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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

$$\Omega \; = \; \Omega_1(U_1) \, \times \, \Omega_2(U_2)$$ (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

$$\sigma \; = \; \sigma_1(U_1) \; + \; \sigma_2(U_2)$$ (15.5)

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