several problems in the mechanical theory of heat” 3rd paragraph, Wiss. Abhand,
Vol II reprint 39.) [Boltzmann refers to thermodynamics by the term “The mechanical theory of heat”]. [Wiss. Abhand refers to Boltzmann’s collected works]. This
relationship is also confirmed by demonstrating that an exact proof of the fundamental theorems of the equilibrium of heat is most easily obtained if one demonstrates that a certain quantity—which I wish to define again as E—has to decrease as
a result of the exchange of the kinetic energy among the gaseous molecules and
therefore reaching its minimum value for the state of the equilibrium of heat.
(Compare my “Additional studies about the equilibrium of heat among gaseous
molecules” Wiss. Abhand, vol I, reprint 22, p 316). The relationship between the
second fundamental theorem and the laws of the equilibrium of heat is made even
more compelling in light of the developments in the second paragraph of my
“Remarks about several problems of the mechanical theory of heat.” There, I
mentioned for the first time the possibility of a unique way of calculating the
equilibrium of heat using the following formulation. “It is clear that every single
uniform state distribution which establishes itself after a certain time given a defined
initial state is equally as probable as every single non-uniform state distribution,
comparable to the situation in game of Lotto [a board game] where every single
quintet is as improbable as the quintet 12345. The higher probability that the state
distribution becomes uniform with time arises only because there are far more
uniform than non-uniform state distributions”. Furthermore: “It is even possible to
calculate the probabilities from the relationships of the number of different state
distributions. This approach would perhaps lead to an interesting method for the
calculation of the equilibrium of heat.” It is thereby indicated that it is possible to
calculate the state of the equilibrium of heat by finding the probability of the different
possible states of the system. The initial state in most cases is bound to be highly
improbable, and from it the system will always rapidly approach a more probable
state until it finally reaches the most probable state, i.e., that of the heat equilibrium.
If we apply this to the second basic theorem, we will be able to identify that quantity
which is usually called entropy with the probability of the particular state. Let us
assume a system of bodies which are in a state of isolation with no interaction with
other bodies, e.g., one body with higher and one body with lower temperature and
one so-called intermediate body which accomplishes the heat transfer between the
two bodies; or choosing another example by assuming a vessel with absolutely even
and rigid walls, one half of which is filled with air of low temperature and pressure
whereas the other half is filled with air of high temperature and pressure. The
hypothetical system of particles is assumed to have a certain state at time zero.
Through the interaction between the particles, the state is changed. According to the
second fundamental theorem, this change has to take place in such a way that the
total entropy of the particles increases. This means according to our present interpretation that nothing changes except that the probability of the overall state for all
particles will get larger and larger. The system of particles always changes from an
improbable state to a probable state. It will become clear later what this means. After
the publication of my last treatise regarding this topic, the same idea was taken up
and developed further by Mr. Oskar Emil Meyer totally independent of me (Die
138
4 Unified Mechanics Theory
Vol II reprint 39.) [Boltzmann refers to thermodynamics by the term “The mechanical theory of heat”]. [Wiss. Abhand refers to Boltzmann’s collected works]. This
relationship is also confirmed by demonstrating that an exact proof of the fundamental theorems of the equilibrium of heat is most easily obtained if one demonstrates that a certain quantity—which I wish to define again as E—has to decrease as
a result of the exchange of the kinetic energy among the gaseous molecules and
therefore reaching its minimum value for the state of the equilibrium of heat.
(Compare my “Additional studies about the equilibrium of heat among gaseous
molecules” Wiss. Abhand, vol I, reprint 22, p 316). The relationship between the
second fundamental theorem and the laws of the equilibrium of heat is made even
more compelling in light of the developments in the second paragraph of my
“Remarks about several problems of the mechanical theory of heat.” There, I
mentioned for the first time the possibility of a unique way of calculating the
equilibrium of heat using the following formulation. “It is clear that every single
uniform state distribution which establishes itself after a certain time given a defined
initial state is equally as probable as every single non-uniform state distribution,
comparable to the situation in game of Lotto [a board game] where every single
quintet is as improbable as the quintet 12345. The higher probability that the state
distribution becomes uniform with time arises only because there are far more
uniform than non-uniform state distributions”. Furthermore: “It is even possible to
calculate the probabilities from the relationships of the number of different state
distributions. This approach would perhaps lead to an interesting method for the
calculation of the equilibrium of heat.” It is thereby indicated that it is possible to
calculate the state of the equilibrium of heat by finding the probability of the different
possible states of the system. The initial state in most cases is bound to be highly
improbable, and from it the system will always rapidly approach a more probable
state until it finally reaches the most probable state, i.e., that of the heat equilibrium.
If we apply this to the second basic theorem, we will be able to identify that quantity
which is usually called entropy with the probability of the particular state. Let us
assume a system of bodies which are in a state of isolation with no interaction with
other bodies, e.g., one body with higher and one body with lower temperature and
one so-called intermediate body which accomplishes the heat transfer between the
two bodies; or choosing another example by assuming a vessel with absolutely even
and rigid walls, one half of which is filled with air of low temperature and pressure
whereas the other half is filled with air of high temperature and pressure. The
hypothetical system of particles is assumed to have a certain state at time zero.
Through the interaction between the particles, the state is changed. According to the
second fundamental theorem, this change has to take place in such a way that the
total entropy of the particles increases. This means according to our present interpretation that nothing changes except that the probability of the overall state for all
particles will get larger and larger. The system of particles always changes from an
improbable state to a probable state. It will become clear later what this means. After
the publication of my last treatise regarding this topic, the same idea was taken up
and developed further by Mr. Oskar Emil Meyer totally independent of me (Die
138
4 Unified Mechanics Theory