Why are cumulants extensive and moments are not?

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Discussion Overview

The discussion centers on the distinction between cumulants and moments in the context of their extensive properties, particularly in relation to quantum systems. Participants explore the implications of these properties for theoretical frameworks and applications in many-body theory.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • One participant states that 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.
  • Another participant emphasizes that size-extensivity is necessary for the applicability of quantum estimates to large quantum systems, suggesting that without it, even noninteracting large systems cannot be described correctly.
  • A request for further resources on the topic is made, indicating a desire for deeper understanding.
  • One participant provides a reference to a paper discussing size-extensivity and related concepts, as well as a recommendation for an online book that covers statistical mechanics.

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus, as participants present different aspects of the topic without resolving the broader implications or potential disagreements regarding the definitions and applications of cumulants and moments.

Contextual Notes

The discussion may be limited by assumptions about the definitions of extensive properties and the specific contexts in which cumulants and moments are applied. There is also a lack of detailed exploration of the mathematical steps involved in establishing these properties.

Who May Find This Useful

Researchers and students interested in statistical mechanics, quantum theory, and many-body systems may find this discussion relevant for understanding the properties of cumulants and moments.

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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