  
pp. 56-65
Author(s):  
Siavash H. Sohrab
Keyword(s):  
Information Theory ◽  
Statistical Mechanics ◽  
Comparative Study ◽  
Shannon Entropy ◽  
Scale Invariant ◽  
Invariant Model ◽  
Shannon's Entropy ◽  
The Third ◽  
Third Law Of Thermodynamics ◽  
Third Law

A scale invariant model of statistical mechanics is applied for a comparative study of Boltzmann’s entropy in thermodynamics versus Shannon’s entropy in information theory. The implications of the model to the objective versus subjective aspects of entropy as well as Nernst-Planck statement of the third law of thermodynamics are also discussed

scholarly journals General forms of statistical mechanics with special reference to the requirements of the new quantum mechanics

1926 ◽  
Vol 113 (764) ◽  
pp. 432-449 ◽  
Keyword(s):  
Statistical Mechanics ◽  
Equations Of Motion ◽  
Ideal Gas ◽  
Absolute Temperature ◽  
Coupled Systems ◽  
Light Quantum ◽  
Classical Form ◽  
Special Cases ◽  
New Form ◽  
Temperature Radiation

It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation d Q = T d S. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of d Q, deduce from d Q the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.

scholarly journals Completeness of Classical Thermodynamics: The Ideal Gas, the Unconventional Systems, the Rubber Band, the Paramagnetic Solid and the Kelly Plasma

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 398 ◽  
Author(s):  
Karen Arango-Reyes ◽  
Gonzalo Ares de Parga
Keyword(s):  
Equations Of State ◽  
Fundamental Equation ◽  
Rubber Band ◽  
Thermodynamic System ◽  
Ideal Gas ◽  
Entropy Method ◽  
Complete Set ◽  
Classical Thermodynamics ◽  
Original Approach ◽  
The Ideal

A method is developed to complete an incomplete set of equations of state of a thermodynamic system. Once the complete set of equations is found, in order to verify the thermodynamic validity of a system, the Hessian and entropy methods are exposed. An original approach called the completeness method in order to complete all the information about the thermodynamic system is exposed. The Hessian method is improved by developing a procedure to calculate the Hessian when it is not possible to have an expression of the internal energy as a fundamental equation. The entropy method is improved by showing how to prove the first-degree homogeneous property of the entropy without having a fundamental expression of it. The completeness method is developed giving a total study of the thermodynamic system by obtaining the set of independent T d S equations and a recipe to obtain all the thermodynamics identities. In order to show the viability of the methods, they are applied to a typical thermodynamic system as the ideal gas. Some well-known and unknown thermodynamic identities are deduced. We also analyze a set of nonphysical equations of state showing that they can represent a thermodynamic system, but in an unstable manner. The rubber band, the paramagnetic solid and the Kelly equation of state for a plasma are corrected using our methods. In each case, a comparison is made between the three methods, showing that the three of them are complementary to the understanding of a thermodynamic system.

Sign in / Sign up

Export Citation Format

Share Document