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Temperature

``The general connection between energy and temperature may only be established by probability considerations. [Two systems] are in statistical equilibrium when a transfer of energy does not increase the probability.''
-- M. Planck

When we put two systems   ${\cal S}_1$  and   ${\cal S}_2$  (with  N1  and  N2  particles, respectively) into ``thermal contact'' so that the (constant) total energy   U = U1 + U2  can redistribute itself randomly between   ${\cal S}_1$  and   ${\cal S}_2$,  the combined system   ${\cal S} = {\cal S}_1 + {\cal S}_2$  will, we postulate, obey the FUNDAMENTAL PRINCIPLE - it is equally likely to be found in any one of its accessible states. The number of accessible states of  ${\cal S}$  (partially constrained by the requirement that  N1,  N2  and   U = U1 + U2  remain constant) is given by

 \begin{displaymath}\Omega \; = \; \Omega_1(U_1) \, \times \, \Omega_2(U_2)
\end{displaymath} (15.4)

where  $\Omega_1$  and  $\Omega_2$  are the MULTIPLICITY FUNCTIONS for   ${\cal S}_1$  and   ${\cal S}_2$  taken separately [both depend upon their internal energies  U1  and  U2] and the overall multiplicity function is the product of the two individual multiplicity functions because the rearrangements within one system are statistically independent of the rearrangements within the other.15.11 Since the ENTROPY is the log of the MULTIPLICITY and the log of a product is the sum of the logs, Eq. (4) can also be written

 \begin{displaymath}\sigma \; = \; \sigma_1(U_1) \; + \; \sigma_2(U_2)
\end{displaymath} (15.5)

-- i.e. the entropy of the combined system is the sum of the entropies of its two subsystems.



 
next up previous
Next: The Most Probable Up: Thermal Physics Previous: Ensembles
Jess H. Brewer
1998-11-22