Why are cumulants extensive and moments are not?

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SUMMARY

Cumulants are extensive because 1-particle operators are additive when applied to disjoint systems, and the logarithm of the expectation of an exponential, which defines cumulants from the partition function, is also additive. Size-extensivity is essential for applying quantum estimates to large quantum systems; without it, even noninteracting large systems cannot be accurately described. Key resources for further understanding include the paper by Nooijen, Shamasundar & Mukherjee and the online book by A. Neumaier and D. Westra.

PREREQUISITES
  • Understanding of cumulants and moments in statistical mechanics
  • Familiarity with quantum mechanics and many-body theory
  • Knowledge of partition functions and their significance
  • Basic grasp of additive properties in mathematical physics
NEXT STEPS
  • Read "Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory" by Nooijen, Shamasundar & Mukherjee
  • Study Part II (Statistical mechanics) of "Classical and Quantum Mechanics via Lie algebras" by A. Neumaier and D. Westra
  • Explore the mathematical foundations of cumulants and their applications in quantum systems
  • Investigate the implications of size-extensivity in thermodynamic limits
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Researchers in theoretical physics, particularly those focused on statistical mechanics and quantum systems, as well as students seeking to deepen their understanding of cumulants and their significance in many-body theory.

annaphys
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And why does it matter if they are extensive or not?
 
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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.
 
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That makes sense thank you. Do you have any resources on this that explains it further?
 
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
 
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