Mechanical Interaction Between Macrosystems

Let us now examine a purely mechanical interaction between macrostates, where one or more of the external parameters is modified, but there is no exchange of heat energy. Consider, for the sake of simplicity, a situation where only one external parameter, $x$ , of the system is free to vary. In general, the number of microstates accessible to the system when the overall energy lies between $E$ and $E+\delta E$ depends on the particular value of $x$ , so we can write ${\mit\Omega} \equiv {\mit\Omega}(E, x)$ .

When $x$ is changed by the amount $dx$ , the energy $E_r(x)$ of a given microstate $r$ changes by $(\partial E_r/\partial x) dx$ . The number of states, $\sigma(E,x)$ , whose energy is changed from a value less than $E$ to a value greater than $E$ , when the parameter changes from $x$ to $x+dx$ , is given by the number of microstates per unit energy range multiplied by the average shift in energy of the microstates. Hence,

$$\sigma (E,x) = \frac{{\mit\Omega}(E,x)}{\delta E} \overline{\frac{\partial E_r}{\partial x}} dx,$$ (5.34)

where the mean value of $\partial E_r/\partial x$ is taken over all accessible microstates (i.e., all states for which the energy lies between $E$ and $E+\delta E$ , and the external parameter takes the value $x$ ). The previous equation can also be written

$$\sigma (E,x) = - \frac{{\mit\Omega}(E,x)}{\delta E} \overline{X} dx,$$ (5.35)

where

$$\overline{X}(E,x) = -\overline{\frac{\partial E_r}{\partial x}}$$ (5.36)

is the mean generalized force conjugate to the external parameter $x$ . (See Section 4.4.)

Consider the total number of microstates whose energies lies between $E$ and $E+\delta E$ . When the external parameter changes from $x$ to $x+dx$ , the number of states in this energy range changes by $(\partial {\mit\Omega}/\partial x) dx$ . This change is due to the difference between the number of states that enter the range because their energy is changed from a value less than $E$ to one greater than $E$ , and the number that leave because their energy is changed from a value less than $E+\delta E$ to one greater than $E+\delta E$ . In symbols,

$$\frac{\partial {\mit\Omega}(E,x)}{\partial x} dx = \sigma(E)- \sigma(E+\delta E)\simeq -\frac{\partial \sigma}{\partial E} \delta E,$$ (5.37)

which yields

$$\frac{\partial {\mit\Omega}}{\partial x} = \frac{\partial({\mit\Omega} \overline{X})}{\partial E},$$ (5.38)

where use has been made of Equation (5.35). Dividing both sides by ${\mit\Omega}$ gives

$$\frac{\partial \ln {\mit\Omega}}{\partial x} = \frac{\partial \ln... ...t\Omega}}{\partial E} \overline{X} +\frac{\partial \overline{X}}{\partial E}.$$ (5.39)

However, according to the usual estimate ${\mit\Omega}\propto E^{ f}$ (see Section 3.8), the first term on the right-hand side is of order $(f/\overline{E}) \overline{X}$ , whereas the second term is only of order $\overline{X}/\overline{E}$ . Clearly, for a macroscopic system with many degrees of freedom, the second term is utterly negligible, so we have

$$\frac{\partial\ln {\mit\Omega}}{\partial x} = \frac{\partial \ln {\mit\Omega}}{\partial E} \overline{X} = \beta \overline{X},$$ (5.40)

where use has been made of Equation (5.30).

When there are several external parameters, $x_1, \cdots, x_n$ , so that ${\mit\Omega}\equiv {\mit\Omega}(E, x_1,\cdots,$ $x_n)$ , the previous derivation is valid for each parameter taken in isolation. Thus,

$$\frac{\partial \ln{\mit\Omega}}{\partial x_\alpha} = \beta \overline{X}_\alpha,$$ (5.41)

where $\overline{X}_\alpha$ is the mean generalized force conjugate to the parameter $x_\alpha$ . (See Section B.2.)