Abstract
People often observe that today’s world is complex. Although this term is not precisely defined, we instinctively distinguish simple from complex situations and often need to tackle complexity.
The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
H. Poincaré
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Notes
- 1.
He states that: “The curve described by a single molecule in air or vapour is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance”, and, more generally, that: “The regularity which astronomy shows us in the movements of the comets doubtless exists also in all phenomena.”
- 2.
We may be allowed to express here a hypothesis concerning the nature of heat. At present, light is generally regarded as the result of a vibratory movement of the ethereal fluid. Light produces heat, or at least accompanies the radiant heat and moves with the same velocity as heat. Radiant heat is therefore a vibratory movement. It would be ridiculous to suppose that it is an emission of matter while the light which accompanies it could only be a movement.
- 3.
Here, the quotation marks hint at the fact that the term “microscopic” may be quite different from our everyday notion of microscopically small. For instance, it could refer to entire galaxies, in cosmological studies, while the term “collective” may be used in nuclear physics to describe the agglomerate of elementary particles in an atomic nucleus, clearly too small compared to any object that we consider “macroscopic”.
- 4.
Conceptually, quantum mechanics does not change the picture (Lebowitz 1993).
- 5.
The impressive range of validity of classical mechanics should not be confused with the validity of Hamiltonian mechanics, the mathematically elegant formulation of Newtonian mechanics. This, indeed, suffers from various limitations, such as the absence of dissipation, which may instead be incorporated via Lagrangian mechanics.
- 6.
If the elementary constituents of the system cannot be approximated by point-like particles, but still obey classical mechanics, one would have to refer to a larger phase space, which has more dimensions, but the basic idea does not change.
- 7.
The existence of such limits was proven by Birkhoff in 1931 for a wide class of systems, while the dependence on \(\varGamma \) poses many more problems, from a mathematical perspective.
- 8.
In his renowned small book on thermodynamics, Fermi says that: Studying the thermodynamical state of a homogeneous fluid of given volume at a given temperature (the pressure is then defined by the equation of state), we observe that there is an infinite number of states of molecular motion that correspond to it. With increasing time, the system exists successively in all these dynamical states that correspond to the given thermodynamical state. From this point of view we may say that a thermodynamical state is the ensemble of all the dynamical states through which, as a result of the molecular motion, the system is rapidly passing.
- 9.
A physical system is in an equilibrium state if all currents—of mass, momentum, heat, etc.—vanish, and the system is uniquely described by a (typically quite small) set of state variables which do not change with time.
- 10.
In their numerical simulations, known as the FPU experiment, Fermi and coworkers showed that a typical Hamiltonian system is not ergodic. This fact was totally unexpected, at that time, and was only later explained in the sophisticated mathematical terms of KAM theory (Cencini et al. 2009).
- 11.
However one should be wary of possible misunderstandings. In particular, ensembles are often described as fictitious collections of macroscopically identical copies of the object of interest, whose microstates differ from each other. While this maybe a convenient perspective, at times, one should not forget that their purpose is to describe the properties of a single system, whose microstate evolves forever. We can say that the word “statistical ensemble” is nothing but a way to indicate the probability density of \(\varGamma \).
- 12.
At that time, \(N_A\) was but a parameter of the atomic theory, whose value was unknown.
- 13.
Up to the order of billions of particles.
- 14.
Assuming that the pollen and the molecules have the same average kinetic energy.
- 15.
Assuming the validity of the Stokes law.
- 16.
Logical supervenience, in our framework, means that it is logically impossible to produce different macroscopic states with identical microscopic states. This requires the microstate to fully determine the properties of the macrostate.
- 17.
The Ising model consists of a discrete set of positions in space representing the lattice of the atoms of a crystalline solid, in each of which the “spin” variable \(S_i\) takes either value 1 or \(-1\). The spins are assumed to interact in pairs of nearest neighbours, with interaction energy that depends on whether the neighbours have same value or opposite values. In the presence of a “magnetic field”, the Hamiltonian of the system is given by:
$$\begin{aligned} H = - J \sum _{\langle i,j \rangle } S_i S_j - h \sum _i S_i \end{aligned}$$where \(J\) is the coupling between spins in different positions, \(h\) is the coupling with the magnetic field and \(\langle i,j \rangle \) denotes summation over nearest neighbours only. If \(J > 0\), the model is described as ferromagnetic; if \(J \,<\, 0\), the model is anti-ferromagnetic. In the first case, spins tend to align, and the “material” to become magnetised. The equilibrium states of this system are described by the canonical ensemble. The Ising model is the first in which phase transitions could be explicitly proven to occur in the thermodynamic limit (cf. below), provided the dimensionality is higher than 1. The solution of the 1-dimensional model was obtained by Ising in 1925, while the much more difficult solution of the 2-dimensional case was derived by Onsager in 1944.
- 18.
In this case, the divergence of the correlation length suggests that near the critical point one must resort to a theory based on long-range collective fluctuations and on Hamiltonians or free energies constrained only by the fundamental symmetries of the system.
- 19.
The Heisenberg ferromagnet has a structure similar to that of the Ising model, with Hamiltonian given by:
$$\begin{aligned} H = -J \sum _{\langle i,j \rangle } \mathbf {S}_i \cdot \mathbf {S}_j - \sum _i \mathbf {S}_i \cdot {\fancyscript{H}} \end{aligned}$$where each \(\mathbf {S}_i\) is a vector of unit length in \(\varvec{R}^2\) or \(\varvec{R}^3\), the dot indicates the scalar product and \({\fancyscript{H}}\) is an external magnetic field.
- 20.
Apart from the tremendous mathematical effort it required, the importance of Onsager’s exact solution is that it first demonstrated that the formalism of SM can describe phase transitions and critical phenomena.
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Chibbaro, S., Rondoni, L., Vulpiani, A. (2014). A First Attempt to Tame Complexity: Statistical Mechanics. In: Reductionism, Emergence and Levels of Reality. Springer, Cham. https://doi.org/10.1007/978-3-319-06361-4_3
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