Why are cumulants extensive and moments are not?

[annaphys]

And why does it matter if they are extensive or not?

 

[A. Neumaier]

They are extensive because 1-particle operators are additive when applied to disjoint systems, and the log of the expectation of an exponential (which defines the cumulants from the partition function) is additive, too.

Size-extensivity is a necessary condition for the applicability of quantum estimates to large quantum systems, since without it not even a noninteracting large system is described correctly.

 

[annaphys]

That makes sense thank you. Do you have any resources on this that explains it further?

 

[A. Neumaier]

A good discussion is in

• Nooijen, Shamasundar & Mukherjee, Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Molecular Physics, 103 (2005), 2277-2298.

For the general background you might also benefit from reading Part II (Statistical mechanics) of my online book

• A. Neumaier and D.Westra, Classical and Quantum Mechanics via Lie algebras, 2011.