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Next: General Interaction Between Macrosystems Up: Statistical Thermodynamics Previous: Temperature

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,

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

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

where

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

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

which yields

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

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

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

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

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

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


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Next: General Interaction Between Macrosystems Up: Statistical Thermodynamics Previous: Temperature
Richard Fitzpatrick 2016-01-25