# Scientists Provide Vital Clues on How the ‘Heating by Cooling’ Effect Functions

Artist impression of aggregating granular gases in space. Professor Nikolai Brilliantov

A Leicester mathematician has developed a theory to explain ‘heating by cooling,’ where the temperature of a granular gas increases while the total energy drops down – a peculiar phenomenon which can be observed both on Earth and in space.

Granular gases are one of the few examples where this scientific mystery can be observed. These systems are widely spread in nature in the form of aerosols and smoke on the Earth, or in the form of interstellar dust, planetary rings, and proto-planetary discs in space.

The stunning ‘heating by cooling’ effect corresponds, in physical terms, to a negative heat capacity. Aggregating granular gases is the second object in the world, after gravitating systems, which manifests this astonishing property.

Partial temperatures of i-mers. Evolution of the partial temperatures, T i, of i-mers for a granular gas of 107 particles for λ1 = λ2 = 4/3, a = 0.1, and ε = 0.99. Initially, the gas of monomers has the temperature, T1(0) = 1. The dashed lines show the limiting cases of a hot gas (regime of non-aggregative cooling) when the temperature follows Haff’s law, T ~ t−2, and cold gas when almost all collisions are aggregative and T ~ t−1/3. Nikolai V. Brilliantov, et al., Nature Communications, doi:10.1038/s41467-017-02803-7

“From secondary school we are taught that temperature means energy — the higher the temperature, the larger the energy. If a system loses energy, its temperature drops down,” says Professor Nikolai Brilliantov from the University of Leicester’s Department of Mathematics, who led the research. “Surprisingly, this is not always true for granular gases.”

The international group of scientists has provided a vital clue on how granular gases function and demonstrate this mysterious quality in a paper published in the journal, Nature Communications, where they have built a solid mathematical foundation of the phenomenon.

They have elaborated a novel mathematical tool – generalized Smoluchowski equations. While classical Smoluchowski equations, which have been known for more than a century, deal with the evolution of agglomerates concentration only, the new equations describe the evolution of the agglomerates temperature as well.

Evolution of temperature for different aggregation mechanisms. The rate of temperature growth in the regime of increasing temperature depends on the aggregation mechanism, quantified by the parameter Λ. This parameter characterizes the dependence of the aggregation barrier on the agglomerate size. a N = 107ε = 0.99, a = 0.1, Λ = 0.4, thus β = −0.173, see Eq. (21); b same but Λ = 1.6, thus β = −0.941. With increasing value of Λ, the increase of temperature becomes steeper, in agreement with the theoretical predictions, T~tβT~tβ. Nikolai V. Brilliantov, et al., Nature Communications, doi:10.1038/s41467-017-02803-7

The direct microscopic modeling of the system, by extensive computer simulations, has confirmed the existence of this surprising regime and other predictions of the theory.

It has also been shown that, in spite of its peculiarity, ‘heating by cooling’ may be observed for many systems at natural conditions. However, the inter-particle forces have to comply with an important prerequisite – the attraction strength should increase with the agglomerates size.

“Understanding different regimes of the evolution of aggregating granular gases is important to comprehend numerous natural phenomena where these systems are involved,” adds Professor Brilliantov.

Reference: “Increasing temperature of cooling granular gases” by Nikolai V. Brilliantov, Arno Formella and Thorsten Pöschel, 23 February 2018,  Nature Communications.
DOI: 10.1038/s41467-017-02803-7