In summary, therefore, according to the now accepted definition, the probability
of this state distribution is
λ!
ð Þ
2
n λ Á
1
w 0 !w 1 !w 2 ! . . . w λ !
Á
1
0!
ð Þ
w 0
Á 1!
ð Þ
w 1
Á 2!
ð Þ
w 2
. . . λ!
ð Þ
w λ !
However, the maximization of this quantity also does not lead to the state
distribution corresponding to thermal equilibrium.
4.3.2.4 Relationship of the Entropy to that Quantity Which I Have
Called the Probability Distribution
When considering this relationship, let us initially deal with the simplest and clearest
case, by first investigating a monoatomic gas on which no external forces act. In this
case, formula (4.103) of Section 2 applies. To give it full generality, however, this
formula must also include the x, y, z coordinates of the position of the molecule [Note
by C. B.: Here it is clear that LB considers the new axis linearly independent of x, y,
z coordinates]. The maximum of such a generalized expression (4.103) then yields
not only the distribution of the velocity components of the gas molecules, which was
sufficient for the case considered there, but also the distribution of the whole mass of
gas in an enclosing vessel, where there it was taken for granted that the gas mass fills
the vessel uniformly.
The generalization of (4.103) for the permutability measure can be easily
obtained from Eq. (4.126) by substituting x, y, z, u, v, w for p 1 , p 2 . . .q r and simply
omitting the terms with accented variables. It reads as follows:
Ω ¼ À
Z Z Z Z Z Z
f x, y, z, u, v, w
ð
Þ ln f x, y, z, u, v, w
ð
Þ dxdydzdudvdw,
ð4:165Þ
where f(x, y, z, u, v, w)dxdydzdudvdw is the number of gas molecules present for
which the six variables x, y, z, u, v, w lie between the limits x and x + dx, y and
y + dy, z and z + dz. . . etc. w and w + dw and the integration limits for the velocity are
between À1 and +1, and for the coordinates over the dimensions of the vessel in
which the gas exists. If the gas was not previously in thermal equilibrium, this
quantity must grow. We want to compute the value this quantity has when the gas
has reached the state of thermal equilibrium. Let V be the total volume of the gas,
T be the average kinetic energy of a gas molecule, and N be the total number of
molecules of the gas; finally m is the mass of a gas molecule. There is then for the
state of thermal equilibrium
174
4 Unified Mechanics Theory
of this state distribution is
λ!
ð Þ
2
n λ Á
1
w 0 !w 1 !w 2 ! . . . w λ !
Á
1
0!
ð Þ
w 0
Á 1!
ð Þ
w 1
Á 2!
ð Þ
w 2
. . . λ!
ð Þ
w λ !
However, the maximization of this quantity also does not lead to the state
distribution corresponding to thermal equilibrium.
4.3.2.4 Relationship of the Entropy to that Quantity Which I Have
Called the Probability Distribution
When considering this relationship, let us initially deal with the simplest and clearest
case, by first investigating a monoatomic gas on which no external forces act. In this
case, formula (4.103) of Section 2 applies. To give it full generality, however, this
formula must also include the x, y, z coordinates of the position of the molecule [Note
by C. B.: Here it is clear that LB considers the new axis linearly independent of x, y,
z coordinates]. The maximum of such a generalized expression (4.103) then yields
not only the distribution of the velocity components of the gas molecules, which was
sufficient for the case considered there, but also the distribution of the whole mass of
gas in an enclosing vessel, where there it was taken for granted that the gas mass fills
the vessel uniformly.
The generalization of (4.103) for the permutability measure can be easily
obtained from Eq. (4.126) by substituting x, y, z, u, v, w for p 1 , p 2 . . .q r and simply
omitting the terms with accented variables. It reads as follows:
Ω ¼ À
Z Z Z Z Z Z
f x, y, z, u, v, w
ð
Þ ln f x, y, z, u, v, w
ð
Þ dxdydzdudvdw,
ð4:165Þ
where f(x, y, z, u, v, w)dxdydzdudvdw is the number of gas molecules present for
which the six variables x, y, z, u, v, w lie between the limits x and x + dx, y and
y + dy, z and z + dz. . . etc. w and w + dw and the integration limits for the velocity are
between À1 and +1, and for the coordinates over the dimensions of the vessel in
which the gas exists. If the gas was not previously in thermal equilibrium, this
quantity must grow. We want to compute the value this quantity has when the gas
has reached the state of thermal equilibrium. Let V be the total volume of the gas,
T be the average kinetic energy of a gas molecule, and N be the total number of
molecules of the gas; finally m is the mass of a gas molecule. There is then for the
state of thermal equilibrium
174
4 Unified Mechanics Theory