The Thermodynamics of the Sandpile and the Avalanche Effect

The Sandpile as a Self-Organized System

The canonical example of a complex dynamic system exhibiting self-organized criticality is the sandpile. Sand grains, pushed by the wind or added slowly, accumulate on the pile, which grows gradually.

The sandpile will inevitably rise until the slope reaches a critical value. At the first critical threshold, several small avalanches may occur—simple grain falls—but the slope continues to grow.

At the next threshold, a larger avalanche may occur, without completely interrupting the growth of the slope. More rarely, a massive avalanche occurs, abruptly readjusting the entire profile of the pile.

This behavior illustrates how a system subjected to a low but constant energy flow can self-regulate around a critical state, characterized by events of varying sizes. It is precisely to avoid this rare but inevitable phenomenon that artificers regularly trigger controlled small avalanches in the mountains to reduce the risk of a major collapse.

The Avalanche Effect and Bifurcations in Physical Systems

The avalanche effect is a phenomenon of physical transformations obeying the laws of thermodynamics. All physical structures follow the same laws because they dissipate energy.

It is observed that physical systems self-organize to maximize the flow of dissipated energy. They all tend to maintain themselves permanently near a critical point that can lead to a break, until they find another critical point.

Whether cosmological, geophysical, biological, or sociological, the system adjusts as it evolves toward criticality. This adjustment, unpredictable and chaotic, can be invisible or catastrophic. The properties of this process are those of continuous phase transitions; in nonlinear dynamics, this is called a bifurcation.

Indeed, avalanche effects produce bifurcations in physical structures (galaxies, stars, planets, water, human societies, etc.), which in turn can trigger avalanches of bifurcations. A bifurcation thus follows an amplification of fluctuations or a symmetry breaking, which can lead to other bifurcations, and so on. These cascades of bifurcations are found everywhere in the observable phenomena of our environment.

Distribution of Events

The smaller the avalanche phenomena, the more frequent they are. For example: minor earthquakes are almost permanent; those of moderate intensity are more spaced out; the strongest are even rarer, while destructive earthquakes remain rare. This behavior follows a so-called \(1/f\) law (where \(f\) is the frequency): "Energy dissipates by producing avalanches whose amplitude is inversely proportional to the frequency." This observation was highlighted by Per Bak (1948-2002), a Danish theoretical physicist specializing in phase transitions.

The avalanche effect, also called the "multiplier effect" or "avalanche multiplication," is a physical phenomenon where a tiny initial event triggers a chain reaction that produces considerable consequences.

N.B.: The avalanche effect also refers to a multiplier phenomenon in electric current within materials that, until triggered, were good insulators. This effect can occur in solid, liquid, or gaseous semiconductors or insulators. When the electric field in the material becomes sufficiently intense, it accelerates electrons; these, by colliding with atoms, release other electrons. The number of free electrons then grows rapidly, triggering a chain reaction comparable to that of a snow avalanche.

Why a Sandpile is Interesting in Thermodynamics

Sometimes, adding a single grain does almost nothing. Sometimes, it triggers a large-scale avalanche. Unlike a fluid, the injected energy (e.g., adding a grain) is quickly dissipated by friction, but the spatial organization of constraints leads to large-amplitude events (avalanches).

The system naturally self-organizes toward a critical state where it is exactly at the boundary between stability and instability. In this state, the distribution of avalanche sizes follows a power law (many small avalanches, few very large ones). There is no characteristic scale.

This critical state is a universal attractor, somewhat like a critical point in the thermodynamics of phase transitions (e.g., liquid-gas critical point). The sandpile becomes a model for studying criticality in much larger complex systems (earthquakes, stock market crashes, ecosystems).

Statistics of Avalanches

Experiments and simulations show that, for certain slow feeding protocols, the distribution \(P(s)\) of avalanche sizes \(s\) often follows a power law: \( P(s) \propto s^{-\tau} \), with a critical exponent \(\tau\). This law expresses that small avalanches are very frequent, while major events can occur without a characteristic scale.

The Primordial Avalanche: From the Big Bang to Cosmic Bifurcations

It must be admitted that the largest avalanche of bifurcations known is the one that gave rise to the Big Bang. This avalanche was so gigantic that it must be considered an extremely rare event in the history of the cosmos.

About 13.77 billion years ago, the amount of matter and antimatter in the Universe was exactly identical. A fundamental question remains: why do we live today in a Universe made almost exclusively of matter?

The system—the primordial Universe—was at a critical point, a quantum fluctuation, which tipped toward a bifurcation giving matter a slight advantage over antimatter. This spontaneous breaking of a fundamental symmetry occurred during the first fractions of a second of the observable Universe.

This initial avalanche triggered other avalanches: formation of protons and neutrons, then nucleosynthesis of light nuclei, appearance of stars, structuring of galaxies, up to the emergence of life and humans. These self-organization processes continue today, in other forms and at other scales.

The work of Yoichiro Nambu (1921-2015), Makoto Kobayashi (1944-), and Toshihide Maskawa (1940-2021), awarded the Nobel Prize in Physics in 2008, explained the existence of this small difference: the spontaneous breaking of symmetry between matter and antimatter, which made the Universe as we know it possible.

Future Avalanches and Planetary Challenges

Avalanches of bifurcations are not limited to cosmic phenomena: they also concern our present and future. Climate change, for example, is a complex system that could cross critical points: irreversible melting of ice caps, lasting disruption of ocean circulation, massive acidification of oceans, etc. These tipping points, analogous to slow but powerful avalanches, could durably reshape the habitability conditions of our planet.