All this points to two quite different formulations of the second law of thermodynamics. In one formulation (Boltzmann), the notions of state and entropy are quite intuitive but the monotonic behaviour of entropy cannot be strictly maintained. In the other (Gibbs), the increase of entropy is a rigorous theorem but the notions of entropy and approach to equilibrium are much less intuitive and direct.
– Mark Kac, [67, p.98]
From both approaches, we learn that there are two important elements to understand how an increase of entropy emerges at the macroscopic world:
• Large number of particles
A clear scale separation between the microscopic and the macroscopic world is obviously required for the argument to work. This sheds new light on the microscopic interpretation of the second law of thermody- namics. We quote from Maxwell [86] in 1878:
The second law is drawn from our experience of bodies con- sisting of an immense number of molecules. [...] it is con- tinually being violated, [...], in any sufficiently small group of molecules [...] As the number [...] is increased [...] the probability of a measurable variation [...] may be regarded as practically an impossibility.
– James C. Maxwell
Or, as Gibbs [51] summarized it earlier in 1875:
In other words, the impossibility of an uncompensated de- crease of entropy seems to be reduced to an improbability.
– Josiah W. Gibbs
For large systems, violations of the second law are no longer an impossi- bility, but merely reduced to an improbability due to the huge difference in phase space volumes.
The large number of constituents of a macroscopic system also explains why it is so difficult experimentally to force a macroscopic system in a mi- crostate which will lead to violations of the second law. Such microstates certainly exist, in fact, computer simulations confirm these violations [74] when the velocities are reversed in a system relaxing to equilibrium. We listen to Thomson [113], who speaks about the relaxation to equilibrium as the “equalization” of a gas:
If we allowed this equalization to proceed for a certain time, and then reversed the motions of all the molecules, we would observe a disequalization. However, if the number of molecules is very large, as it is in a gas, any slight de- viation from absolute precision in the reversal will greatly shorten the time during which disequalization occurs.
– William Thomson (Lord Kelvin)
By the latter, Thomson refers to the fact that even the slightest mistake in the operational inversion of the momenta, i.e., experimentally apply- ing the operation π(1.3), will send the microscopic evolution in totally the wrong direction. After all, from the huge difference in phase space volumes, the trajectory has to be aimed at a very small region in phase space.
• Assumptions on the initial condition
For the increase of Boltzmann entropy, and thus thermodynamic entropy, it is necessary that the initial condition is special, in the sense that it has a relatively low phase space volume. From a fundamental point of view, nothing more needs to be done [13]:
Once one has remarked that, a priori, there is no contra- diction between irreversibility and the fundamental laws, one could stop the discussion. It all depends on the ini- tial conditions, period. [...] Luckily, much more can be said. It is perfectly possible to give a natural account of ir- reversible phenomena on the basis of reversible fundamen- tal laws, and of suitable assumptions on initial conditions. This was essentially done a century ago by Boltzmann.
– Jean Bricmont
Indeed, Boltzmann [10, p.170] realized the importance of appropriate initial conditions for macroscopic irreversibility:
Since in the differential equations of mechanics themselves, there is absolutely nothing analogous to the Second Law of thermodynamics, the latter can be mechanically represented only by means of assumptions regarding initial conditions.
– Ludwig Boltzmann
All this seems to shift the problem to the question why there are such special initial conditions in the first place, so we can start the argument from sections 2.4.2–2.4.3. Ultimately, the problem comes down to the initial conditions of the universe. This goes beyond the scope of this text, yet, to emphasize the scale of the problem of the initial condition of the universe, it is interesting to quote the vivid explanation of Roger Penrose from his book [94, p.343], see also figure 2.6:
How big was the original phase space volume W that the Creator had to aim for in order to provide a universe com- patible with the second law of thermodynamics and with what we now observe? [...] [A short calculation] tells us how precise the Creator’s aim must have been: namely to an accuracy of
one part in 1010123.
This is an extraordinary figure. One could not possibly even write the number down in full, in the ordinary denary notation: it would be ‘1’ followed by 10123 successive ‘0’s!
Even if we were to write a ‘0’ on each separate proton and on each separate neutron in the universe – and we could throw in all the other particles as well for good measure – we should fall far short of writing down the figure needed.
– Roger Penrose
As a final comment, see also the discussion in reference [31], there remains the question how useful an attempt at mechanically deriving the second law can
Figure 2.6: The Penrose [94] representation of how well-chosen the initial condition of the universe is. In order to reproduce a universe resembling the one in which we live, the Creator would have to aim for an absurdly tiny volume of the phase space of the possible universes. The pin and the spot aimed for, are not drawn to scale!
be today. Mathematically, an H-theorem is useful in the sense that it gives a Lyapunov function for a dynamical system. Physically, an H-theorem gives an extension and microscopic derivation of the second law of thermodynamics. It is important to emphasize that it is fruitful to reverse the logic, as was done in years that followed Boltzmann’s pioneering work, particularly by Einstein. The statistical definition of entropy starts from a specific choice of microstates. If for that choice, the corresponding macroscopic evolution is not satisfying an H-theorem, then we know that our picture of the microstructure of the system may be inadequate. In other words, we can obtain information about the microscopic structure and dynamics from the autonomous macroscopic behaviour. Then, instead of concentrating on the derivation of the macroscopic evolutions with associated H-theorem, we use the phenomenology to discover crucial features about the microscopic world. That was already the strategy of Einstein in 1905 when he formulated the photon hypothesis.