ly don’t exist when the interactions are studied individually. As Aristotle said: “The whole is greater than the sum of its parts”. As we will see later, emergent properties are in the core of the field of complex systems.
One of the remarkable results of the kinetic theory is that the law of ideal gases can be derived from it. That is, the laws of movement can explain a thermodynamic law.
Kinetic theory was one of the touchstones of the new field of statistical mechanics. Since its foundations were developed in the second half of the 19th century (mainly due to Clausius, Maxwell and Boltzmann), statistical mechanics harvested one success after another.
The details of the theory of statistical mechanics are far beyond the scope of this article. As a summary, I’ll just say that problems in classical statistical mechanics usually have these elements:
A huge amount of particles.
Some common well known dynamic properties (for example, all particles can be inside a container, have the same mass, or be under the influence of a gravitational field).
Some common partially-unknown dynamic properties, introduced via probability density functions (for example, the kinetic energy distribution given the temperature).
Or, stated in a much simpler way:
A description of the system in terms of its components.
What we know about the components.
What we don’t know so well.
Of course, there is a wide range of phenomena that can be naturally modelled as a system whose behavior is only partially known. And in such cases, the methods of statistical mechanics become a useful tool.
For example, think of the stock market. A very simple model of it can be fitted into our scheme as follows:
The buyers and sellers (the individual elements).
The fact that all of them want to maximize their earnings (the common “driving force”).
The random fluctuations and the weight of bad decisions (the unknown, the random noise).
Most of these “exotic” applications of statistical mechanics have very serious problems. Being the following the most important:
Usually, the deterministic part of our problem (the Hamiltonian or driving force) of our elements is not clearly defined.
The systems are, by far, smaller than the usual systems in statistical mechanics. So the statistical approach loses a lot of accuracy when compared with classical examples, with moles and moles of elements.
Despite these and other flaws, even such an oversimplified description is still an interesting approach to this kind of problems. In fact, statistical mechanics was the seed for the methods of the so-called complex systems science. Complex systems science is a subfield halfway between physics and applied mathematics, which evolved mostly (but not only) from statistical mechanics.
Currently, the field of complex systems covers an enormously wide range of applications in almost every field of knowledge (such as ecology 1, demography 2, economy 3 and, going back to our example of the raging crowd, even riot control 4 and law-enforcement 5).
W. Greiner, L. Neise, H. Stöcker. Thermodynamics and statistical mechanics. New York. Springer – Verlag. 1997. ISBN 978-1-4612-0827-3
J. Aguilar. Curso de termodinámica. Pearson Educación. ISBN 9788420513829.
1 Just for storing this data, assuming 2 bytes per number and 6 numbers per particle (position and momentum), will be required about 1016 Gb. This is roughly equivalent to 10.000 computers per person in the world. Not to talk about the horrible task of typing it all.
- Garcia-Algarra, Galeano, Pastor, Iriondo, Ramasco. Rethinking the logistic approach for population dynamics of mutualistic interactions. Journal of Theoretical Biology (2013). ArXiv ID: 1305.5411 ↩
- Castellano, Castellano, Fortunato et al. Statistical physics of social dynamics. Reviews of Modern Physics (2007). ArXiv ID: 0710.3256 ↩
- Yakovenko. (2012). Applications of statistical mechanics to economics: Entropic origin of the probability distributions of money, income, and energy consumption. ArXiv ID: 1204.6483. ↩
- Helbing, Johansson, Al-Abideen. Crowd turbulence: the physics of cro