where k is the Boltzmann’s constant and Ω is the number of microstates
corresponding to a given state with the macroscopic constraints (Callen 1985;
DeHoff 1993).
Sharp and Matschinky (2015) point out that with Boltzmann’s (permutabilitat)
measure of permutability method for counting possible microstates, there is no need
for a posteriori division by N! “to correct” the derivation using the “somewhat
mystical arguments of Gibbs (1902) and Planck.” Van Kampen (1984), Ehrenfest
and Tikal (1921), Van kampen (1984), Jaynes (1992), and Swendsen (2006) have
pointed out that correct counting of microstates a la Boltzmann precludes the need
for the spurious indistinguishability term N!. “This fact has been ignored in most text
books.”
Translation by Sharp and Matchinsky (2015) also clarifies one very important
point about Boltzmann’s derivation regarding nonequilibrium states. lnΩ is not the
logarithm of a volume in momentum-coordinate-phase space [Boltzmann here refers
to a five dimensional space, the fifth axis being momentum], occupied by the system.
Boltzmann notes, using Liouville’s theorem, that dv remains constant in time.
Therefore, it cannot describe the entropy increase upon approach to the equilibrium
that Boltzmann was so concerned with. He thus avoids at the outset the considerable
difficulty Gibbs had been accounting for changes in entropy with time. Boltzmann
gave us for the first time a definition of entropy applicable to every state at
equilibrium or not. “Then the entropy of the initial and final states is not defined,
but one can still calculate the quantity which we have called the permutability
measure.” By extension, every complexion can then be assigned an entropy, using
the permutability measure of the state distribution to which that complexion belongs
(Lebowitz 1993), opening the door to the statistical mechanics of nonequilibrium
states and irreversible processes.
Boltzmann’s work has historically been misinterpreted, assumed applicable to
gasses only and ignored in continuum mechanics field. Therefore, we feel compelled
to include his original paper in this book. The following section is the English
translation of Boltzmann (1877) by Sharp and Matschinky (2015).
4.3.1 On the Relationship Between the Second Fundamental
Theorem of the Mechanical Theory of Heat
and Probability Calculations Regarding the Conditions
for Thermal Equilibrium, by Ludwig Boltzmann (1877)
The relationship between the second fundamental theorem and calculations of
probability became clear for the first time when I demonstrated that the theorem’s
analytical proof is only possible based on probability calculations. (I refer to my
publication “Analytical proof of the second fundamental theorem of the mechanical
theory of heat derived from the laws of equilibrium for kinetic energy” Wien. Ber.
63, p 8 reprinted ass Wiss. Abhand. Vol I, reprint 20, pp 295 and my “Remarks about
4.3 Evolution of Thermodynamic State Index (Φ)
137
corresponding to a given state with the macroscopic constraints (Callen 1985;
DeHoff 1993).
Sharp and Matschinky (2015) point out that with Boltzmann’s (permutabilitat)
measure of permutability method for counting possible microstates, there is no need
for a posteriori division by N! “to correct” the derivation using the “somewhat
mystical arguments of Gibbs (1902) and Planck.” Van Kampen (1984), Ehrenfest
and Tikal (1921), Van kampen (1984), Jaynes (1992), and Swendsen (2006) have
pointed out that correct counting of microstates a la Boltzmann precludes the need
for the spurious indistinguishability term N!. “This fact has been ignored in most text
books.”
Translation by Sharp and Matchinsky (2015) also clarifies one very important
point about Boltzmann’s derivation regarding nonequilibrium states. lnΩ is not the
logarithm of a volume in momentum-coordinate-phase space [Boltzmann here refers
to a five dimensional space, the fifth axis being momentum], occupied by the system.
Boltzmann notes, using Liouville’s theorem, that dv remains constant in time.
Therefore, it cannot describe the entropy increase upon approach to the equilibrium
that Boltzmann was so concerned with. He thus avoids at the outset the considerable
difficulty Gibbs had been accounting for changes in entropy with time. Boltzmann
gave us for the first time a definition of entropy applicable to every state at
equilibrium or not. “Then the entropy of the initial and final states is not defined,
but one can still calculate the quantity which we have called the permutability
measure.” By extension, every complexion can then be assigned an entropy, using
the permutability measure of the state distribution to which that complexion belongs
(Lebowitz 1993), opening the door to the statistical mechanics of nonequilibrium
states and irreversible processes.
Boltzmann’s work has historically been misinterpreted, assumed applicable to
gasses only and ignored in continuum mechanics field. Therefore, we feel compelled
to include his original paper in this book. The following section is the English
translation of Boltzmann (1877) by Sharp and Matschinky (2015).
4.3.1 On the Relationship Between the Second Fundamental
Theorem of the Mechanical Theory of Heat
and Probability Calculations Regarding the Conditions
for Thermal Equilibrium, by Ludwig Boltzmann (1877)
The relationship between the second fundamental theorem and calculations of
probability became clear for the first time when I demonstrated that the theorem’s
analytical proof is only possible based on probability calculations. (I refer to my
publication “Analytical proof of the second fundamental theorem of the mechanical
theory of heat derived from the laws of equilibrium for kinetic energy” Wien. Ber.
63, p 8 reprinted ass Wiss. Abhand. Vol I, reprint 20, pp 295 and my “Remarks about
4.3 Evolution of Thermodynamic State Index (Φ)
137