# Elementary Principles in Statistical Mechanics/Chapter II

CHAPTER II.
APPLICATION OF THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE TO THE THEORY OF ERRORS.
Let us now proceed to combine the principle which has been demonstrated in the preceding chapter and which in its different applications and regarded from different points of view has been variously designated as the conservation of density-in-phase, or of extension-in-phase, or of probability of phase, with those approximate relations which are generally used in the 'theory of errors.'
We suppose that the differential equations of the motion of a system are exactly known, but that the constants of the integral equations are only approximately determined. It is evident that the probability that the momenta and coördinates at the time $t'$ fall between the limits $p_{1}'$ and $p_{1}'+dp_{1}'$, $q_{1}'$ and $q_{1}'+dq_{1}'$, etc., may be expressed by the formula
| $e^{\eta '}\,dp_{1}'\ldots dq_{1}',$ | (48) |
$\eta '$
Let $Q_{1}'$, $P_{1}'$, etc. be the values of the coördinates and momenta which give the maximum value to $\eta '$, and let the general value of $\eta '$ be developed by Taylor's theorem according to ascending powers and products of the differences $p_{1}'-P_{1}'$, $q_{1}'-Q_{1}'$, etc. and let us suppose that we have a sufficient approximation without going beyond terms of the second degree in these differences. We may therefore set
| $\eta '=c-F',$ | (49) |
$c$
$p_{1}'-P_{1}'$
$q_{1}'-Q_{1}'$
$F'$
$F'$
| $C=e^{c},$ | (50) |
| $Ce^{-F'}\,dp_{1}'\ldots dq_{1}'.$ | (51) |
$C$
In regard to the degree of approximation represented by these formulæ, it is to be observed that we suppose, as is usual in the 'theory of errors' that the determination (explicit or implicit) of the constants of motion is of such precision that the coefficient of probability $e^{\eta '}$ or $Ce^{-F'}$ is practically zero except for very small values of the differences $p_{1}'-P_{1}'$, $q_{1}'-Q_{1}'$, etc. For very small values of these differences the approximation is evidently in general sufficient, for larger values of these differences the value of $Ce^{-F'}$ will be sensibly zero, as it should be, and in this sense the formula will represent the facts.
We shall suppose that the forces to which the system is subject are functions of the coördinates either alone or with the time. The principle of conservation of probability of phase will therefore apply, which requires that at any other time ($t''$) the maximum value of the coefficient of probability shall be the same as at the time $t'$ and that the phase $(P_{1}'',Q_{1}'',\mathrm {etc.} )$ which has this greatest probability-coefficient, shall be that which corresponds to the phase $(P_{1}',Q_{1}',\mathrm {etc.} )$, i. e., which is calculated from the same values of the constants of the integral equations of motion.
We may therefore write for the probability that the phase at the time $t''$ falls within the limits $p_{1}''$ and $p_{1}''+dp_{1}''$, $q_{1}''$ and $q_{1}''+dq_{1}''$, etc.,
| $Ce^{-F''}\,dp_{1}''\ldots dq_{1}'',$ | (52) |
$C$
$F''$
$p_{1}''-P_{1}''$
$q_{1}''-Q_{1}''$
$(P_{1}'',Q_{1}'',\mathrm {etc.} )$
$t''$
$(P_{1}',Q_{1}',\mathrm {etc.} )$
$t'$
Now we have necessarily
| $\int \ldots \int Ce^{-F'}\,dp_{1}'\ldots dq_{1}'=\int \ldots \int Ce^{-F''}\,dp_{1}''\ldots dq_{1}''=1,$ | (53) |
$\pm \infty$
$\pm \infty$
| ${\frac {C\pi ^{n}}{\sqrt {f'}}}={\frac {C\pi ^{n}}{\sqrt {f''}}}=1,$ | (54) |
$f'$
$F'$
$f''$
$F''$
$C$
$C^{2}$
Let us see precisely how the functions $F'$ and $F''$ are related. The principle of the conservation of the probability-coefficient requires that any values of the coördinates and momenta at the time $t'$ shall give the function $F'$ the same value as the corresponding coördinates and momenta at the time $t''$ give to $F''$. Therefore $F''$ may be derived from $F'$ by substituting for $p_{1}',\ldots q_{1}'$ their values in terms of $p_{1}'',\ldots q_{1}''$. Now we have approximately ​
| $\left.{\begin{array}{c}p_{1}'-P_{1}'={\frac {dP_{1}'}{dP_{1}''}}(p_{1}''-P_{1}'')\ldots +{\frac {dP_{1}'}{dQ_{n}''}}(q_{n}''-Q_{n}'')\\\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\q_{n}'-Q_{n}'={\frac {dQ_{n}'}{dP_{1}''}}(p_{1}''-P_{1}'')\ldots +{\frac {dQ_{n}'}{dQ_{n}''}}(q_{n}''-Q_{n}'')\\\end{array}}\right\$ | (55) |
$F''$
$F''$
$F'$
At the time $t'$ the phases satisfying the equation
| $F'=k,$ | (56) |
$k$
$Ce^{-k}$
$t''$
| $F''=k,$ | (57) |
$Ce^{-k}$
$t'$
$F'=k$
$F''=k$
$t''$
$F=k$
$U$
$R$
$F=1$
$U_{1}$
| $U=\mathop {\int \ldots \int } ^{F=k}dp_{1}\ldots dq_{n},$ | (58) |
| $R=\mathop {\int \ldots \int } ^{F=k}Ce^{-F}\,dp_{1}\ldots dq_{n},$ | (59) |
| $U_{1}=\mathop {\int \ldots \int } ^{F=1}dp_{1}\ldots dq_{n}.$ | (60) |
$F$
* $p_{1}-P_{1},p_{2}-P_{2},\ldots q_{n}-Q_{n},$
* ${\begin{aligned}&\mathop {\int \ldots \int } ^{F=k}d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n})\\&=\mathop {\int \ldots \int } ^{kF=k}k^{n}\,d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n})\\&=k^{n}\mathop {\int \ldots \int } ^{F=1}d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n}).\end{aligned}}$
| $U=k^{n}U_{1},$ | (61) |
| $dU=U_{1}nk^{n-1}\,dk.$ | (62) |
$k$
| $dU=\mathop {\int \ldots \int } _{F=k}^{F=k+dk}dp_{1}\ldots dq_{n}$ | (63) |
| $dR=\mathop {\int \ldots \int } _{F=k}^{F=k+dk}Ce^{-F}\,dp_{1}\ldots dq_{n}$ | (64) |
Since the factor $Ce^{-F}$ has the constant value $Ce^{-k}$ in the last multiple integral, we have
| $dR=Ce^{-k}dU=CU_{1}ne^{-k}k^{n-1}\,dk,$ | (65) |
| $R=-CU_{1}n!e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right)+\mathrm {const.}$ | (66) |
$R$
$k$
| $R=CU_{1}n!-CU_{1}n!e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right).$ | (67) |
$U_{1}$
$R=1$
$k=\infty$
$CU_{1}n!=1$
| $R=1-e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right),$ | (68) |
| $U={\frac {k^{n}}{Cn!}}.$ | (69) |
It is worthy of notice that the form of these equations depends only on the number of degrees of freedom of the system, being in other respects independent of its dynamical nature, except that the forces must be functions of the coördinates either alone or with the time.
If we write
* $k_{R={\tfrac {1}{2}}}$
$k$
$R={\tfrac {1}{2}}$
| $F=k_{R={\tfrac {1}{2}}}$ | (70) |
The probability that the phase falls within the limits formed by these phases is greater than the probability that it falls within any other limits enclosing an equal extension-in-phase. It is equal to the probability that the phase falls without the same limits.
These properties are analogous to those which in the theory of errors in the determination of a single quantity belong to values expressed by $A\pm a$, when $A$ is the most probable value, and $a$ the 'probable error.'