View source for Eight Lectures on Theoretical Physics/III

{{header
 | title      = [[Eight Lectures on Theoretical Physics]]
 | author     = Max Planck
 | translator = 
 | section    = The Atomic Theory of Matter
 | previous   = [[../II|Second Lecture]]
 | next       = [[../IV|Fourth Lecture]]
 | notes      = 
}}

{{c|{{x-larger|{{UC|THIRD LECTURE.}}}}

{{larger|{{sc|The Atomic Theory of Matter.}}}}}}{{anchor|Lect3}}


The problem with which we shall be occupied in the present
lecture is that of a closer investigation of the atomic theory of
matter. It is, however, not my intention to introduce this
theory with nothing further, and to set it up as something apart
and disconnected with other physical theories, but I intend above
all to bring out the peculiar significance of the atomic theory as
related to the present general system of theoretical physics; for
in this way only will it be possible to regard the whole system
as one containing within itself the essential compact unity, and
thereby to realize the principal object of these lectures.

Consequently it is self evident that we must rely on that sort
of treatment which we have recognized in last week's lecture as
fundamental. That is, the division of all physical processes into
reversible and irreversible processes. Furthermore, we shall be
convinced that the accomplishment of this division is only possible
through the atomic theory of matter, or, in other words,
that irreversibility leads of necessity to atomistics.

I have already referred at the close of the first lecture to the
fact that in pure thermodynamics, which knows nothing of an
atomic structure and which regards all substances as absolutely
continuous, the difference between reversible and irreversible
processes can only be defined in one way, which a priori carries
a provisional character and does not withstand penetrating analysis.
This appears immediately evident when one reflects that
the purely thermodynamic definition of irreversibility which
proceeds from the impossibility of the realization of certain
changes in nature, as, e. g., the transformation of heat into
work without compensation, has at the outset assumed a definite
limit to man's mental capacity, while, however, such a
limit is not indicated in reality. On the contrary: mankind is
making every endeavor to press beyond the present boundaries
of its capacity, and we hope that later on many things will be
attained which, perhaps, many regard at present as impossible
of accomplishment. Can it not happen then that a process,
which up to the present has been regarded as irreversible, may
be proved, through a new discovery or invention, to be reversible?
In this case the whole structure of the second law would undeniably
collapse, for the irreversibility of a single process conditions
that of all the others.

It is evident then that the only means to assure to the second
law real meaning consists in this, that the idea of irreversibility
be made independent of any relationship to man and especially of
all technical relations.

Now the idea of irreversibility harks back to the idea of entropy;
for a process is irreversible when it is connected with an increase
of entropy. The problem is hereby referred back to a proper
improvement of the definition of entropy. In accordance with
the original definition of Clausius, the entropy is measured by
means of a certain reversible process, and the weakness of this
definition rests upon the fact that many such reversible processes,
strictly speaking all, are not capable of being carried out in
practice. With some reason it may be objected that we have
here to do, not with an actual process and an actual physicist,
but only with ideal processes, so-called thought experiments, and
with an ideal physicist who operates with all the experimental
methods with absolute accuracy. But at this point the difficulty
is encountered: How far do the physicist's ideal measurements
of this sort suffice? It may be understood, by passing to the
limit, that a gas is compressed by a pressure which is equal to
the pressure of the gas, and is heated by a heat reservoir which
possesses the same temperature as the gas, but, for example,
that a saturated vapor shall be transformed through isothermal
compression in a reversible manner to a liquid without at any
time a part of the vapor being condensed, as in certain thermodynamic 
considerations is supposed, must certainly appear
doubtful. Still more striking, however, is the liberty as regards
thought experiments, which in physical chemistry is granted the
theorist. With his semi-permeable membranes, which in reality
are only realizable under certain special conditions and then
only with a certain approximation, he separates in a reversible
manner, not only all possible varieties of molecules, whether or
not they are in stable or unstable conditions, but he also separates
the oppositely charged ions from one another and from the
undissociated molecules, and he is disturbed, neither by the
enormous electrostatic forces which resist such a separation, nor
by the circumstance that in reality, from the beginning of the
separation, the molecules become in part dissociated while the
ions in part again combine. But such ideal processes are necessary
throughout in order to make possible the comparison of
the entropy of the undissociated molecules with the entropy of
the dissociated molecules; for the law of thermodynamic equilibrium
does not permit in general of derivation in any other way,
in case one wishes to retain pure thermodynamics as a basis. It
must be considered remarkable that all these ingenious thought
processes have so well found confirmation of their results in
experience, as is shown by the examples considered by us in the
last lecture.

If now, on the other hand, one reflects that in all these results
every reference to the possibility of actually carrying out each
ideal process has disappeared—there are certainly left relations
between directly measurable quantities only, such as temperature,
heat effect, concentration, etc.—the presumption forces
itself upon one that perhaps the introduction as above of such
ideal processes is at bottom a round-about method, and that
the peculiar import of the principle of increase of entropy with
all its consequences can be evolved from the original idea of
irreversibility or, just as well, from the impossibility of perpetual
motion of the second kind, just as the principle of conservation
of energy has been evolved from the law of impossibility of
perpetual motion of the first kind.

This step: to have completed the emancipation of the entropy
idea from the experimental art of man and the elevation of the
second law thereby to a real principle, was the scientific life's
work of Ludwig Boltzmann. Briefly stated, it consisted in
general of referring back the idea of entropy to the idea of
probability. Thereby is also explained, at the same time, the
significance of [[Eight Lectures on Theoretical Physics/I#png27lab1|the above]] auxiliary term used by me;
“preference” of nature for a definite state. Nature prefers the
more probable states to the less probable, because in nature
processes take place in the direction of greater probability. Heat
goes from a body at higher temperature to a body at lower
temperature because the state of equal temperature distribution
is more probable than a state of unequal temperature distribution.

Through this conception the second law of thermodynamics
is removed at one stroke from its isolated position, the mystery
concerning the preference of nature vanishes, and the entropy
principle reduces to a well understood law of the calculus of
probability.

The enormous fruitfulness of so “objective” a definition of
entropy for all domains of physics I shall seek to demonstrate
in the following lectures. But today we have principally to do
with the proof of its admissibility; for on closer consideration we
shall immediately perceive that the new conception of entropy
at once introduces a great number of questions, new requirements
and difficult problems. The first requirement is the introduction
of the atomic hypothesis into the system of physics. For, if one
wishes to speak of the probability of a physical state, i. e., if he
wishes to introduce the probability for a given state as a definite
quantity into the calculation, this can only be brought about, as
in cases of all probability calculations, by referring the state back
to a variety of possibilities; i. e., by considering a finite number
of a priori equally likely configurations (complexions) through
each of which the state considered may be realized. The greater
the number of complexions, the greater is the probability of the
state. Thus, e. g., the probability of throwing a total of four
with two ordinary six-sided dice is found through counting the
complexions by which the throw with a total of four may be
realized. Of these there are three complexions:
{{block center/s}}
{| 
|-
|with the first die, 1, with the second die, 3,
|-
|with the first die, 2, with the second die, 2,
|-
|with the first die, 3, with the second die, 1.
|}
{{block center/e}}
On the other hand, the throw of two is only realized through
a single complexion. Therefore, the probability of throwing a
total of four is three times as great as the probability of throwing
a total of two.

Now, in connection with the physical state under consideration,
in order to be able to differentiate completely from one another
the complexions realizing it, and to associate it with a definite
reckonable number, there is obviously no other means than to
regard it as made up of numerous discrete homogeneous elements—for
in perfectly continuous systems there exist no reckonable
elements—and hereby the atomistic view is made a fundamental
requirement. We have, therefore, to regard all bodies in nature,
in so far as they possess an entropy, as constituted of atoms, and
we therefore arrive in physics at the same conception of matter as
that which obtained in chemistry for so long previously.

But we can immediately go a step further yet. The conclusions
reached hold, not only for thermodynamics of material
bodies, but also possess complete validity for the processes of
heat radiation, which are thus referred back to the second law
of thermodynamics. That radiant heat also possesses an entropy
follows from the fact that a body which emits radiation into a surrounding
diathermanous medium experiences a loss of heat and,
therefore, a decrease of entropy. Since the total entropy of
a physical system can only increase, it follows that one part
of the entropy of the whole system, consisting of the body and the
diathermanous medium, must be contained in the radiated heat.
If the entropy of the radiant heat is to be referred back to the
notion of probability, we are forced, in a similar way as above, to
the conclusion that for radiant heat the atomic conception
possesses a definite meaning. But, since radiant heat is not
directly connected with matter, it follows that this atomistic conception
relates, not to matter, but only to energy, and hence,
that in heat radiation certain energy elements play an essential
rôle. Even though this conclusion appears so singular and even
though in many circles today vigorous objection is strongly urged
against it, in the long run physical research will not be able
to withhold its sanction from it, and the less, since it is confirmed
by experience in quite a satisfactory manner. We shall return
to this point in the lectures on heat radiation. I desire here
only to mention that the novelty involved by the introduction
of atomistic conceptions into the theory of heat radiation is by no
means so revolutionary as, perhaps, might appear at the first
glance. For there is, in my opinion at least, nothing which makes
necessary the consideration of the heat processes in a complete
vacuum as atomic, and it suffices to seek the atomistic features at
the source of radiation, i. e., in those processes which have
their play in the centres of emission and absorption of radiation.
Then the Maxwellian electrodynamic differential equations can
retain completely their validity for the vacuum, and, besides,
the discrete elements of heat radiation are relegated exclusively
to a domain which is still very mysterious and where there is
still present plenty of room for all sorts of hypotheses.

Returning to more general considerations, the most important
question comes up as to whether, with the introduction of atomistic
conceptions and with the reference of entropy to probability,
the content of the principle of increase of entropy is exhaustively
comprehended, or whether still further physical hypotheses are required
in order to secure the full import of that principle. If this
important question had been settled at the time of the introduction
of the atomic theory into thermodynamics, then the
atomistic views would surely have been spared a large number of
conceivable misunderstandings and justifiable attacks. For it
turns out, in fact—and our further considerations will confirm 
this conclusion—that there has as yet nothing been done with
atomistics which in itself requires much more than an essential
generalization, in order to guarantee the validity of the
second law.

We must first reflect that, in accordance with the central
idea laid down in the [[Eight Lectures on Theoretical Physics/I#png17lab1|first lecture]], the second law must
possess validity as an objective physical law, independently of
the individuality of the physicist. There is nothing to hinder
us from imagining a physicist—we shall designate him a “microscopic”
observer—whose senses are so sharpened that he
is able to recognize each individual atom and to follow it in
its motion. For this observer each atom moves exactly in
accordance with the elementary laws which general dynamics
lays down for it, and these laws allow, so far as we know, of an
inverse performance of every process. Accordingly, here again
the question is neither one of probability nor of entropy and its
increase. Let us imagine, on the other hand, another observer,
designated a “macroscopic” observer, who regards an
ensemble of atoms as a homogeneous gas, say, and consequently
applies the laws of thermodynamics to the mechanical and thermal
processes within it. Then, for such an observer, in accordance
with the second law, the process in general is an irreversible
process. Would not now the first observer be justified in saying:
“The reference of the entropy to probability has its origin in
the fact that irreversible processes ought to be explained through
reversible processes. At any rate, this procedure appears to me
in the highest degree dubious. In any case, I declare each change
of state which takes place in the ensemble of atoms designated
a gas, as reversible, in opposition to the macroscopic observer.”
There is not the slightest thing, so far as I know, that one can
urge against the validity of these statements. But do we not
thereby place ourselves in the painful position of the judge who
declared in a trial the correctness of the position of each separately
of two contending parties and then, when a third contends that
only one of the parties could emerge from the process victorious,
was obliged to declare him also correct? Fortunately we find ourselves
in a more favorable position. We can certainly mediate
between the two parties without its being necessary for one or
the other to give up his principal point of view. For closer
consideration shows that the whole controversy rests upon a misunderstanding—a
new proof of how necessary it is before one
begins a controversy to come to an understanding with his
opponent concerning the subject of the quarrel. Certainly, a
given change of state cannot be both reversible and irreversible.
But the one observer connects a wholly different idea with the
phrase “change of state” than the other. What is then, in
general, a change of state? The state of a physical system cannot
well be otherwise defined than as the aggregate of all those physical
quantities, through whose instantaneous values the time
changes of the quantities, with given boundary conditions, are
uniquely determined. If we inquire now, in accordance with
the import of this definition, of the two observers as to what
they understand by the state of the collection of atoms or the
gas considered, they will give quite different answers. The
microscopic observer will mention those quantities which determine
the position and the velocities of all the individual atoms.
There are present in the simplest case, namely, that in which
the atoms may be considered as material points, six times as many
quantities as atoms, namely, for each atom the three coordinates
and the three velocity components, and in the case of combined
molecules, still more quantities. For him the state and the
progress of a process is then first determined when all these
various quantities are individually given. We shall designate
the state defined in this way the “micro-state.” The macroscopic
observer, on the other hand, requires fewer data. He will
say that the state of the homogeneous gas considered by him is
determined by the density, the visible velocity and the temperature
at each point of the gas, and he will expect that, when these
quantities are given, their time variations and, therefore, the progress
of the process, to be completely determined in accordance
with the two laws of thermo-dynamics, and therefore accompanied
by an increase in entropy. In this connection he can call upon
all the experience at his disposal, which will fully confirm his expectation.
If we call this state the “macro-state,” it is clear that
the two laws: “the micro-changes of state are reversible” and
“the macro-changes of state are irreversible,” lie in wholly
different domains and, at any rate, are not contradictory.

But now how can we succeed in bringing the two observers to
an understanding? This is a question whose answer is obviously
of fundamental significance for the atomic theory. First of all,
it is easy to see that the macro-observer reckons only with mean
values; for what he calls density, visible velocity and temperature
of the gas are, for the micro-observer, certain mean values, statistical
data, which are derived from the space distribution and from
the velocities of the atoms in an appropriate manner. But the
micro-observer cannot operate with these mean values alone, for,
if these are given at one instant of time, the progress of the process
is not determined throughout; on the contrary: he can easily
find with given mean values an enormously large number of
individual values for the positions and the velocities of the atoms,
all of which correspond with the same mean values and which, in
spite of this, lead to quite different processes with regard to the
mean values. It follows from this of necessity that the micro-observer
must either {{anchor|png59lab1}}give up the attempt to understand the unique
progress, in accordance with experience, of the macroscopic
changes of state—and this would be the end of the atomic theory—or
that he, through the introduction of a special physical
hypothesis, restrict in a suitable manner the manifold of micro-states
considered by him. There is certainly nothing to prevent
him from assuming that not all conceivable micro-states are
realizable in nature, and that certain of them are in fact thinkable,
but never actually realized. In the formularization of such a
hypothesis, there is of course no point of departure to be found
from the principles of dynamics alone; for pure dynamics leaves
this case undetermined. But on just this account any dynamical
hypothesis, which involves nothing further than a closer specification
of the micro-states realized in nature, is certainly permissible.
Which hypothesis is to be given the preference can only
be decided through comparison of the results to which the
different possible hypotheses lead in the course of experience.

In order to limit the investigation in this way, we must obviously
fix our attention only upon all imaginable configurations and
velocities of the individual atoms which are compatible with
determinate values of the density, the velocity and the temperature
of the gas, or in other words: we must consider all the
micro-states which belong to a determinate macro-state, and
must investigate the various kinds of processes which follow in
accordance with the fixed laws of dynamics from the different
micro-states. Now, precise calculation has in every case always
led to the important result that an enormously large number of
these different micro-processes relate to one and the same macro-process,
and that only proportionately few of the same, which are
distinguished by quite special exceptional conditions concerning
the positions and the velocities of neighboring atoms, furnish
exceptions. Furthermore, it has also shown that one of the
resulting macro-processes is that which the macroscopic observer
recognizes, so that it is compatible with the second law
of thermodynamics.

Here, manifestly, the bridge of understanding is supplied. The
micro-observer needs only to assimilate in his theory the physical
hypothesis that all those special cases in which special exceptional
conditions exist among the neighboring configurations of interacting
atoms do not occur in nature, or, in other words, that the
micro-states are in elementary disorder. Then the uniqueness
of the macroscopic process is assured and with it, also, the fulfillment
of the principle of increase of entropy in all directions.

Therefore, it is not the atomic distribution, but rather the
hypothesis of elementary disorder, which forms the real kernel of
the principle of increase of entropy and, therefore, the preliminary
condition for the existence of entropy. Without elementary 
disorder there is neither entropy nor irreversible process.<ref>To those physicists who, in spite of all this, regard the hypothesis of elementary disorder as gratuitous or as incorrect, I wish to refer the simple fact that in every calculation of a coefficient of friction, of diffusion, or of heat conduction, from molecular considerations, the notion of elementary disorder is employed, whether tacitly or otherwise, and that it is therefore essentially more correct to stipulate this condition instead of ignoring or concealing it. But he who regards the hypothesis of elementary disorder as self-evident, should be reminded that, in accordance with a law of H. Poincaré, the precise investigation concerning the foundation of which would here lead us too far, the assumption of this hypothesis for all times is unwarranted for a closed space with absolutely smooth walls,—an important conclusion, against which can only be urged the fact that absolutely smooth walls do not exist in nature.</ref>
Therefore, a single atom can never possess an entropy; for we
cannot speak of disorder in connection with it. But with a
fairly large number of atoms, say <math>100</math> or <math>1,000</math>, the matter is
quite different. Here, one can certainly speak of a disorder, in
case that the values of the coordinates and the velocity components
are distributed among the atoms in accordance with the
laws of accident. Then it is possible to calculate the probability
for a given state. But how is it with regard to the increase of
entropy? May we assert that the motion of <math>100</math> atoms is irreversible?
Certainly not; but this is only because the state of
<math>100</math> atoms cannot be defined in a thermodynamic sense, since the
process does not proceed in a unique manner from the standpoint
of a macro-observer, and this requirement forms, as we have seen
above, the foundation and preliminary condition for the definition
of a thermodynamic state.

If one therefore asks: How many atoms are at least necessary
in order that a process may be considered irreversible?, the answer
is: so many atoms that one may form from them definite mean
values which define the state in a macroscopic sense. One must
reflect that to secure the validity of the principle of increase of
entropy there must be added to the condition of elementary disorder
still another, namely, that the number of the elements
under consideration be sufficiently large to render possible the
formation of definite mean values. The second law has a
meaning for these mean values only; but for them, it is quite
exact, just as exact as the law of the calculus of probability, that
the mean value, so far as it may be defined, of a sufficiently large
number of throws with a six-sided die, is <math>3\tfrac{1}{2}</math>.

These considerations are, at the same time, capable of throwing
light upon questions such as the following: Does the principle of
increase of entropy possess a meaning for the so-called Brownian
molecular movement of a suspended particle? Does the kinetic
energy of this motion represent useful work or not? The entropy
principle is just as little valid for a single suspended particle as
for an atom, and therefore is not valid for a few of them, but
only when there is so large a number that definite mean values
can be formed. That one is able to see the particles and not
the atoms makes no material difference; because the progress of a
process does not depend upon the power of an observing instrument.
The question with regard to useful work plays no rôle
in this connection; strictly speaking, this possesses, in general, no
objective physical meaning. For it does not admit of an answer
without reference to the scheme of the physicist or technician
who proposes to make use of the work in question. The second
law, therefore, has fundamentally nothing to do with the idea of
useful work (cf. [[Eight Lectures on Theoretical Physics/I#png25lab1|first lecture]]).

But, if the entropy principle is to hold, a further assumption is
necessary, concerning the various disordered elements,—an
assumption which tacitly is commonly made and which we
have not previously definitely expressed. It is, however, not
less important than those referred to above. The elements must
actually be of the same kind, or they must at least form a number
of groups of like kind, e. g., constitute a mixture in which each
kind of element occurs in large numbers. For only through the
similarity of the elements does it come about that order and law
can result in the larger from the smaller. If the molecules of a
gas be all different from one another, the properties of a gas can
never show so simple a law-abiding behavior as that which is
indicated by thermodynamics. In fact, the calculation of the
probability of a state presupposes that all complexions which
correspond to the state are a priori equally likely. Without
this condition one is just as little able to calculate the probability
of a given state as, for instance, the probability of a given throw
with dice whose sides are unequal in size. In summing up we
may therefore say: the second law of thermodynamics in its
objective physical conception, freed from anthropomorphism,
relates to certain mean values which are formed from a large
number of disordered elements of the same kind.

The validity of the principle of increase of entropy and of the
irreversible progress of thermodynamic processes in nature is
completely assured in this formularization. After the introduction
of the hypothesis of elementary disorder, the microscopic
observer can no longer confidently assert that each process considered
by him in a collection of atoms is reversible; for the
motion occurring in the reverse order will not always obey the
requirements of that hypothesis. In fact, the motions of single
atoms are always reversible, and thus far one may say, as before,
that the irreversible processes appear reduced to a reversible
process, but the phenomenon as a whole is nevertheless irreversible,
because upon reversal the disorder of the numerous
individual elementary processes would be eliminated. Irreversibility
is inherent, not in the individual elementary processes
themselves, but solely in their irregular constitution. It is
this only which guarantees the unique change of the macroscopic
mean values.

Thus, for example, the reverse progress of a frictional process
is impossible, in that it would presuppose elementary arrangement
of interacting neighboring molecules. For the collisions between
any two molecules must thereby possess a certain distinguishing
character, in that the velocities of two colliding molecules
depend in a definite way upon the place at which they meet.
In this way only can it happen that in collisions like directed
velocities ensue and, therefore, visible motion.

Previously we have only referred to the principle of elementary
disorder in its application to the atomic theory of matter. But
it may also be assumed as valid, as I wish to indicate at this
point, on quite the same grounds as those holding in the case of
matter, for the theory of radiant heat. Let us consider, e. g.,
two bodies at different temperatures between which exchange of
heat occurs through radiation. We can in this case also imagine
a microscopic observer, as opposed to the ordinary macroscopic
observer, who possesses insight into all the particulars
of electromagnetic processes which are connected with emission
and absorption, and the propagation of heat rays. The microscopic
observer would declare the whole process reversible
because all electrodynamic processes can also take place in the
reverse direction, and the contradiction may here be referred
back to a difference in definition of the state of a heat ray. Thus,
while the macroscopic observer completely defines a monochromatic
ray through direction, state of polarization, color, and
intensity, the microscopic observer, in order to possess a complete
knowledge of an electromagnetic state, necessarily requires the
specification of all the numerous irregular variations of amplitude
and phase to which the most homogeneous heat ray is actually
subject. That such irregular variations actually exist follows
immediately from the well known fact that two rays of the same
color never interfere, except when they originate in the same source
of light. But until these fluctuations are given in all particulars,
the micro-observer can say nothing with regard to the progress
of the process. He is also unable to specify whether the exchange
of heat radiation between the two bodies leads to a decrease or
to an increase of their difference in temperature. The principle
of elementary disorder first furnishes the adequate criterion of
the tendency of the radiation process, i. e., the warming of the
colder body at the expense of the warmer, just as the same principle
conditions the irreversibility of exchange of heat through conduction.
However, in the two cases compared, there is indicated
an essential difference in the kind of the disorder. While in
heat conduction the disordered elements may be represented
as associated with the various molecules, in heat radiation there
are the numerous vibration periods, connected with a heat ray,
among which the energy of radiation is irregularly distributed.
In other words: the disorder among the molecules is a material
one, while in heat radiation it is one of energy distribution. This
is the most important difference between the two kinds of disorder;
a common feature exists as regards the great number of
uncoordinated elements required. Just as the entropy of a body
is defined as a function of the macroscopic state, only when the
body contains so many atoms that from them definite mean
values may be formed, so the entropy principle only possesses
a meaning with regard to a heat ray when the ray comprehends
so many periodic vibrations, i. e., persists for so long a time, that
a definite mean value for the intensity of the ray may be obtained
from the successive irregular fluctuating amplitudes.

Now, after the principle of elementary disorder has been
introduced and accepted by us as valid throughout nature, the
fundamental question arises as to the calculation of the probability
of a given state, and the actual derivation of the entropy
therefrom. From the entropy all the laws of thermodynamic
states of equilibrium, for material substances, and also for
energy radiation, may be uniquely derived. With regard to
the connection between entropy and probability, this is inferred
very simply from the law that the probability of two independent
configurations is represented by the product of the individual
probabilities:
:<math>\begin{align}&{\color{White}.(00)}\qquad&&
W = W_{1} \cdot W_{2},
\end{align}</math>
while the entropy <math>S</math> is represented by the sum of the individual
entropies:
:<math>\begin{align}&{\color{White}.(00)}\qquad&&
S = S_{1} + S_{2}.
\end{align}</math>
Accordingly, the entropy is proportional to the logarithm of the
probability:
:<math>\begin{align}&(12){\color{White}.}\qquad&&
S = k \log W.
\end{align}</math>
<math>k</math> is a universal constant. In particular, it is the same for atomic
as for radiation configurations, for there is nothing to prevent
us assuming that the configuration designated by <math>1</math> is atomic,
while that designated by <math>2</math> is a radiation configuration. If <math>k</math> has
been calculated, say with the aid of radiation measurements,
then <math>k</math> must have the same value for atomic processes. Later
we shall follow this procedure, in order to utilize the laws of heat
radiation in the kinetic theory of gases. Now, there remains, as
the last and most difficult part of the problem, the calculation of
the probability <math>W</math> of a given physical configuration in a given
macroscopic state. We shall treat today, by way of preparation
for the quite general problem to follow, the simple problem: to
specify the probability of a given state for a single moving
material point, subject to given conservative forces. Since the
state depends upon <math>6</math> variables: the <math>3</math> generalized coordinates
<math>\varphi_{1}</math>, <math>\varphi_{2}</math>, <math>\varphi_{3}</math>, and the three corresponding velocity components
<math>\dot{\varphi}_{1}</math>, <math>\dot{\varphi}_{2}</math>, <math>\dot{\varphi}_{3}</math>, and since all possible values of these <math>6</math> variables constitute
a continuous manifold, the probability sought is, that
these <math>6</math> quantities shall lie respectively within certain infinitely
small intervals, or, if one thinks of these <math>6</math> quantities as the
rectilinear orthogonal coordinates of a point in an ideal six-dimensional
space, that this ideal “state point” shall fall within
a given, infinitely small “state domain.” Since the domain is
infinitely small, the probability will be proportional to the magnitude
of the domain and therefore proportional to
:<math>\begin{align}&{\color{White}.(00)}\qquad&&
\int d\varphi_{1} \cdot d\varphi_{2} \cdot d\varphi_{3} \cdot d\dot{\varphi}_{1} \cdot d\dot{\varphi}_{2} \cdot d\dot{\varphi}_{3}.
\end{align}</math>

But this expression cannot serve as an absolute measure of
the probability, because in general it changes in magnitude with
the time, if each state point moves in accordance with the laws
of motion of material points, while the probability of a state
which follows of necessity from another must be the same for
the one as the other. Now, as is well known, another integral
quite similarly formed, may be specified in place of the one
above, which possesses the special property of not changing in
value with the time. It is only necessary to employ, in addition
to the general coordinates <math>\varphi_{1}</math>, <math>\varphi_{2}</math>, <math>\varphi_{3}</math>, the three so-called momenta
<math>\psi_{1}</math>, <math>\psi_{2}</math>, <math>\psi_{3}</math>, in place of the three velocities <math>\dot{\varphi}_{1}</math>, <math>\dot{\varphi}_{2}</math>, <math>\dot{\varphi}_{3}</math> as the determining
coordinates of the state. These are defined in the
following way:
{{anchor|png67lab1}}
:<math>\begin{align}&{\color{White}.(00)}\qquad&&
\psi_{1} = \left(\frac{\partial H}{\partial \dot{\varphi}_{1}}\right)_{\varphi},\quad
\psi_{2} = \left(\frac{\partial H}{\partial \dot{\varphi}_{2}}\right)_{\varphi},\quad
\psi_{3} = \left(\frac{\partial H}{\partial \dot{\varphi}_{3}}\right)_{\varphi},
\end{align}</math>
wherein <math>H</math> denotes the kinetic potential (Helmholz). Then, in
Hamiltonian form, the equations of motion are:
{{anchor|png67lab2}}
:<math>\begin{align}&{\color{White}.(00)}\qquad&&
\dot{\psi}_{1} = \frac{d\psi_{1}}{dt} = -\left(\frac{\partial E}{\partial \varphi_{1}}\right)_{\psi},\ \cdots,\quad
\dot{\varphi}_{1} = \frac{d\varphi_{1}}{dt} =  \left(\frac{\partial E}{\partial \psi_{1}}\right)_{\varphi},\ \cdots,
\end{align}</math>
(<math>E</math> is the energy), and from these equations follows the “condition
of incompressibility”:
:<math>\begin{align}&{\color{White}.(00)}\qquad&&
\frac{\partial \dot{\varphi}_{1}}{\partial \varphi_{1}} + \frac{\partial \dot{\psi}_{1}}{\partial \psi_{1}} + \cdots = 0.
\end{align}</math>
Referring to the six-dimensional space represented by the coordinates
<math>\varphi_{1}</math>, <math>\varphi_{2}</math>, <math>\varphi_{3}</math>, <math>\psi_{1}</math>, <math>\psi_{2}</math>, <math>\psi_{3}</math>, this equation states that the magnitude
of an arbitrarily chosen state domain, viz.:
{{anchor|png67lab3}}
:<math>\begin{align}&{\color{White}.(00)}\qquad&&
\int d\varphi_{1} \cdot d\varphi_{2} \cdot d\varphi_{3} \cdot d\psi_{1} \cdot d\psi_{2} \cdot d\psi_{3}
\end{align}</math>
does not change with the time, when each point of the domain
changes its position in accordance with the laws of motion of
material points. Accordingly, it is made possible to take the
magnitude of this domain as a direct measure for the probability
that the state point falls within the domain.

From the last expression, which can be easily generalized for
the case of an arbitrary number of variables, we shall {{anchor|png67lab4}}calculate
later the probability of a thermodynamic state, for the
case of radiant energy as well as that for material substances.

{{rule}}{{smallrefs}}