Elementary Principles in Statistical Mechanics/Chapter VI

by Josiah Willard Gibbs

CHAPTER VI.

EXTENSION IN CONFIGURATION AND EXTENSION IN VELOCITY.

The formulae relating to canonical ensembles in the closing paragraphs of the last chapter suggest certain general notions and principles, which we shall consider in this chapter, and which are not at all limited in their application to the canonical law of distribution.^[1]

We have seen in Chapter IV. that the nature of the distribution which we have called canonical is independent of the system of coördinates by which it is described, being determined entirely by the modulus. It follows that the value represented by the multiple integral (142), which is the fractional part of the ensemble which lies within certain limiting configurations, is independent of the system of coördinates, being determined entirely by the limiting configurations with the modulus. Now $\psi$, as we have already seen, represents a value which is independent of the system of coördinates by which it is defined. The same is evidently true of $\psi _{p}$ by equation (140), and therefore, by (141), of $\psi _{q}$. Hence the exponential factor in the multiple integral (142) represents a value which is independent of the system of coördinates. It follows that the value of a multiple integral of the form

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}$

(148)

is independent of the system of coördinates which is employed for its evaluation, as will appear at once, if we suppose the multiple integral to be broken up into parts so small that the exponential factor may be regarded as constant in each.

In the same way the formulae (144) and (145) which express the probability that a system (in a canonical ensemble) of given configuration will fall within certain limits of velocity, show that multiple integrals of the form

$\int \ldots \int \Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n}$

(149)

or

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}$

(150)

relating to velocities possible for a given configuration, when the limits are formed by given velocities, have values independent of the system of coördinates employed.

These relations may easily be verified directly. It has already been proved that

${\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}={\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}={\frac {d(q_{1},\ldots q_{n})}{d(Q_{1},\ldots Q_{n})}}$

where $q_{1},\ldots q_{n},p_{1},\ldots q_{n}$ and $Q_{1},\ldots Q_{n},P_{1},\ldots P_{n}$ are two systems of coördinates and momenta.^[2] It follows that

${\begin{aligned}&\int \ldots \int {\bigg (}{\frac {d(p_{1},\ldots p_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}{\bigg )}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}\\&\quad =\int \ldots \int {\bigg (}{\frac {d(p_{1},\ldots p_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}{\bigg )}^{\frac {1}{2}}{\frac {d(q_{1},\ldots q_{n})}{d(Q_{1},\ldots Q_{n})}}\,dQ_{1}\ldots dQ_{n}\\&=\int \ldots \int {\bigg (}{\frac {d(p_{1},\ldots p_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}{\bigg )}^{\frac {1}{2}}{\bigg (}{\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}{\bigg )}^{\frac {1}{2}}{\bigg (}{\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}{\bigg )}^{\frac {1}{2}}\,dQ_{1}\ldots dQ_{n}\\&\quad =\int \ldots \int {\bigg (}{\frac {d(P_{1},\ldots P_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}\,dQ_{1}\ldots dQ_{n},\end{aligned}}$

and

${\begin{aligned}&\int \ldots \int {\bigg (}{\frac {d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}{d(P_{1},\ldots P_{n})}}{\bigg )}^{\frac {1}{2}}\,dP_{1}\ldots dP_{n}\\&\quad =\int \ldots \int {\bigg (}{\frac {d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}{d(P_{1},\ldots P_{n})}}{\bigg )}^{\frac {1}{2}}{\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}\,dp_{1}\ldots dp_{n}\\&=\int \ldots \int {\bigg (}{\frac {d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}{d(P_{1},\ldots P_{n})}}{\bigg )}^{\frac {1}{2}}{\bigg (}{\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}{\bigg )}^{\frac {1}{2}}{\bigg (}{\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}{\bigg )}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n}\\&\quad =\int \ldots \int {\bigg (}{\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d(p_{1},\ldots p_{n})}}\,dp_{1}\ldots dp_{n}.\end{aligned}}$

The multiple integral

$\int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},$

(151)

which may also be written

$\int \ldots \int \Delta _{\dot {q}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}\,dq_{1}\ldots dq_{n},$

(152)

and which, when taken within any given limits of phase, has been shown to have a value independent of the coördinates employed, expresses what we have called an extension-in-phase.^[3] In like manner we may say that the multiple integral (148) expresses an extension-in-configuration, and that the multiple integrals (149) and (150) express an extension-in-velocity. We have called

$dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},$

(153)

which is equivalent to

$\Delta _{\dot {q}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}\,dq_{1}\ldots dq_{n},$

(154)

an element of extension-in-phase. We may call

$\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}$

(155)

an element of extension-in-configuration, and

$\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},$

(156)

or its equivalent

$\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n},$

(157)

an element of extension-in-velocity.

An extension-in-phase may always be regarded as an integral of elementary extensions-in-configuration multiplied each by an extension-in-velocity. This is evident from the formulae (151) and (152) which express an extension-in-phase, if we imagine the integrations relative to velocity to be first carried out.

The product of the two expressions for an element of extension-in-velocity (149) and (150) is evidently of the same dimensions as the product

$p_{1}\ldots p_{n}{\dot {q}}_{1}\ldots {\dot {q}}_{n}$

that is, as the $n$th power of energy, since every product of the form $p_{1}{\dot {q}}_{1}$ has the dimensions of energy. Therefore an extension-in-velocity has the dimensions of the square root of the $n$th power of energy. Again we see by (155) and (156) that the product of an extension-in-configuration and an extension-in-velocity have the dimensions of the $n$th power of energy multiplied by the $n$th power of time. Therefore an extension-in-configuration has the dimensions of the $n$th power of time multiplied by the square root of the $n$th power of energy.

To the notion of extension-in-configuration there attach themselves certain other notions analogous to those which have presented themselves in connection with the notion of extension-in-phase. The number of systems of any ensemble (whether distributed canonically or in any other manner) which are contained in an element of extension-in-configuration, divided by the numerical value of that element, may be called the density-in-configuration. That is, if a certain configuration is specified by the coördinates $q_{1}\ldots q_{n}$, and the number of systems of which the coördinates fall between the limits $q_{1}$ and $q_{1}+dq_{1}$,...$q_{n}$ and $q_{n}+dq_{n}$ is expressed by

$D_{q}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(158)

$D_{q}$ will be the density-in-configuration. And if we set

$e^{\eta _{q}}={\frac {D_{q}}{N}},$

(159)

where $N$ denotes, as usual, the total number of systems in the ensemble, the probability that an unspecified system of the ensemble will fall within the given limits of configuration, is expressed by

$e^{\eta _{q}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}.$

(160)

We may call $e^{\eta _{q}}$ the coefficient of probability of the configuration, and $\eta _{q}$ the index of probability of the configuration.

The fractional part of the whole number of systems which are within any given limits of configuration will be expressed by the multiple integral

$\int \ldots \int e^{\eta _{q}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}.$

(161)

The value of this integral (taken within any given configurations) is therefore independent of the system of coördinates which is used. Since the same has been proved of the same integral without the factor $e^{\eta _{q}}$, it follows that the values of $\eta _{q}$ and $D_{q}$ for a given configuration in a given ensemble are independent of the system of coördinates which is used.

The notion of extension-in-velocity relates to systems having the same configuration.^[4] If an ensemble is distributed both in configuration and in velocity, we may confine our attention to those systems which are contained within certain infinitesimal limits of configuration, and compare the whole number of such systems with those which are also contained within certain infinitesimal limits of velocity. The second of these numbers divided by the first expresses the probability that a system which is only specified as falling within the infinitesimal limits of configuration shall also fall within the infinitesimal limits of velocity. If the limits with respect to velocity are expressed by the condition that the momenta shall fall between the limits $p_{1}$ and $p_{1}+dp_{1}$,...$p_{n}$ and $p_{n}+dp_{n}$ the extension-in-velocity within those limits will be

$\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},$

and we may express the probability in question by

$e^{\eta _{p}}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n}.$

(162)

This may be regarded as defining $\eta _{p}$.

The probability that a system which is only specified as having a configuration within certain infinitesimal limits shall also fall within any given limits of velocity will be expressed by the multiple integral

$\int \ldots \int e^{\eta _{p}}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},$

(163)

or its equivalent

$\int \ldots \int e^{\eta _{p}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n},$

(164)

taken within the given limits.

It follows that the probability that the system will fall within the limits of velocity, ${\dot {q}}_{1}$ and ${\dot {q}}_{1}+d{\dot {q}}_{1}$,...${\dot {q}}_{n}$ and ${\dot {q}}_{n}+d{\dot {q}}_{n}$ is expressed by

$e^{\eta _{p}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}.$

(165)

The value of the integrals (163), (164) is independent of the system of coördinates and momenta which is used, as is also the value of the same integrals without the factor $e^{\eta _{p}}$; therefore the value of $\eta _{p}$ must be independent of the system of coördinates and momenta. We may call $e^{\eta _{p}}$ the coefficient of probability of velocity, and $\eta _{p}$ the index of probability of velocity.

Comparing (160) and (162) with (40), we get

$e^{\eta _{q}}e^{\eta _{p}}=P=e^{\eta }$

(XXX)

or

$\eta _{q}+\eta _{p}=\eta .$

(XXX)

That is: the product of the coefficients of probability of configuration and of velocity is equal to the coefficient of probability of phase; the sum of the indices of probability of configuration and of velocity is equal to the index of probability of phase.

It is evident that $e^{\eta _{q}}$ and $e^{\eta _{p}}$ have the dimensions of the reciprocals of extension-in-configuration and extension-in-velocity respectively, i. e., the dimensions of $t^{-n}\epsilon ^{-{\frac {n}{2}}}$ and $\epsilon ^{-{\frac {n}{2}}}$, where $t$ represent any tine, and $\epsilon$ any energy. If, therefore, the unit of time is multiplied by $c_{t}$, and the unit of energy by $c_{\epsilon }$, every $\eta _{q}$ will be increased by the addition of

$n\log c_{t}+{\tfrac {1}{2}}n\log c_{\epsilon },$

(168)

and every $\eta _{p}$ by the addition of

${\tfrac {1}{2}}n\log c_{\epsilon }.$[5]

(169)

It should be observed that the quantities which have been called extension-in-configuration and extension-in-velocity are not, as the terms might seem to imply, purely geometrical or kinematical conceptions. To express their nature more fully, they might appropriately have been called, respectively, the dynamical measure of the extension in configuration, and the dynamical measure of the extension in velocity. They depend upon the masses, although not upon the forces of the system. In the simple case of material points, where each point is limited to a given space, the extension-in-configuration is the product of the volumes within which the several points are confined (these may be the same or different), multiplied by the square root of the cube of the product of the masses of the several points. The extension-in-velocity for such systems is most easily defined as the extension-in-configuration of systems which have moved from the same configuration for the unit of time with the given velocities.

In the general case, the notions of extension-in-configuration and extension-in-velocity may be connected as follows.

If an ensemble of similar systems of $n$ degrees of freedom have the same configuration at a given instant, but are distributed throughout any finite extension-in-velocity, the same ensemble after an infinitesimal interval of time $\delta t$ will be distributed throughout an extension in configuration equal to its original extension-in-velocity multiplied by $\delta t^{n}$.

In demonstrating this theorem, we shall write $q_{1}{}',\ldots q_{n}{}'$ for the initial values of the coördinates. The final values will evidently be connected with the initial by the equations

$q_{1}-q_{1}{}'={\dot {q}}_{1}\delta t,\ldots q_{n}-q_{n}{}'={\dot {q}}_{n}\delta t.$

(170)

Now the original extension-in-velocity is by definition represented by the integral

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n},$

(171)

where the limits may be expressed by an equation of the form

$F({\dot {q}}_{1},\ldots {\dot {q}}_{n})=0.$

(172)

The same integral multiplied by the constant $\delta t^{n}$ may be written

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,d({\dot {q}}_{1}\delta t),\ldots d({\dot {q}}_{n}\delta t),$

(173)

and the limits may be written

$F({\dot {q}}_{1},\ldots {\dot {q}}_{n})=f({\dot {q}}_{1}\delta t,\ldots {\dot {q}}_{n}\delta t)=0.$

(174)

(It will be observed that $\delta t$ as well as $\Delta _{\dot {q}}$ is constant in the integrations.) Now this integral is identically equal to

$\int \ldots \int \Delta _{q}{}^{\frac {1}{2}}\,d(q_{1}-q_{1}{}')\ldots d(q_{n}-q_{n}{}'),$

(175)

or its equivalent

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(176)

with limits expressed by the equation

$f(q_{1}-q_{1}{}',\ldots q_{n}-q_{n}{}')=0.$

(177)

But the systems which initially had velocities satisfying the equation (172) will after the interval $\delta t$ have configurations satisfying equation (177). Therefore the extension-in-configuration represented by the last integral is that which belongs to the systems which originally had the extension-in-velocity represented by the integral (171).

Since the quantities which we have called extensions-in-phase, extensions-in-configuration, and extensions-in-velocity are independent of the nature of the system of coördinates used in their definitions, it is natural to seek definitions which shall be independent of the use of any coördinates. It will be sufficient to give the following definitions without formal proof of their equivalence with those given above, since they are less convenient for use than those founded on systems of coördinates, and since we shall in fact have no occasion to use them.

We commence with the definition of extension-in- velocity. We may imagine $n$ independent velocities, $V_{1},\ldots V_{n}$ of which a system in a given configuration is capable. We may conceive of the system as having a certain velocity $V_{0}$ combined with a part of each of these velocities $V_{1}\ldots V_{n}$. By a part of $V_{1}$ is meant a velocity of the same nature as $V_{1}$ but in amount being anything between zero and $V_{1}$. Now all the velocities which may be thus described may be regarded as forming or lying in a certain extension of which we desire a measure. The case is greatly simplified if we suppose that certain relations exist between the velocities $V_{1},\ldots V_{n}$, viz: that the kinetic energy due to any two of these velocities combined is the sum of the kinetic energies due to the velocities separately. In this case the extension-in-motion is the square root of the product of the doubled kinetic energies due to the $n$ velocities $V_{1},\ldots V_{n}$ taken separately.

The more general case may be reduced to this simpler case as follows. The velocity $V_{2}$ may always be regarded as composed of two velocities $V_{2}{}'$ and $V_{2}{}''$, of which $V_{2}{}'$ is of the same nature as $V_{1}$, (it may be more or less in amount, or opposite in sign,) while $V_{2}{}''$ satisfies the relation that the kinetic energy due to $V_{1}$ and $V_{2}{}''$ combined is the sum of the kinetic energies due to these velocities taken separately. And the velocity $V_{3}$ may be regarded as compounded of three, $V_{3}{}'$, $V_{3}{}''$, $V_{3}{}'''$, of which $V_{3}{}'$ is of the same nature as $V_{1}$, $V_{3}{}''$ of the same nature as $V_{2}{}''$, while $V_{3}{}'''$ satisfies the relations that if combined either with $V_{1}$ or $V_{2}{}''$ the kinetic energy of the combined velocities is the sum of the kinetic energies of the velocities taken separately. When all the velocities $V_{2},\ldots V_{n}$ have been thus decomposed, the square root of the product of the doubled kinetic energies of the several velocities $V_{1}$, $V_{2}{}''$, $V_{3}{}'''$, etc., will be the value of the extension-in-velocity which is sought.

This method of evaluation of the extension-in- velocity which we are considering is perhaps the most simple and natural, but the result may be expressed in a more symmetrical form. Let us write $\epsilon _{12}$ for the kinetic energy of the velocities $V_{1}$ and $V_{2}$ combined, diminished by the sum of the kinetic energies due to the same velocities taken separately. This may be called the mutual energy of the velocities $V_{1}$ and $V_{2}$. Let the mutual energy of every pair of the velocities $V_{1},\ldots V_{n}$ be expressed in the same way. Analogy would make $\epsilon _{11}$ represent the energy of twice $V_{1}$ diminished by twice the energy of $V_{1}$, i. e., $\epsilon _{11}$ would represent twice the energy of $V_{1}$, although the term mutual energy is hardly appropriate to this case. At all events, let $\epsilon _{11}$ have this signification, and $\epsilon _{22}$ represent twice the energy of $V_{2}$, etc. The square root of the determinant

${\begin{vmatrix}\epsilon _{11}&\epsilon _{12}&\ldots &\epsilon _{1n}\\\epsilon _{21}&\epsilon _{22}&\ldots &\epsilon _{2n}\\\ldots &\ldots &\ldots &\ldots \\\epsilon _{n1}&\epsilon _{n2}&\ldots &\epsilon _{nn}\end{vmatrix}}$

represents the value of the extension-in-velocity determined as above described by the velocities $V_{1},\ldots V_{n}$.

The statements of the preceding paragraph may be readily proved from the expression (157) on page 60, viz.,

$\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}$

by which the notion of an element of extension-in-velocity was originally defined. Since $\Delta _{\dot {q}}$ in this expression represents the determinant of which the general element is

${\frac {d^{2}\epsilon }{d{\dot {q}}_{i}\,d{\dot {q}}_{j}}}$

the square of the preceding expression represents the determinant of which the general element is

${\frac {d^{2}\epsilon }{d{\dot {q}}_{i}\,d{\dot {q}}_{j}}}\,d{\dot {q}}_{i}\,d{\dot {q}}_{j}.$

Now we may regard the differentials of velocity $d{\dot {q}}_{i}$, $d{\dot {q}}_{j}$ as themselves infinitesimal velocities. Then the last expression represents the mutual energy of these velocities, and

${\frac {d^{2}\epsilon }{d{\dot {q}}_{i}{}^{2}}}\,d{\dot {q}}_{i}{}^{2}$

represents twice the energy due to the velocity $d{\dot {q}}_{i}$.

The case which we have considered is an extension-in-velocity of the simplest form. All extensions-in-velocity do not have this form, but all may be regarded as composed of elementary extensions of this form, in the same manner as all volumes may be regarded as composed of elementary parallelepipeds.

Having thus a measure of extension-in-velocity founded, it will be observed, on the dynamical notion of kinetic energy, and not involving an explicit mention of coördinates, we may derive from it a measure of extension-in-configuration by the principle connecting these quantities which has been given in a preceding paragraph of this chapter.

The measure of extension-in-phase may be obtained from that of extension-in-configuration and of extension-in-velocity. For to every configuration in an extension-in-phase there will belong a certain extension-in-velocity, and the integral of the elements of extension-in-configuration within any extension-in-phase multiplied each by its extension-in-velocity is the measure of the extension-in-phase.

---

1. These notions and principles are in fact such as a more logical arrangement of the subject would place in connection with those of Chapter I., to which they are closely related. The strict requirements of logical order have been sacrificed to the natural development of the subject, and very elementary notions have been left until they have presented themselves in the study of the leading problems.
2. See equation (29).
3. See Chapter I, p. 10.
4. Except in some simple cases, such as a system of material points, we cannot compare velocities in one configuration with velocities in another, and speak of their identity or difference except in a sense entirely artificial. We may indeed say that we call the velocities in one configuration the same as those in another when the quantities ${\dot {q}}_{1},\ldots {\dot {q}}_{n}$ have the same values in the two cases. But this signifies nothing until the system of coördinates has been defined. We might identify the velocities in the two cases which make the quantities $p_{1},\ldots p_{n}$ the same in each. This again would signify nothing independently of the system of coördinates employed.
5. Compare (47) in Chapter I.