Transfer entropy characterization of causal relations in heat transport near equilibrium

The causal relationships between the physical fields that intervene in the description of heat transport in rigid heat-conducting solids are analyzed in the light of the transfer of information. The analysis runs through three heat transport models, namely, the Fourier (F), the Maxwell-Cattaneo-Vernotte (MCV), and the Jeffreys-Guyer-Krumhansl (J-GK) models. It is argued that the MCV model can be obtained (to first order) by introducing a delay time into the F model, and, similarly, the GK model can be obtained from the MCV model. This implicit causality allows to identify two sets of causal physical fields. The causal relationship of each of the pairs is then characterized by the transfer of information measured with the so-called transfer entropy. Validation experiments are used to interpret the results, which leads to our main conclusion: information flows from spatial inhomogeneities to heat flow and temporal temperature variation, thus clarifying causal relationships in the system. Such causality vanishes for long timescales.