in energy eigenstate is given by: Since the system can be in any energy eigenstate within an interval of , we define the generalized force for the system as the expectation value of the above expression: To evaluate the average, we partition the energy eigenstates by counting how many of them have a value for within a range between and . Calling this number , we have: The average defining the generalized force can now be written: We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between and . Let's focus again on the energy eigenstates for which lies within the range between and . Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E − Y dx to E move from below E to above E. There are such energy eigenstates. If , all these energy eigenstates will move into the range between and and contribute to an increase in . The number of energy eigenstates that move from below to above is, of course, given by . The difference is thus the net contribution to the increase in . Note that if Y dx is larger than there will be energy eigenstates that move from below to above . They are counted in both and , therefore the above expression is also valid in that case. Expressing the above expression as a derivative with respect to E and summing over Y yields the expression: The logarithmic derivative of with respect to x is thus given by: The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and thus vanishes in the thermodynamic limit. We have thus found that: Combining this with Gives: which we can write as: Derivation of statistical mechanical principles from the fundamental thermodynamic relation It has been shown that the fundamental thermodynamic relation together with the following three postulates is sufficient to build the theory of statistical mechanics without the equal a priori probability postulate. For example, in order to derive the Boltzmann distribution, we assume the probability density