Термодинамикалық теңестіксіздіктегі энтропияның ауытқу теоремасы

The fluctuation theorem (FT), which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium (i. e., maximum entropy) will increase or decrease over a given amount of time. While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously decrease; the fluctuation theorem precisely quantifies this probability.

The FT was first proposed and tested using computer simulations, by Denis Evans, E. G. D. Cohen and Gary Morriss in 1993. The first derivation was given by Evans and Debra Searles in 1994. Since then, much mathematical and computational work has been done to show that the FT applies to a variety of statistical ensembles. The first laboratory experiment that verified the validity of the FT was carried out in 2002. In this experiment, a plastic bead was pulled through a solution by a laser. Fluctuations in the velocity were recorded that were opposite to what the second law of thermodynamics would dictate for macroscopic systems. In 2020, observations at high spatial and spectral resolution of the solar photosphere have shown that solar turbulent convection satisfies the symmetries predicted by the fluctuation relation at a local level.

There are many important implications from the fluctuation theorem. One is that small machines (such as nanomachines or even mitochondria in a cell) will spend part of their time actually running in "reverse". What we mean by "reverse" is that it is possible to observe that these small molecular machines are able to generate work by taking heat from the environment. This is possible because there exists a symmetry relation in the work fluctuations associated with the forward and reverse changes a system undergoes as it is driven away from thermal equilibrium by the action of an external perturbation, which is a result predicted by the Crooks fluctuation theorem. The environment itself continuously drives these molecular machines away from equilibrium and the fluctuations it generates over the system are very relevant because the probability of observing an apparent violation of the second law of thermodynamics becomes significant at this scale. This is counterintuitive because, from a macroscopic point of view, it would describe complex processes running in reverse. For example, a jet engine running in reverse, taking in ambient heat and exhaust fumes to generate kerosene and oxygen. Nevertheless the size of such a system makes this observation almost impossible to occur. Such a process is possible to be observed microscopically because, as it has been stated above, the probability of observing a "reverse" trajectory depends on system size and is significant for molecular machines if an appropriate measurement instrument is available. This is the case with the development of new biophysical instruments such as the optical tweezers or the atomic force microscope. Crooks fluctuation theorem has been verified through RNA folding experiments.

The second law of thermodynamics, which predicts that the entropy of an isolated system out of equilibrium should tend to increase rather than decrease or stay constant, stands in apparent contradiction with the time reversible equations of motion for classical and quantum systems. The time reversal symmetry of the equations of motion show that if one films a given time dependent physical process, then playing the movie of that process backwards does not violate the laws of mechanics. It is often argued that for every forward trajectory in which entropy increases, there exists a time reversed anti trajectory where entropy decreases, thus if one picks an initial state randomly from the system's phase space and evolves it forward according to the laws governing the system, decreasing entropy should be just as likely as increasing entropy. It might seem that this is incompatible with the second law of thermodynamics which predicts that entropy tends to increase. The problem of deriving irreversible thermodynamics from time symmetric fundamental laws is referred to as Loschmidt's paradox. The mathematical derivation of the fluctuation theorem and in particular the second law inequality shows that, for a nonequilibrium process, the ensemble averaged value for the dissipation function will be greater than zero. This result requires causality, i. e. that cause (the initial conditions) precede effect (the value taken on by the dissipation function). This is clearly demonstrated in section 6 of that paper, where it is shown how one could use the same laws of mechanics to extrapolate backwards from a later state to an earlier state, and in this case the fluctuation theorem would lead us to predict the ensemble average dissipation function to be negative, an anti second law. This second prediction, which is inconsistent with the real world, is obtained using an anti causal assumption. That is to say that effect (the value taken on by the dissipation function) precedes the cause (here the later state has been incorrectly used for the initial conditions). The fluctuation theorem shows how the second law is a consequence of the assumption of causality. When we solve a problem we set the initial conditions and then let the laws of mechanics evolve the system forward in time, we don't solve problems by setting the final conditions and letting the laws of mechanics run backwards in time.