Energy transformations for small systems

How does the description of the energy transformation processes presented so far change when we deal with small systems? To answer this question we start considering an important aspect when we deal with physical systems: the condition of being an isolated system. If we say that a system is not isolated, we intend that it has interactions of some kind with something that we consider external to the systems itself. If this is not the case (isolated system) all the dynamics is self-determined by the system itself and we can deal with it by addressing the equations of motion for each particle coupled to each other particle in the system. At this aim we may use the standard Newton laws (or in the quantum case the Schrodinger equation). If the system is not isolated the situation is generally more complex and we need to take into account the interaction of our system with the "external world". In principle however any system can be considered isolated provided that we include in the system all the sources of interactions. In the extreme case we can consider the universe itself as an isolated system. For this reason we will limit our consideration to systems that are isolated.

Before answering the question about the energy transformations in small systems we should be more precise in defining what a small system is. When we deal with real physical systems we cannot ignore that all the matter, as we know it, is composed by atoms. These are more or less individual particles whose interactions determine most of the properties that characterize the matter. The ordinary devices that we are used to deal with are composed by very a large assembly of atoms, numbers are of the order of the Avogadro number, i. e. NA = 6.022 * 1023. Thus when we are dealing with small systems, in general we intend systems composed by a number of atoms N that is small compare to NA. Clearly, due to the extremely large value of NA, a system composed by few thousands of atoms (or molecules or "particles") can still be considered small. This is the case for example of the nanodevices like last generation transis­tors. Unfortunately in this case the small systems are not isolated because they exchange energy and information with the outside. On the other hand small isolated systems are quite rare. An example of the small isolated system can be found in the realm of what is generally called "high energy physics": here the particles are most of the time just few (small system) and isolated from the external. Back to the realm of the physics of matter we have frequently to deal with systems that are usually not small but can be considered in good approximation isolated. What do we do in these cases?

One possibility is to do what we did just before, when we dealt with the movable set in contact with the gas of N particles. Here N is of the order of NA. Overall our system is composed by 3N+1 degrees of freedom (dof): 3 for each of the N particles and 1 for the movable set position coordinate x. This is clearly not a small system, although isolated, because all the interactions are inside the 3N+1 dof. In this case we played a trick: we focused our attention on the single degree x and summarized the role of the remaining 3N dof by introducing the dissipative and the fluctuating force like external forces. By the moment that both these forces are necessary to account for the observed dynamics and by the moment that both are born out of the neglected 3N dof, it comes out that they are connected to each other and the connection is nothing else that the FDT that we discussed. Our equation of motion is not anymore the deterministic Newton (Schrodinger) equation, instead it is the novel stochastic Langevin equation where there is a both friction and fluctuation caused by the added forces due to the neglected dof. Thus the trick we played was to exchange the dynamics of a not small isolated system with small not isolated system. Such an approach has different names (adiabatic elimination, coarse graining, .) and it is considered a very useful tool in describing the properties of dynamical systems composed by many dof.

To summarize our approach: we have transformed a non small isolated system into a small non isolated system. What is the advantage? Easy to say: the dynamics of a non small isolated system can be described in terms of 3N+1 dof by 3N+1 coupled motion equations and when N is of the order of Na this is a practically impossible approach. Thus the advantage was to drastically reduce the number of equations of motion (in this case to just 1) but the price we had to pay is the introduction of dissipation and fluctuation. What we have found is that dissipative and fluctuating effects appear only if we neglect some (usually many) dof through some coarse graining approximation to the system dynamics. In this perspective the dissipation of energy appears to be only an illusion due to our choice of dynamical description.

On the other hand we know that if we perform a real experiment with our movable set, indeed we observe a decrease in the oscillation amplitude of the set until it reaches the stop and then it does start to fluctuate around the equilibrium position. This is not and illusion. The potential energy initially stored in the spring is now dissipated due to the presence of the gas particles. How does this fit with what we just said about the dissipation being an illusion? The answer is that the total energy (the kinetic energy of the gas particles + the potential energy initially stored in the spring) is conserved because the (not small) system is isolated. What happened is that the potential energy of the movable set has been progressively transformed into additional kinetic energy of the N particles that now have a slightly larger average velocity (the temperature of our gas slightly increased). Thus during the dynamics the energy is transferred from some (few) dof to others (many) dof. This is what we called before energy dissipation and now it appears to be nothing more than energy re-distribution. Before we have seen that dissipative effects during a transformation are associated with an increase of entropy. Indeed this energy distribution process is an aspect of the tendency of the system to reach the maximum entropy (while conserving the energy). This is what we have called a spontaneous

transformation: the increase of the entropy up to the point where no more energy distribution process takes place, i. e. the thermal equilibrium.

Is this the end of the story? Actually it is not. There is a quite subtle aspect that is associated with the conservation of energy. It is known as Poincare recurrence theorem. It states that in a system that conserves energy the dynamics evolve in such a way that, after a sufficiently long time, it returns to a state arbitrarily close to the initial state. The time that we have to wait in order to have this recurrence is called the Poincare recurrence time. In simple words this means that not only the dissipation of energy is an illusion because the energy is simply re­distributed among all the dof but also that this redistribution is not final (i. e. on the long term the equilibrium does not exist). If we wait long enough we will see that after some time the energy will flow back to its initial distribution and our movable set will get its potential energy back (with the gas particle becoming slightly colder). This is quite surprising indeed because it implies that in this way we can reverse the entropy increase typical of processes with friction and thus fail the second principle. Although this may appear a paradox this answer was already included in the description of entropy proposed by Boltzmann and specifically in its intrinsic probabilistic character. The decrease of entropy for a system composed by many dof is not impossible: it is simply extremely improbable. It goes like this: for any finite observation time the dynamic system evolves most probably in a direction where the entropy increases because according to Boltzmann this is the most probable evolution. However if we wait long enough also the less probable outcome will be realized and thus the second principle violated. How much time should we wait? The answer depends on the dof of our isolated (energy conserving) system. The larger the number of dof the longer the time to wait. exponentially longer.

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