Content deleted Content added
No edit summary |
No edit summary |
||
Line 45:
* Qualitatively, Z grows when the temperature rises, because then the exponential weights increase for states of larger energy. Roughly, Z is a measure of how many different energy states are populated appreciably in thermal equilibrium (at least when we suppose the ground state energy to be zero).
* Given Z as a function of temperature, we may calculate the average [[energy]] as
Line 56 ⟶ 57:
* From these two relations, the entropy S may be obtained as
:<math>S=k_B \sum_j P(j) \ln P(j)=(E-F)/T=k_B T^2 {d \over dT} {\ln Z \over T}</math>
* Alternatively, with <math>\beta\equiv 1/(k_B T)</math>, we have <math>E=-{d \over d \beta} \ln Z</math> and <math>F=-\beta^{-1} \ln Z</math>, as well as <math>S=-k_B \beta^2 {d \over d \beta} (\beta^{-1} \ln Z)</math>.
| |||