​

In the simple but important case of a system of material points, if we use rectangular coördinates, we have for the product of the differentials of the coördinates

${\displaystyle dx_{1}\,dy_{1}\,dz_{1}\ldots dx_{\nu }\,dy_{\nu }\,dz_{\nu },}$

and for the product of the differentials of the momenta

${\displaystyle m_{1}\,d{\dot {x}}_{1}\,m_{1}\,d{\dot {y}}_{1}\,m_{1}\,d{\dot {z}}_{1}\ldots m_{\nu }\,d{\dot {x}}_{\nu }\,m_{\nu }\,d{\dot {y}}_{\nu }\,m_{\nu }\,d{\dot {z}}_{\nu }.}$

The product of these expressions, which represents an element of extension-in-phase, may be briefly written

${\displaystyle m_{1}\,d{\dot {x}}_{1}\ldots m_{\nu }\,d{\dot {z}}_{\nu }\,dx_{1}\ldots dz_{\nu };}$

and the integral

${\displaystyle \int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}m_{1}\,d{\dot {x}}_{1}\ldots m_{\nu }\,d{\dot {z}}_{\nu }\,dx_{1}\ldots dz_{\nu }}$

(118)

will represent the probability that a system taken at random from an ensemble canonically distributed will fall within any given limits of phase.

In this case

${\displaystyle \epsilon =\epsilon _{q}+{\tfrac {1}{2}}m_{1}{\dot {x}}_{1}{}^{2}\ldots {\tfrac {1}{2}}m_{\nu }{\dot {z}}_{\nu }{}^{2},}$

(119)

and

${\displaystyle e^{\frac {\psi -\epsilon }{\Theta }}=e^{\frac {\psi -\epsilon _{q}}{\Theta }}e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}\ldots e^{-{\frac {m_{\nu }{\dot {z}}_{\nu }{}^{2}}{2\Theta }}}.}$

(120)

The potential energy (${\displaystyle \epsilon _{q}}$) is independent of the velocities, and if the limits of integration for the coördinates are independent of the velocities, and the limits of the several velocities are independent of each other as well as of the coördinates, ​the multiple integral may be resolved into the product of integrals

${\displaystyle \int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}d{\dot {x}}_{1}\ldots dz_{\nu }\int e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}m_{1}\,d{\dot {x}}_{1}\ldots \int e^{-{\frac {m_{1}{\dot {z}}_{n}u{}^{2}}{2\Theta }}}m_{1}\,d{\dot {z}}_{\nu }.}$

(121)

This shows that the probability that the configuration lies within any given limits is independent of the velocities, and that the probability that any component velocity lies within any given limits is independent of the other component velocities and of the configuration.

Since

${\displaystyle \int _{-\infty }^{+\infty }e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}m_{1}\,d{\dot {x}}_{1}={\sqrt {2\pi m_{1}\Theta }},}$

(122)

and

${\displaystyle \int _{-\infty }^{+\infty }m_{1}{\dot {x}}_{1}{}^{2}e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}m_{1}\,d{\dot {x}}_{1}={\sqrt {{\tfrac {1}{2}}\pi m_{1}\Theta ^{3}}},}$

(123)

the average value of the part of the kinetic energy due to the velocity ${\displaystyle {\dot {x}}_{1}}$, which is expressed by the quotient of these integrals, is ${\displaystyle {\tfrac {1}{2}}\Theta }$. This is true whether the average is taken for the whole ensemble or for any particular configuration, whether it is taken without reference to the other component velocities, or only those systems are considered in which the other component velocities have particular values or lie within specified limits.

The number of coördinates is ${\displaystyle 3\nu }$ or ${\displaystyle n}$. We have, therefore, for the average value of the kinetic energy of a system

${\displaystyle {\overline {\epsilon }}_{p}={\tfrac {3}{2}}\nu \Theta ={\tfrac {1}{2}}n\Theta .}$

(124)

This is equally true whether we take the average for the whole ensemble, or limit the average to a single configuration.

The distribution of the systems with respect to their component velocities follows the 'law of errors'; the probability that the value of any component velocity lies within any given limits being represented by the value of the corresponding integral in (121) for those limits, divided by ${\displaystyle (2\pi m\Theta )^{\frac {1}{2}}}$, ​which is the value of the same integral for infinite limits. Thus the probability that the value of ${\displaystyle {\dot {x}}_{1}}$ lies between any given limits is expressed by

${\displaystyle {\bigg (}{\frac {m_{1}}{2\pi \Theta }}{\bigg )}^{\frac {1}{2}}\int e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}\,d{\dot {x}}_{1}.}$

(125)

The expression becomes more simple when the velocity is expressed with reference to the energy involved. If we set

${\displaystyle s={\bigg (}{\frac {m_{1}}{2\Theta }}{\bigg )}^{\frac {1}{2}}{\dot {x}}_{1},}$

the probability that ${\displaystyle s}$ lies between any given limits is expressed by

${\displaystyle {\frac {1}{\sqrt {\pi }}}\int e^{-s^{2}}\,ds.}$

(126)

Here ${\displaystyle s}$ is the ratio of the component velocity to that which would give the energy ${\displaystyle \Theta }$; in other words, ${\displaystyle s^{2}}$ is the quotient of the energy due to the component velocity divided by ${\displaystyle \Theta }$. The distribution with respect to the partial energies due to the component velocities is therefore the same for all the component velocities.

The probability that the configuration lies within any given limits is expressed by the value of

${\displaystyle M^{3/2}(2\pi \Theta )^{\frac {3\nu }{2}}\int \cdots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}\,dx_{1}\ldots dz_{\nu }}$

(127)

for those limits, where ${\displaystyle M}$ denotes the product of all the masses. This is derived from (121) by substitution of the values of the integrals relating to velocities taken for infinite limits.

Very similar results may be obtained in the general case of a conservative system of ${\displaystyle n}$ degrees of freedom. Since ${\displaystyle \epsilon _{p}}$ is a homogeneous quadratic function of the ${\displaystyle p}$'s, it may be divided into parts by the formula

${\displaystyle \epsilon _{p}={\tfrac {1}{2}}p_{1}{\frac {d\epsilon _{p}}{dp_{1}}}\ldots +{\tfrac {1}{2}}p_{n}{\frac {d\epsilon _{p}}{dp_{n}}}}$

(128)

​where ${\displaystyle \epsilon }$ might be written for ${\displaystyle \epsilon _{p}}$ in the differential coefficients without affecting the signification. The average value of the first of these parts, for any given configuration, is expressed by the quotient

${\displaystyle {\frac {\int _{-\infty }^{+\infty }\ldots \int _{-\infty }^{+\infty }{\tfrac {1}{2}}p_{1}{\frac {d\epsilon }{dp_{1}}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dp_{n}}{\int _{-\infty }^{+\infty }\ldots \int _{-\infty }^{+\infty }e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dp_{n}}}}$

(129)

Now we have by integration by parts

${\displaystyle \int _{-\infty }^{+\infty }p_{1}e^{\frac {\psi -\epsilon }{\Theta }}{\frac {d\epsilon }{dp_{1}}}\,dp_{1}=\Theta \int _{-\infty }^{+\infty }e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}.}$

(130)

By substitution of this value, the above quotient reduces to ${\displaystyle {\frac {\Theta }{2}}}$, which is therefore the average value of ${\displaystyle {\tfrac {1}{2}}p_{1}{\frac {d\epsilon }{dp_{1}}}}$ for the given configuration. Since this value is independent of the configuration, it must also be the average for the whole ensemble, as might easily be proved directly. (To make the preceding proof apply directly to the whole ensemble, we have only to write ${\displaystyle dp_{1}\ldots dq_{n}}$ for ${\displaystyle dp_{1}\ldots dp_{n}}$ in the multiple integrals.) This gives ${\displaystyle {\tfrac {1}{2}}n\Theta }$ for the average value of the whole kinetic energy for any given configuration, or for the whole ensemble, as has already been proved in the case of material points.

The mechanical significance of the several parts into which the kinetic energy is divided in equation (128) will be apparent if we imagine that by the application of suitable forces (different from those derived from ${\displaystyle \epsilon _{q}}$ and so much greater that the latter may be neglected in comparison) the system was brought from rest to the state of motion considered, so rapidly that the configuration was not sensibly altered during the process, and in such a manner also that the ratios of the component velocities were constant in the process. If we write

${\displaystyle F_{1}\,dq_{1}\ldots +F_{n}\,dq_{n}}$

​for the moment of these forces, we have for the period of their action by equation (3)

${\displaystyle {\dot {p}}_{1}=-{\frac {d\epsilon _{p}}{dq_{1}}}-{\frac {d\epsilon _{q}}{dq_{1}}}+F_{1}=-{\frac {d\epsilon }{dq_{1}}}+F_{1}.}$

The work done by the force ${\displaystyle F_{1}}$ may be evaluated as follows:

${\displaystyle \int F_{1}\,dq_{1}=\int {\dot {p}}_{1}\,dq_{1}+\int {\frac {d\epsilon }{dq_{1}}}\,dq_{1},}$

where the last term may be cancelled because the configuration does not vary sensibly during the application of the forces. (It will be observed that the other terms contain factors which increase as the time of the action of the forces is diminished.) We have therefore,

${\displaystyle \int F_{1}\,dq_{1}=\int {\dot {p}}_{1}{\dot {q}}_{1}\,dt=\int {\dot {q}}_{1}\,dp_{1}={\frac {{\dot {q}}_{1}}{p_{1}}}\int p_{1}\,dp_{1}.}$

(131)

For since the ${\displaystyle p}$'s are linear functions of the ${\displaystyle {\dot {q}}}$'s (with coefficients involving the ${\displaystyle q}$'s) the supposed constancy of the ${\displaystyle q}$'s and of the ratios of the ${\displaystyle {\dot {q}}}$'s will make the ratio ${\displaystyle {\dot {q}}_{1}/p_{1}}$ constant. The last integral is evidently to be taken between the limits zero and the value of ${\displaystyle p_{1}}$ in the phase originally considered, and the quantities before the integral sign may be taken as relating to that phase. We have therefore

${\displaystyle \int F_{1}\,dq_{1}={\tfrac {1}{2}}p_{1}{\dot {q}}_{1}={\tfrac {1}{2}}p_{1}{\frac {d\epsilon }{dp_{1}}}.}$

(132)

That is: the several parts into which the kinetic energy is divided in equation (128) represent the amounts of energy communicated to the system by the several forces ${\displaystyle F_{1},\ldots F_{n}}$ under the conditions mentioned.

The following transformation will not only give the value of the average kinetic energy, but will also serve to separate the distribution of the ensemble in configuration from its distribution in velocity.

Since ${\displaystyle 2\epsilon _{p}}$ is a homogeneous quadratic function of the ${\displaystyle p}$'s, which is incapable of a negative value, it can always be expressed (and in more than one way) as a sum of squares of ​linear functions of the ${\displaystyle p}$'s.^[1] The coefficients in these linear functions, like those in the quadratic function, must be regarded in the general case as functions of the ${\displaystyle q}$'s. Let

${\displaystyle 2\epsilon _{p}=u_{1}{}^{2}+u_{2}{}^{2}\ldots +u_{n}{}^{2}}$

where ${\displaystyle u_{1}\ldots u_{n}}$ are such linear functions of the ${\displaystyle p}$'s. If we write

${\displaystyle {\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}}$

for the Jacobian or determinant of the differential coefficients of the form ${\displaystyle dp/du}$, we may substitute

${\displaystyle {\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}\,du_{1}\ldots du_{n}}$

for

${\displaystyle dp_{1}\ldots dp_{n}}$

under the multiple integral sign in any of our formulæ. It will be observed that this determinant is function of the ${\displaystyle q}$'s alone. The sign of such a determinant depends on the relative order of the variables in the numerator and denominator. But since the suffixes of the ${\displaystyle u}$'s are only used to distinguish these functions from one another, and no especial relation is supposed between a ${\displaystyle p}$ and a ${\displaystyle u}$ which have the same suffix, we may evidently, without loss of generality, suppose the suffixes so applied that the determinant is positive.

Since the ${\displaystyle u}$'s are linear functions of the ${\displaystyle p}$'s, when the integrations are to cover all values of the ${\displaystyle p}$'s (for constant ${\displaystyle q}$'s) once and only once, they must cover all values of the ${\displaystyle u}$'s once and only once, and the limits will be ${\displaystyle \pm \infty }$ for all the ${\displaystyle u}$'s. Without the supposition of the last paragraph the upper limits would not always be ${\displaystyle +\infty }$, as is evident on considering the effect of changing the sign of a ${\displaystyle u}$. But with the supposition which we have made (that the determinant is always positive) we may make the upper limits ${\displaystyle +\infty }$ and the lower ${\displaystyle -\infty }$ for all the ${\displaystyle u}$'s. Analogous considerations will apply where the integrations do not cover all values of the ${\displaystyle p}$'s and therefore of ​the ${\displaystyle u}$'s. The integrals may always be taken from a less to a greater value of a ${\displaystyle u}$.

The general integral which expresses the fractional part of the ensemble which falls within any given limits of phase is thus reduced to the form

${\displaystyle \int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}{\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}e^{-{\frac {u_{1}{}^{2}\ldots u_{n}{}^{2}}{2\Theta }}}\,du_{1}\ldots du_{n}\,dq_{1}\ldots dq_{n}.}$

(134)

For the average value of the part of the kinetic energy which is represented by ${\displaystyle {\tfrac {1}{2}}u_{1}{}^{2}}$ whether the average is taken for the whole ensemble, or for a given configuration, we have therefore

${\displaystyle {\tfrac {1}{2}}{\overline {u}}_{1}{}^{2}={\frac {\int _{-\infty }^{+\infty }{\tfrac {1}{2}}u_{1}{}^{2}e^{-{\frac {u_{1}{}^{2}}{2\Theta }}}\,du_{1}}{\int _{-\infty }^{+\infty }e^{-{\frac {u_{1}{}^{2}}{2\Theta }}}\,du_{1}}}={\frac {({\tfrac {1}{2}}\pi \Theta ^{3})^{\frac {1}{2}}}{(2\pi \Theta )^{\frac {1}{2}}}}={\frac {\Theta }{2}},}$

(135)

and for the average of the whole kinetic energy, ${\displaystyle {\tfrac {1}{2}}n\Theta }$, as before.

The fractional part of the ensemble which lies within any given limits of configuration, is found by integrating (184) with respect to the ${\displaystyle u}$'s from ${\displaystyle -\infty }$ to ${\displaystyle +\infty }$. This gives

${\displaystyle (2\pi \Theta )^{\frac {n}{2}}\int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}{\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}\,dq_{1}\ldots dq_{n}}$

(136)

which shows that the value of the Jacobian is independent of the manner in which ${\displaystyle 2\epsilon _{p}}$ is divided into a sum of squares. We may verify this directly, and at the same time obtain a more convenient expression for the Jacobian, as follows.

It will be observed that since the ${\displaystyle u}$'s are linear functions of the ${\displaystyle p}$'s, and the ${\displaystyle p}$'s linear functions of the ${\displaystyle {\dot {q}}}$'s, the ${\displaystyle u}$'s will be linear functions of the ${\displaystyle {\dot {q}}}$'s, so that a differential coefficient of the form ${\displaystyle du/d{\dot {q}}}$ will be independent of the ${\displaystyle {\dot {q}}}$'s, and function of the ${\displaystyle q}$'s alone. Let us write ${\displaystyle dp_{x}/du_{y}}$ for the general element of the Jacobian determinant. We have ​

${\displaystyle {\begin{aligned}{\frac {dp_{x}}{du_{y}}}&={\frac {d}{du_{y}}}{\frac {d\epsilon }{d{\dot {q}}_{x}}}={\frac {d}{du_{y}}}\sum _{r=1}^{r=n}{\frac {d\epsilon }{du_{r}}}{\frac {du_{r}}{d{\dot {q}}_{x}}}\\&=\sum _{r=1}^{r=n}{\bigg (}{\frac {d^{2}\epsilon }{du_{y}\,du_{r}}}{\frac {du_{r}}{d{\dot {q}}_{x}}}{\bigg )}={\frac {d}{d{\dot {q}}_{x}}}{\frac {d\epsilon }{du_{y}}}={\frac {du_{y}}{d{\dot {q}}_{x}}}\end{aligned}}}$

(137)

Therefore

${\displaystyle {\frac {d(p,\ldots p_{n})}{d(u,\ldots u_{n})}}={\frac {d(u,\ldots u_{n})}{d({\dot {q}},\ldots {\dot {q}}_{n})}}}$

(138)

and

${\displaystyle {\bigg (}{\frac {d(p,\ldots p_{n})}{d(u,\ldots u_{n})}}{\bigg )}^{2}={\bigg (}{\frac {d(u,\ldots u_{n})}{d({\dot {q}},\ldots {\dot {q}}_{n})}}{\bigg )}^{2}={\frac {d(p,\ldots p_{n})}{d({\dot {q}},\ldots {\dot {q}}_{n})}}}$

(139)

These determinants are all functions of the ${\displaystyle q}$'s alone.^[2] The last is evidently the Hessian or determinant formed of the second differential coefficients of the kinetic energy with respect to ${\displaystyle {\dot {q}}_{1},\ldots {\dot {q}}_{n}}$. We shall denote it by ${\displaystyle \Delta _{\dot {q}}}$. The reciprocal determinant

${\displaystyle {\frac {d({\dot {q}}_{1}\ldots {\dot {q}}_{n})}{d(p_{1}\ldots p_{n})}},}$

which is the Hessian of the kinetic energy regarded as function of the ${\displaystyle p}$'s, we shall denote by ${\displaystyle \Delta _{p}}$.

If we set

${\displaystyle {\begin{aligned}e^{-{\frac {\psi _{p}}{\Theta }}}&=\int \limits _{-\infty }^{+\infty }\ldots \int \limits _{-\infty }^{+\infty }e^{-{\frac {\epsilon _{p}}{\Theta }}}\,\Delta _{p}{}^{\frac {1}{2}}\,dp,\ldots dp_{n}\\&=\int \limits _{-\infty }^{+\infty }\ldots \int \limits _{-\infty }^{+\infty }e^{\frac {-u_{1}{}^{2}\ldots -u_{n}{}^{2}}{2\Theta }}du_{1}\ldots du_{n}=(2\pi \Theta )^{\frac {n}{2}},\end{aligned}}}$

(140)

and

${\displaystyle \psi _{q}=\psi -\psi _{p},}$

(141)

​the fractional part of the ensemble which lies within any given limits of configuration (136) may be written

${\displaystyle \int \ldots \int e^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\,\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},}$

(142)

where the constant ${\displaystyle \psi _{q}}$ may be determined by the condition that the integral extended over all configurations has the value unity.^[3]

​When an ensemble of systems is distributed in configuration in the manner indicated in this formula, i. e., when its distribution in configuration is the same as that of an ensemble canonically distributed in phase, we shall say, without any reference to its velocities, that it is canonically distributed in configuration.

For any given configuration, the fractional part of the systems which lie within any given limits of velocity is represented by the quotient of the multiple integral

${\displaystyle \int \ldots \int e^{-{\frac {\epsilon _{p}}{\Theta }}}\,dp_{1}\ldots dp_{n},}$

or its equivalent

${\displaystyle \int \ldots \int e^{\frac {-u_{1}{}^{2}\ldots -u_{n}{}^{2}}{2\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,du_{1}\ldots du_{n},}$

taken within those limits divided by the value of the same integral for the limits ${\displaystyle \pm \infty }$. But the value, of the second multiple integral for the limits ${\displaystyle \pm \infty }$ is evidently

${\displaystyle \Delta _{\dot {q}}^{\frac {1}{2}}(2\pi \Theta )^{\frac {n}{2}}.}$

We may therefore write

${\displaystyle \int \ldots \int e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\,du_{1}\ldots du_{n},}$

(143)

​or

${\displaystyle \int \ldots \int e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},}$

(144)

or again

${\displaystyle \int \ldots \int e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n},}$

(145)

for the fractional part of the systems of any given configuration which lie within given limits of velocity.

When systems are distributed in velocity according to these formulae, i. e., when the distribution in velocity is like that in an ensemble which is canonically distributed in phase, we shall say that they are canonically distributed in velocity.

The fractional part of the whole ensemble which falls within any given limits of phase, which we have before expressed in the form

${\displaystyle \int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},}$

(146)

may also be expressed in the form

${\displaystyle \int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}\Delta _{\dot {q}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}\,dq_{1}\ldots dq_{n}.}$

(147)