Thermodynamic state index is the normalized form of the second law of thermodynamics. Derivation of the (1 À Φ) term will be provided in the following section.
The additional axes Thermodynamics State Index (TSI) is necessary to locate the
thermodynamic state of the particle. Coordinates of a point can be defined by
Newton’s laws of motion in the space-time coordinate system. However, thermodynamic state index axis coordinate cannot be defined by space-time coordinate
system. We find it necessary to give the following example. Let us assume there is
a 5-year-old boy in Istanbul and 100-year-old man in New York. Using the spacetime Cartesian coordinate system, we can define their location by x, y, z coordinates
and age on the time axis. However, this does not give any information about their
thermodynamic state. Let us assume 5-year-old boy has stage 4 cancer and is
expected to die in a few days and 100-year-old is also expected to die in few days.
This information cannot be represented in x, y, z-time-space coordinate system.
However, on TSI axis both will be at Φ ¼ 0.999 coordinate (Fig. 4.4).
Another example is, going back to Newton’s second law, if a soccer ball is given
an initial acceleration with a kick force of F, it will move but eventually will come to
a stop. Assume the ball was stationary initially in the terrain shown in Fig. 4.6.
Depending on the path it follows, it will come to a stop at one of the valleys. Again
the initial acceleration of the ball is governed by the second law of Newton and
slowing process is governed by the laws of thermodynamics, which is represented by
(1 À Φ) term. Laws of thermodynamics govern the motion of the ball after the initial
kicking second.
4.2.2 Third Law of Unified Mechanics Theory
All forces between two objects exist in equal magnitude and opposite direction.
However, resulting deformation in two objects will change over time because of
irreversible entropy generation. The resulting equation can be given by,
F 12 ¼
dU 21
du 21
¼
d
du 21
1
2
k 21 1 À Φ
½
u
2
21
h
i
ð4:37Þ
If we assume change in stiffness and TSI are smaller than other derivative terms
by order of magnitude as differential in displacement goes to zero, we can write the
following simple relation
F 12 ¼ F 21 ¼ k 21 Á u 21 1 À Φ
ð
Þ
ð4:38Þ
where U 21 is the strain energy of the reactionary member, Φ is the Thermodynamic
State Index, k is the stiffness and U 21 is the displacement in the reacting member.
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4 Unified Mechanics Theory
The additional axes Thermodynamics State Index (TSI) is necessary to locate the
thermodynamic state of the particle. Coordinates of a point can be defined by
Newton’s laws of motion in the space-time coordinate system. However, thermodynamic state index axis coordinate cannot be defined by space-time coordinate
system. We find it necessary to give the following example. Let us assume there is
a 5-year-old boy in Istanbul and 100-year-old man in New York. Using the spacetime Cartesian coordinate system, we can define their location by x, y, z coordinates
and age on the time axis. However, this does not give any information about their
thermodynamic state. Let us assume 5-year-old boy has stage 4 cancer and is
expected to die in a few days and 100-year-old is also expected to die in few days.
This information cannot be represented in x, y, z-time-space coordinate system.
However, on TSI axis both will be at Φ ¼ 0.999 coordinate (Fig. 4.4).
Another example is, going back to Newton’s second law, if a soccer ball is given
an initial acceleration with a kick force of F, it will move but eventually will come to
a stop. Assume the ball was stationary initially in the terrain shown in Fig. 4.6.
Depending on the path it follows, it will come to a stop at one of the valleys. Again
the initial acceleration of the ball is governed by the second law of Newton and
slowing process is governed by the laws of thermodynamics, which is represented by
(1 À Φ) term. Laws of thermodynamics govern the motion of the ball after the initial
kicking second.
4.2.2 Third Law of Unified Mechanics Theory
All forces between two objects exist in equal magnitude and opposite direction.
However, resulting deformation in two objects will change over time because of
irreversible entropy generation. The resulting equation can be given by,
F 12 ¼
dU 21
du 21
¼
d
du 21
1
2
k 21 1 À Φ
½
u
2
21
h
i
ð4:37Þ
If we assume change in stiffness and TSI are smaller than other derivative terms
by order of magnitude as differential in displacement goes to zero, we can write the
following simple relation
F 12 ¼ F 21 ¼ k 21 Á u 21 1 À Φ
ð
Þ
ð4:38Þ
where U 21 is the strain energy of the reactionary member, Φ is the Thermodynamic
State Index, k is the stiffness and U 21 is the displacement in the reacting member.
134
4 Unified Mechanics Theory