Why might you expect the canonical partition function to scale exponentially with the system size, thus always ensuring that the Helmholtz free energy is extensive?

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allstylekvsvi · Accepted Answer

The canonical partition function is scaled exponentially because the canonical ensemble is classical and discrete.  So n classical mechanics the position and momentum variable of the particle vary continuously. So the set of microstates become uncountable. So it becomes very difficult to express all the discrete terms in the form of summation. Therefore, it scales exponentially with the size of the system.Also,It is a probability distribution function. The general assumptions about large systems with definite temperature are defined as,The first assumption is-The probability of state is in some very small regions of state space is proportional to the product of the size (volume) of that region and probability density in the region.The second assumption is-The Sum of all probabilities is 1.The third assumption is-The energy of the system is not constant but is exchanged between the system and the reservoir. That means energy is conserved.The fourth assumption is-Subsystems can be assumed to be independent in the sense that probability is the product of probabilities.The only class of function that obeys the above four assumptions is the only the exponential function.And it makes instinctive sense that micro-states having higher energy takes place with a lower probability and the exponential function has reasonable properties for that.

Why might you expect the canonical partition function to scale

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