4.3 Evolution of Thermodynamic State Index (Φ)
Materials, under externally applied loading, change their thermodynamic state. This
process will follow the laws of thermodynamics. Simply put, first law of thermodynamics will govern the conservation of energy. The second law of thermodynamics
will govern the entropy generation rate according to the fundamental equation of the
system. Evolution of entropy and disorder in the system is given by Boltzmann
equation. When the entropy is maximum and entropy generation rate is zero for a
closed and isolated system, the change in TSI will come to a stop.
While author of this book and many others have proven validity of Boltzmann’s
equation in solids in the last 20+ years extensively, Boltzmann’s (1877) original
paper and its interpretation on disorder was misinterpreted until recently as being
only applicable to gasses. Recently there has been a great deal of interest in
translating Boltzmann’s original papers into English. One recent such paper is by
Sharp and Matschinsky (2015). We find it essential to include their translation in this
section. It is important to point out that, Sharp and Matschinsky (2015) also state that
“what Boltzmann actually wrote on these subjects is rarely quoted directly, his
methods are not fully appreciated and key concepts have been misinterpreted.”
The following section about Boltzmann’s work is a direct quotation from their paper.
“The translation provided here is for Boltzmann (1877). Previous work of
Maxwell and Boltzmann’s derivations was based on mechanical laws of motion
and particle interaction of gasses. However, this work by Boltzmann is much more
general. His formulation require only that particles can exchange kinetic energy, but
they do not specify the mechanism. As Boltzmann predicted in the final sentences of
this paper, his approach was applicable not just gasses, but to liquids and solids of
any composition. Indeed the Boltzmann distribution has also passed almost
unchanged into the quantum world.”
Boltzmann developed the theoretical basis for statistical mechanics with great
clarity. Boltzmann used three levels of hierarchy to describe the processes. At the
highest level, there is macro-state, where the thermodynamic state variables such as
temperature and pressure can be observed directly. The second level in the hierarchy
is where energy or velocity components of each molecule can be specified. He calls
this Komplexion, which Sharp and Matschinky (2015) translate as complexion.
Finally there is the third level at which the number of molecules with each its energy
level, velocity, and their position is specified. Here Boltzmann does not make any
assumption about the type of the molecules (Boltzmann’sw 0 , w 1 , etc.). Boltzmann
calls a particular set of w-values a Zustandeverteilung, translated as a “state distribution” or “distribution of states.” It is important to point out, as discussed by Sharp
and Matschinsky (2015) microstate is divided into two separate hierarchies in
Boltzmann’s formulation, both complexion and state distribution. “Boltzmann then
uses permutation mathematics to determine distribution of states, which he denotes
by p.” “Boltzmann then shows how to find the distribution of states (w
max
0 , w
max
1 , ⋯)
with the largest number of complexious p
max subject to constraints on the temperature and number of molecules. Boltzmann’s postulate is that (w
max
0 , w
max
1 , . . .) is the
4.3 Evolution of Thermodynamic State Index (Φ)
135
Materials, under externally applied loading, change their thermodynamic state. This
process will follow the laws of thermodynamics. Simply put, first law of thermodynamics will govern the conservation of energy. The second law of thermodynamics
will govern the entropy generation rate according to the fundamental equation of the
system. Evolution of entropy and disorder in the system is given by Boltzmann
equation. When the entropy is maximum and entropy generation rate is zero for a
closed and isolated system, the change in TSI will come to a stop.
While author of this book and many others have proven validity of Boltzmann’s
equation in solids in the last 20+ years extensively, Boltzmann’s (1877) original
paper and its interpretation on disorder was misinterpreted until recently as being
only applicable to gasses. Recently there has been a great deal of interest in
translating Boltzmann’s original papers into English. One recent such paper is by
Sharp and Matschinsky (2015). We find it essential to include their translation in this
section. It is important to point out that, Sharp and Matschinsky (2015) also state that
“what Boltzmann actually wrote on these subjects is rarely quoted directly, his
methods are not fully appreciated and key concepts have been misinterpreted.”
The following section about Boltzmann’s work is a direct quotation from their paper.
“The translation provided here is for Boltzmann (1877). Previous work of
Maxwell and Boltzmann’s derivations was based on mechanical laws of motion
and particle interaction of gasses. However, this work by Boltzmann is much more
general. His formulation require only that particles can exchange kinetic energy, but
they do not specify the mechanism. As Boltzmann predicted in the final sentences of
this paper, his approach was applicable not just gasses, but to liquids and solids of
any composition. Indeed the Boltzmann distribution has also passed almost
unchanged into the quantum world.”
Boltzmann developed the theoretical basis for statistical mechanics with great
clarity. Boltzmann used three levels of hierarchy to describe the processes. At the
highest level, there is macro-state, where the thermodynamic state variables such as
temperature and pressure can be observed directly. The second level in the hierarchy
is where energy or velocity components of each molecule can be specified. He calls
this Komplexion, which Sharp and Matschinky (2015) translate as complexion.
Finally there is the third level at which the number of molecules with each its energy
level, velocity, and their position is specified. Here Boltzmann does not make any
assumption about the type of the molecules (Boltzmann’sw 0 , w 1 , etc.). Boltzmann
calls a particular set of w-values a Zustandeverteilung, translated as a “state distribution” or “distribution of states.” It is important to point out, as discussed by Sharp
and Matschinsky (2015) microstate is divided into two separate hierarchies in
Boltzmann’s formulation, both complexion and state distribution. “Boltzmann then
uses permutation mathematics to determine distribution of states, which he denotes
by p.” “Boltzmann then shows how to find the distribution of states (w
max
0 , w
max
1 , ⋯)
with the largest number of complexious p
max subject to constraints on the temperature and number of molecules. Boltzmann’s postulate is that (w
max
0 , w
max
1 , . . .) is the
4.3 Evolution of Thermodynamic State Index (Φ)
135