Cumulant Expansion: How Does It Work?

  • Context: Graduate 
  • Thread starter Thread starter aaaa202
  • Start date Start date
  • Tags Tags
    Expansion
Click For Summary
SUMMARY

The discussion focuses on the mechanics of cumulant expansion, specifically addressing the logarithmic transformation of the expansion in equation (29-26). Participants express confusion regarding the transition from equation (29-26) to (29-27) and emphasize the importance of verifying the derivatives of the logarithm of the generating function, G(k), at k=0. The derivative is confirmed as , aligning with the expected outcome for the cumulants Ξr. This highlights the necessity of understanding the underlying mathematical principles for accurate application.

PREREQUISITES
  • Understanding of cumulant expansions in statistical mechanics
  • Familiarity with generating functions, specifically G(k)
  • Knowledge of derivatives and their applications in mathematical analysis
  • Proficiency in logarithmic transformations of functions
NEXT STEPS
  • Study the derivation and properties of cumulant expansions in statistical physics
  • Learn about the applications of generating functions in probability theory
  • Explore advanced calculus techniques for evaluating derivatives of logarithmic functions
  • Investigate the implications of cumulants in data analysis and statistical inference
USEFUL FOR

Researchers in statistical mechanics, mathematicians focusing on probability theory, and anyone involved in advanced data analysis requiring a deep understanding of cumulant expansions.

aaaa202
Messages
1,144
Reaction score
2
Can anyone tell me how a cumulant expansion, like the one on the picture, works? If I take the logarithm of the expansion in eq. (29-26) I don't get what is stated below. What should I do to get that?
 

Attachments

  • cumulant.png
    cumulant.png
    20.8 KB · Views: 595
Physics news on Phys.org
aaaa202 said:
If I take the logarithm of the expansion in eq. (29-26) I don't get what is stated below. What should I do to get that?
I also don't agree with the transfer from (29-26) to (29-27). Either there is something very wrong with the statement or there is something going on backstage that we are not shown.
 
You need to verify that the derivatives of ##\log(G(k))## evaluated at ##k=0## are exactly the Ξr
For example: ##\frac{d}{dk}\log(G(k)) = \frac{G^\prime(k)}{G(k)}##. Remember that ##G(0) = 1## and ##G^\prime(0) = <Y>##. Hence the derivative evaluated at ##0## is ##<Y>##, as desired. I will let you compute the other Ξr.
 

Similar threads

Graduate Function expansion in Cartesian and spherical tensors
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Series expansion of ##(1-cx)^{1/x}##
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad How do I use induction more rigorously when making Taylor expansions?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad How is a binomial expansion done?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Cumulative Distribution function in terms of Error function
  • · Replies 1 ·
Replies
1
Views
2K
Statistical Mechanics- moments/cumulants, log expansion
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad How do you get to the conclusion that there is accelerating expansion?
  • · Replies 23 ·
Replies
23
Views
3K
Graduate Series Expansion of $\frac{1}{n^{k (k + 1)/(2 n)}(2 k n - k (1 + k) \ln n)^2}$
  • · Replies 1 ·
Replies
1
Views
2K
Quantum - Postulate of getting specific momentum p = |A_p|^2
  • · Replies 7 ·
Replies
7
Views
938
High School How did expansion take 380,000 years to let light travel freely?
  • · Replies 13 ·
Replies
13
Views
3K