Thus, another condition must now be introduced which will allow the calculation
of S, and to accomplish this it is necessary to look more deeply into the meaning of
the concept of entropy. Consideration of the untenability of the hypothesis made
formerly will help to orient our thoughts in the direction indicated by the above
discussion. In the following a method will be described which yields a new, simpler
expression for entropy and thus provides also a new radiation equation which does
not seem to conflict with any facts so far determined.
4.5.1 Calculations of the Entropy of a Resonator
as a Function of Its Energy
§1.
Entropy depends on disorder and this disorder, according to the electromagnetic
theory of radiation for the monochromatic vibrations of a resonator when situated in
a permanent stationary radiation field, depends on the irregularity with which it
constantly changes its amplitude and phase, provided one considers time intervals
large compared to the time of one vibration but small compared to the duration of a
measurement. If amplitude and phase both remained absolutely constant, which
means completely homogeneous vibrations, no entropy could exist and the vibrational energy would have to be completely free to be converted into work. The
constant energy U of a single stationary vibrating resonator accordingly is to be
taken as time average, or what is the same thing, as a simultaneous average of the
energies of a large number N of identical resonators, situated in the same stationary
radiation field, and which are sufficiently separated so as not to influence each other
directly. It is in this sense that we shall refer to the average energy U of a single
resonator. Then to the total energy
U N ¼ NU
ð4:188Þ
Of such a system of N resonators, there corresponds a certain total entropy
S N ¼ NS
ð4:189Þ
Of the same system, where S represents the average entropy of a single resonator
and the entropy S N depends on the disorder with which the total energy U N is
distributed among the individual resonators.
§2. We now set the entropy S N of the system proportional to the logarithm of its
probability w, within an arbitrary additive constant, so that the N resonators together
have the energy E N :
4.5 On the Law of Distribution of Energy in the Normal Spectrum, By Max Planck,. . .
183
of S, and to accomplish this it is necessary to look more deeply into the meaning of
the concept of entropy. Consideration of the untenability of the hypothesis made
formerly will help to orient our thoughts in the direction indicated by the above
discussion. In the following a method will be described which yields a new, simpler
expression for entropy and thus provides also a new radiation equation which does
not seem to conflict with any facts so far determined.
4.5.1 Calculations of the Entropy of a Resonator
as a Function of Its Energy
§1.
Entropy depends on disorder and this disorder, according to the electromagnetic
theory of radiation for the monochromatic vibrations of a resonator when situated in
a permanent stationary radiation field, depends on the irregularity with which it
constantly changes its amplitude and phase, provided one considers time intervals
large compared to the time of one vibration but small compared to the duration of a
measurement. If amplitude and phase both remained absolutely constant, which
means completely homogeneous vibrations, no entropy could exist and the vibrational energy would have to be completely free to be converted into work. The
constant energy U of a single stationary vibrating resonator accordingly is to be
taken as time average, or what is the same thing, as a simultaneous average of the
energies of a large number N of identical resonators, situated in the same stationary
radiation field, and which are sufficiently separated so as not to influence each other
directly. It is in this sense that we shall refer to the average energy U of a single
resonator. Then to the total energy
U N ¼ NU
ð4:188Þ
Of such a system of N resonators, there corresponds a certain total entropy
S N ¼ NS
ð4:189Þ
Of the same system, where S represents the average entropy of a single resonator
and the entropy S N depends on the disorder with which the total energy U N is
distributed among the individual resonators.
§2. We now set the entropy S N of the system proportional to the logarithm of its
probability w, within an arbitrary additive constant, so that the N resonators together
have the energy E N :
4.5 On the Law of Distribution of Energy in the Normal Spectrum, By Max Planck,. . .
183