Skip to main content

Moment Generating and Free Energy Functionals

  • Chapter
  • First Online:
Information Geometry and Population Genetics

Part of the book series: Understanding Complex Systems ((UCS))

  • 1903 Accesses

Abstract

In this section, we will construct the moment generating function for the Wright–Fisher model and derive a partial differential equation that it satisfies. This differential equation encodes all the moment evolution equation s from the Sect. 4.3

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)

    Google Scholar 

  2. Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348. Springer, Cham (2014)

    Google Scholar 

  3. Calogero, S.: Exponential convergence to equilibrium for kinetic Fokker-Planck equations. Commun. Partial Differ. Equ. 37 (8), 1357–1390 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cattiaux, P., Guillin, A.: Trends to equilibrium in total variation distance. Ann. Inst. Henri Poincaré Probab. Stat. 45 (1), 117–145 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ethier, S.N., Kurtz, T.G.: Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)

    Google Scholar 

  6. Ewens, W.J.: Mathematical Population Genetics. I, 2nd edn. Interdisciplinary Applied Mathematics, vol. 27. Springer, New York (2004)

    Google Scholar 

  7. Fukushima, M., Stroock, D.: Reversibility of solutions to martingale problems. Adv. Math. Suppl. Stud. 9, 107–123 (1986)

    MathSciNet  MATH  Google Scholar 

  8. Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97 (4), 1061–1083 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems. London Mathematical Society Student Texts, vol. 7. Cambridge University Press, Cambridge (1988). Mathematical aspects of selection, Translated from the German

    Google Scholar 

  10. Houchmandzadeh, B., Vallade, M.: Alternative to the diffusion equation in population genetics. Phys. Rev. E 82, 051913 (2010)

    Article  Google Scholar 

  11. Iwasa, Y.: Free fitness that always increases in evolution. J. Theor. Biol. 135 (3), 265–281 (1988)

    Article  MathSciNet  Google Scholar 

  12. Jordan, R., Kinderlehrer, D., Otto, F.: Free energy and the Fokker-Planck equation. Physica D Nonlin. Phenom. 107 (2–4), 265–271 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jost, J.: Riemannian Geometry and Geometric Analysis, 6th edn. Universitext. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  14. Jost, J.: Mathematical Methods in Biology and Neurobiology. Universitext. Springer, London (2014)

    Book  MATH  Google Scholar 

  15. Kolmogoroff, A.: Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Ann. 113 (1), 766–772 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lessard, S., Lahaie, P.: Fixation probability with multiple alleles and projected average allelic effect on selection. Theor. Popul. Biol. 75 (4), 266–277 (2009)

    Article  MATH  Google Scholar 

  17. Littler, R.A.: Loss of variability at one locus in a finite population. Math. Biosci. 25 (1–2), 151–163 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nagylaki, T.: The decay of genetic variability in geographically structured populations. Proc. Natl. Acad. Sci. USA 71, 2932–2936 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Song, Y.S., Steinrücken, M.: A simple method for finding explicit analytic transition densities of diffusion processes with general diploid selection. Genetics 190 (3), 1117–1129 (2012)

    Article  Google Scholar 

  20. Tran, T.D., Hofrichter, J., Jost, J.: A general solution of the Wright–Fisher model of random genetic drift. In: Differential Equations and Dynamical Systems, pp. 1–26. Springer, Berlin (2016)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Hofrichter, J., Jost, J., Tran, T.D. (2017). Moment Generating and Free Energy Functionals. In: Information Geometry and Population Genetics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-52045-2_6

Download citation

Publish with us

Policies and ethics