Quantifying magma mixing with the Shannon entropy: application to simulations and experiments

We introduce a new quantity to petrology, the Shannon entropy, as a tool for quantifying mixing as well as the rate of production of hybrid compositions in the mixing system. The Shannon entropy approach is applied to time series numerical simulations and high-temperature experiments performed with natural melts. We note that in both cases the Shannon entropy increases linearly during the initial stages of mixing and then saturates toward constant values. Furthermore, chemical elements with different mobilities display different rates of increase of the Shannon entropy. This indicates that the hybrid composition for the different elements is attained at different times generating a wide range of spatio-compositional domains which further increase the apparent complexity of the mixing process. Results from the application of the Shannon entropy analysis are compared with the concept of Relaxation of Concentration Variance (RCV), a measure recently introduced in petrology to quantify chemical exchanges during magma mixing. We derive a linear expression relating the change of concentration variance during mixing and the Shannon entropy. We show that the combined use of Shannon entropy and RCV provides the most complete information about the space and time complexity of magma mixing. As a consequence, detailed information about this fundamental petrogenetic and volcanic process can be gathered. In particular, the Shannon entropy can be used as complement to the RCV method to quantify the mobility of chemical elements in magma mixing systems, to obtain information about the rate of production of compositional heterogeneities, and to derive empirical relationships linking the rate of chemical exchanges between interacting magmas and mixing time.

💡 Research Summary

The paper introduces Shannon entropy as a novel quantitative metric for assessing magma mixing and the rate at which hybrid compositions are produced. By treating the compositional space of a mixing system as a probability distribution, the authors compute entropy (S = ‑∑ p_i log p_i) where p_i is the fraction of data points falling within a given compositional bin. Two complementary experimental approaches are employed: (1) high‑temperature laboratory mixing of natural melts (e.g., basaltic and rhyolitic compositions) at ~1200 °C, with systematic sampling and EPMA analysis of major and trace elements; and (2) three‑dimensional numerical simulations of turbulent mixing using a Lévy‑type flow model, where concentration fields are tracked over time. In both the laboratory and numerical datasets, entropy exhibits a characteristic temporal evolution: an initial linear increase as the system moves from a highly ordered state (few compositional domains) toward a more disordered, evenly populated compositional space, followed by a saturation plateau once statistical equilibrium is approached.

A key observation is that elements with high diffusivity (e.g., Na, K) reach the entropy plateau more rapidly than low‑diffusivity species (e.g., Ti, Zr). Consequently, the “hybrid composition” for each element is attained at different times, generating a spectrum of spatio‑compositional domains and increasing the apparent complexity of the mixing process.

The study also revisits the recently proposed Relaxation of Concentration Variance (RCV), which quantifies the decay of concentration variance (σ²) during mixing. By analytically linking the change in variance to entropy, the authors derive a simple linear relationship: Δσ² = ‑α S + β, where α and β are constants that depend on the specific element and initial variance. This equation demonstrates that as entropy grows (i.e., the system becomes more mixed), the variance of concentrations diminishes in a predictable manner.

Importantly, the authors argue that Shannon entropy and RCV are complementary. Entropy captures the overall disorder of the compositional field, while RCV provides a focused measure of how quickly individual element concentrations converge toward a hybrid value. When used together, they deliver a comprehensive picture of both the spatial complexity (through entropy) and the temporal kinetics (through RCV) of magma mixing. This combined framework enables researchers to:

1. Quantify the mobility of different chemical species in a mixing system.
2. Estimate the rate at which compositional heterogeneities are generated and subsequently erased.
3. Derive empirical relationships linking mixing time, chemical exchange rates, and the evolution of compositional variance.

The paper concludes that incorporating Shannon entropy into petrological analyses enriches our understanding of fundamental petrogenetic processes, improves the interpretation of volcanic eruption precursors, and offers a robust quantitative tool for integrating chemical mixing dynamics into volcanic hazard models. Future work is suggested to extend the methodology to more complex flow regimes, multi‑component melt systems, and real‑time field measurements of eruptive products.