On thermodynamic interpretation of transfer entropy

Mikhail Prokopenko*, Joseph T. Lizier, Don C. Price

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

58 Citations (Scopus)


We propose a thermodynamic interpretation of transfer entropy near equilibrium, using a specialised Boltzmann's principle. The approach relates conditional probabilities to the probabilities of the corresponding state transitions. This in turn characterises transfer entropy as a difference of two entropy rates: the rate for a resultant transition and another rate for a possibly irreversible transition within the system affected by an additional source. We then show that this difference, the local transfer entropy, is proportional to the external entropy production, possibly due to irreversibility. Near equilibrium, transfer entropy is also interpreted as the difference in equilibrium stabilities with respect to two scenarios: a default case and the case with an additional source. Finally, we demonstrated that such a thermodynamic treatment is not applicable to information flow, a measure of causal effect.

Original languageEnglish
Pages (from-to)524-543
Number of pages20
Issue number2
Publication statusPublished - Feb 2013
Externally publishedYes


  • Boltzmann's principle
  • Causal effect
  • Entropy production
  • Information transfer
  • Irreversibility
  • Kullback-Leibler divergence
  • Thermodynamic equilibrium
  • Transfer entropy


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