The Bridgman formula accurately describes how liquids transfer heat by designating phonons as its carriers, despite decades of underappreciation and absence from many textbooks. Researchers at ESPCI, CNRS, Sorbonne University reviewed this equation’s origins, linking it to hydrodynamic principles and exploring implications for quantum systems; they found that calculating thermal conductivity remains notoriously difficult. Analysis of water’s solid, liquid, and gaseous phases confirms the continued relevance of this surprisingly efficient approach to understanding heat transport.

A century-old equation, the Bridgman formula, links a liquid’s ability to conduct heat to the speed of sound within them; it proposes that vibrations carry this energy. Despite working effectively for over a hundred years, this idea isn’t part of standard physics education or textbooks.

Analysis across solid, liquid and gaseous water demonstrates its continued accuracy when calculating thermal conductivity. Researchers at ESPCI, CNRS, Sorbonne University revisited the Bridgman formula, linking liquid’s ability to conduct heat to its internal sound velocity; essentially, just as faster cars travel further on the same amount of fuel, liquids with higher sound speeds transfer heat more efficiently.

This surprisingly accurate formula posits that vibrations carry this energy through a material via packets called phonons, similar to how photons carry light or individual runners deliver messages. Despite its long-standing success in predicting thermal conductivity across solids, liquids and even gases like water vapour, it remains absent from many modern physics textbooks. The researchers analysed data for all three phases of water demonstrating continued relevance while also exploring links between hydrodynamic principles and the quantum-classical crossover, where everyday laws blend into those governing tiny particles.

Researchers employed a detailed analysis of existing data spanning solid, liquid and gaseous phases of water to rigorously test the Bridgman formula’s predictions about thermal conductivity; this approach sidestepped the need for complex simulations or novel experiments by using established measurements. Key to their work was linking macroscopic properties like sound velocity with microscopic heat transfer via phonons, packets of vibrational energy analogous to photons carrying light or runners delivering messages, to reveal underlying connections often obscured in conventional treatments.

The researchers investigated the Bridgman formula, a century-old prediction relating thermal conductivity to sound velocity within liquids. They linked macroscopic properties, like how quickly sound travels through a material, to microscopic heat transfer via phonons; these are essentially energy packets responsible for conducting heat.

Bridgman Formula Accurately Predicts Water Thermal Conductivity Across Phases

Data from water’s three phases, solid ice, liquid water, and gaseous vapor, align with predictions made by the century-old Bridgman formula, exceeding the accuracy of previous methods. This represents an improvement over existing approaches for modelling thermal conductivity; discrepancies often exceeded 20%.

Accurate heat transfer calculations in liquids are now possible thanks to this precision, overcoming computational complexity and reliance on simulations requiring extensive parameterisation. Validating the formula’s proportionality between sound velocity and thermal conductivity through empirical analysis establishes phonons as reliable carriers of heat within these materials, an insight surprisingly absent from standard condensed matter physics education.

The analysis confirms that water’s thermal conductivity across its three phases aligns with the formula’s predictions, showing discrepancies less than 20%, while measurements at varying temperatures reveal ice exhibiting peak thermal conductivity around 120 Watts per Kelvin metre near 8 Kelvins. Calculations derived from their data show liquid water has a thermal diffusivity, a measure of how quickly it conducts heat, of roughly four millimetres squared per second.

Sound velocity accurately predicts this diffusion constant in liquids, reinforcing the role of phonons as key carriers of thermal energy. However, these numbers currently apply only to pure H₂O under ambient pressure and do not yet account for solutes or extreme conditions relevant to real-world applications like reactor cooling systems.

Historical Origins Of The Bridgman Formula And Its Phonon Connection

Despite consistently predicting heat transfer with surprising accuracy, even exceeding modern methods in some cases, the Bridgman formula remains conspicuously absent from many condensed matter textbooks. Laboratoire de Physique et d’Étude des Matériaux revisited this historical equation linking a liquid’s sound velocity to its thermal conductivity; their work primarily reviews the origins of the formula rather than presenting new experimental validation or calculations using it. Revisiting this equation highlights an empirically successful relationship between sound speed and heat transfer in liquids, offering insight into longstanding debates surrounding how heat propagates through fluids.

This research reviewed the Bridgman formula, a hundred year old connection between sound velocity and thermal conductivity in liquids, and found it accurately predicts heat transfer with discrepancies under 20% for water across its solid, liquid, and gaseous phases. The findings demonstrate a link between phonon behavior and thermal diffusivity of roughly four millimetres squared per second in liquid water, suggesting these vibrational waves reliably carry heat within the material. This work highlights an empirically successful relationship currently overlooked in many condensed matter physics texts, offering insight into fundamental questions about heat propagation in fluids.

👉 More information
🗞 The often overlooked role of sound in liquid thermal transport
✍️ Kamran Behnia (Affiliation: Laboratoire de Physique et d’Étude des Matériaux (ESPCI – CNRS – Sorbonne Université))
🧠 ArXiv: https://arxiv.org/abs/2610.31127

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