The distribution of energy states in liquids, 100 years after Einstein and Debye

In an article published in the journal PNAS by Matteo Baggioli, from our Department, and Alessio Zaccone of University of Milano, the law which governs the distribution of vibrational energy states of liquids has been derived mathematically for the first time.

In solid materials, elastic waves propagate as vibrations that we all know from our everyday life as acoustic waves. Also in liquids, like water, elastic waves can propagate as acoustic waves, which allows us to hear noise when we are swimming under the water level. There are important differences, though: in solids, both longitudinal acoustic waves (oscillating parallel to the wave propagation direction) as well as transverse acoustic waves (oscillating perpendicularly to the wave propagation direction) are allowed. In liquids, instead, only longitudinal acoustic waves are supported. One of the most important quantities in condensed matter physics is the distribution of frequencies (or vibrational energy states) of vibrational waves that can propagate through the material. This is particularly important because it is the input to calculate and understand some fundamental properties of matter, such as the specific heat (i.e., the energy that a body can store as we increase its temperature), the thermal conductivity (which tells us how effectively heat can be transferred through a layer of the material), and the light-matter interaction.

Around 1910, some of the best known physicists of the 20th century got engaged with calculating the distribution of vibrational states in condensed matter, among them were Albert Einstein and Peter Debye. This was a crucial bottleneck to be able to determine the specific heat of materials as a function of temperature. After a first attempt by Einstein in 1911, the correct and complete calculation was done by Debye (Nobel prize recipient in 1936), in an article which represents a milestone in modern physics as it was pivotal to demonstrate the atomic quantum nature of matter.

Debye mathematically derived the distribution of vibrational frequencies of solids, which goes with the square of the frequency, for acoustic waves. This result allowed Debye to predict that the specific heat of solids grows with the cube of temperature, and effect widely confirmed experimentally.

While this problem, for solids, was solved by Debye in 1912, the same problem of mathematically determining the distribution of vibrational energy states for liquids has remained elusive and unsolved for over 100 years. The big problem with liquids (already recognized by Debye, Einstein and Max Born) is due to the fact that, besides acoustic waves, there are also other types of vibrational excitations in liquids, which are related to the constant disordered motions of atoms and molecules – excitations that are instead basically absent in solids. These excitations are typically short-lived and associated with dynamical chaos of molecular motions, but they are nevertheless very numerous and important especially at the lower energies. Mathematically, these excitations (known as “instantaneous normal modes” or INMs, in the specialized literature) are very difficult to deal with mathematically, as they correspond to energy states described by imaginary numbers.

The new mathematical theory by Zaccone & Baggioli solved the problem of obtaining the distribution of these complex energy states by combining the so-called Zeldovich regularization for unstable quantum states with the mathematical theory of distributions by L. Schwartz. The final result provides an equation in closed form for the distribution of energy states in liquids, which goes linear in the frequency, as opposed to the Debye law for solids, which goes quadratic in frequency, and also correctly predicts its temperature dependence.

This result is important for various technological applications and for environmental sustainability. First of all, this result allows us to explain why the specific heat of liquids decreases upon increasing the temperature (whereas the specific heat of solids increases with increasing temperature!), and also the thermal conductivity of liquids as a function of molecular structure. These progresses, among other things, may lead to discovering new heat-transfer fluids with optimized thermal properties to solve the huge environmental problem related to the cooling of big-data servers, which currently presents a growing and concerning negative impact on our environment.