Untitled Document
Worlds out of equilibrium
J.P. PETIT
Former research director at CNRS
December 2012
English version, translated by François Brault
An article that academician Robert Dautray had committed to support for publication in Pour la Science.
But, after months of silence, I lost hope that it would be done
When the average person thinks of a system in equilibrium, he imagines a ball at the bottom of a depression, or something similar.
The notion of thermodynamic equilibrium involves something more subtle: that of a dynamic equilibrium. The simplest example is the air we breathe. Its molecules are agitated in all directions, representing an average thermal agitation speed of 400 meters per second. At a frantic rate, these molecules collide, interact. These impacts change their speeds. Yet the physicist would say that this results in a certain stationarity, statistically speaking. Imagine a little sprite who, at some point in space, can measure at any given moment the speed of air molecules agitated in a given direction, say, in such or such direction, within a narrow angular range. At each moment he counts, and recount, how many molecules have a speed, in algebraic value, between V and V + ΔV. He records the results of his measurements on a graph, and sees a nice Gaussian curve appear, with a peak near this average speed of 400 meters per second. Then, the more the count concerns faster or slower molecules, the lower their population becomes.
He repeats this operation by pointing his measuring instrument in all directions of space and, to his surprise, arrives at the same result. The molecular agitation of the air in the room is isotropic. Moreover, nothing disturbs this dynamic equilibrium, provided the temperature of this gas remains constant, since its absolute temperature is precisely the measure of the average value of the kinetic energy corresponding to this thermal agitation.
The physicist would say that this gas is in a state of thermodynamic equilibrium. This situation has other facets. The air molecules are not spherical objects. Diatomic molecules, of oxygen or helium, have peanut-like shapes. Those composing carbon dioxide and water vapor are yet different. Nevertheless, these objects can, by rotating, store energy like tiny flywheels. These molecules can also vibrate. The concept of equipartition of these energies prescribes that the energy be equally distributed among these different "modes". During a collision, kinetic energy can cause a molecule to vibrate or rotate. But the reverse phenomenon is also possible. Everything then becomes a question of statistics and our sprite can count how many molecules are in a given state, have a certain kinetic energy, are in a vibrational state. Still in the air we breathe, this census leads to the stationarity of this state. It is said that this medium is in a relaxed state of thermodynamic equilibrium.
Imagine a magician who has the ability to stop the motion of these molecules, in time, to freeze their different rotational and vibrational movements, and who modifies them at will, creating a different statistic, deforming this beautiful Gaussian curve, or even having fun creating some anisotropic situation, where the thermal agitation speeds would be, for example, twice as high in one direction than in the transverse directions. Then he would let this system evolve, at the mercy of collisions.
How many of these would be needed for the system to return to its state of thermodynamic equilibrium? Answer: a few. The mean free path time of a molecule, between two collisions, gives the order of magnitude of the relaxation time in a gas, of its time of E. Are there out-of-equilibrium media, where the statistics of the agitation speeds of the elements significantly deviate from this reassuring isotropy and these beautiful Gaussian curves?
Yes, and this is even the majority of cases in the universe! A galaxy, this "island universe," composed of hundreds of billions of stars, of masses, all fairly similar, is comparable to a gaseous ensemble, whose molecules would be ... the stars. In this specific case, we discover an extremely disconcerting world where the mean free path time of a star, with respect to a meeting with its neighbors, is ten thousand times the age of the universe. But what do we mean by meeting? Would it be a collision where the two stars would crash into each other? Not even. In a field of theoretical physics called the kinetic theory of gases, it is considered that there is a collision when the trajectory of the stars is simply modified in a significant way when it crosses a neighbor. However, calculations show that these events are extremely rare and that the system of 100 and some billion stars orbiting in a galaxy can be considered as a practically non-colliding system. Thus, for billions of years, the trajectory of our Sun has been very regular, almost circular. If this Sun could be endowed with consciousness, in the absence of changes in its trajectory, due to encounters, it would be unaware that it has neighbors. It only perceives the gravitational field in its "smooth" form. It moves as if in a depression it would not perceive the tiny irregularities created by the other stars.
The corollary emerges immediately. Place our sprite, now an astronomer, near the Sun, in our galaxy and ask him to perform a statistics on the relative speeds of all the neighboring stars, in all directions. One thing becomes perfectly evident. The medium is, dynamically speaking, very anisotropic. There is a direction along which the stellar agitation speeds (called by astronomers residual velocities, relative to an average motion of 230 km/s near the Sun, along a nearly circular trajectory) are on average almost twice as high as in the transverse directions. In the air we breathe, we spoke of a velocity ellipsoid. There, it becomes an ellipsoid of velocities.
Well. What impact does this have on our way of conceiving the world, of understanding it? It changes everything. Because we simply do not know how to handle, on a theoretical level, systems that are so categorically out of equilibrium. If we set aside the paradoxical situations in which galaxies find themselves, with this annoying missing mass effect, discovered by the Swiss-born American Fritz Zwicky, we could not produce a model of self-gravitating point-mass systems (orbiting in their own gravitational field). Our physics is always near a state of thermodynamic equilibrium. Of course, any deviation from this or that represents a deviation from equilibrium, for example, a temperature difference between two regions...