of pebbles with partitions or arranging pebbles and partitions The last picture is the most telling. The number of arrangements of objects is . So the number of possible arrangements of pebbles and partitions is . However, if partition #3 and partition #5 trade places, no one would notice. The same argument goes for quanta. To obtain the number of possible distinguishable arrangements one has to divide the total number of arrangements by the number of indistinguishable arrangements. There are identical quanta arrangements, and identical partition arrangements. Therefore, multiplicity of the system is given by which, as mentioned before, is the number of ways to deposit quanta of energy into oscillators. Entropy of the system has the form is a huge number—subtracting one from it has no overall effect whatsoever: With the help of Stirling's approximation, entropy can be simplified: Total energy of the solid is given by since there are q energy quanta in total in the system in addition to the ground state energy of each oscillator. Some authors, such as Schroeder, omit this ground state energy in their definition of the total energy of an Einstein solid. We are now ready to compute the temperature Elimination of q between the two preceding formulas gives for U: The first term is associated with zero point energy and does not contribute to specific heat. It will therefore be lost in the next step. Differentiating with respect to temperature to find we obtain: or Although the Einstein model of the solid predicts the heat capacity accurately at high temperatures, and in this limit , which is equivalent to Dulong–Petit law. Nevertheless, the heat capacity noticeably deviates from experimental values at low temperatures. See Debye model for how to calculate accurate low-temperature heat capacities. Canonical ensemble Heat capacity is obtained through the use of the canonical partition function of a simple quantum harmonic oscillator. where substituting this into the partition function formula yields This is the partition function of one harmonic oscillator. Because, statistically, heat capacity, energy, and entropy of the solid are equally distributed among its atoms, we can work with this partition function to obtain those quantities and then simply multiply them by to get the total. Next, let's compute the average energy of each oscillator where Therefore, Heat capacity of one oscillator is then Up to now, we calculated the heat capacity of a unique degree of freedom,