Abstract
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.
Highlights
Transfer entropy has been introduced as an information-theoretic measure that quantifies the statistical coherence between systems evolving in time [1]
For instance, the source Y is such that the system X is independent of it, there is no difference in the extents of disturbances to the equilibrium, and the transfer entropy is zero
In this paper we proposed a thermodynamic interpretation of transfer entropy: an information-theoretic measure introduced by Schreiber [1] as the average information contained in the source about the state of the destination in the context of what was already contained in the destination’s past
Summary
Transfer entropy has been introduced as an information-theoretic measure that quantifies the statistical coherence between systems evolving in time [1]. This task is not trivial, and needs to be approached carefully Another contribution of this paper is a clarification that similar thermodynamic treatment is not applicable to information flow—a measure introduced by Ay and Polani [18] in order to capture causal effect. This allows us to define components of transfer entropy with the entropy rate of (i) the resultant transition and (ii) the internal entropy production.
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