### Entropy in Tribology: in the Search for Applications

Michael Nosonovsky
2010 Entropy
The paper discusses the concept of entropy as applied to friction and wear. Friction and wear are classical examples of irreversible dissipative processes, and it is widely recognized that entropy generation is their important quantitative measure. On the other hand, the use of thermodynamic methods in tribology remains controversial and questions about the practical usefulness of these methods are often asked. A significant part of entropic tribological research was conducted in Russia since
more » ... d in Russia since the 1970s. Surprisingly, many of these studies are not available in English and still not well known in the West. The paper reviews various views on the role of entropy and self-organization in tribology and it discusses modern approaches to wear and friction, which use the thermodynamic entropic method as well as the application of the mathematical concept of entropy to the dynamic friction effects (e.g., the running-in transient process, stick-slip motion, etc.) and a possible connection between the thermodynamic and information approach. The paper also discusses non-equilibrium thermodynamic approach to friction, wear, and self-healing. In general, the objective of this paper is to answer the frequently asked question -is there any practical application of the thermodynamics in the study of friction and wear?‖ and to show that the thermodynamic methods have potential for both fundamental study of friction and wear and for the development of new (e.g., self-lubricating) materials. 1346 Entropy 2010, 12 1347 A different approach to entropy was suggested by L. Boltzmann in 1877, who defined it using the concept of microstates, which correspond to a given macrostate. Microstates are arrangements of energy and matter in the system, which are distinguishable at the atomic or molecular level; however, they are indistinguishable at the macroscopic level. If  microstates correspond to a given macrostate, then the entropy of the macrostate is given by where k is Boltzmann's constant. The microstates have equal probabilities, and the system tends to evolve to a more probable macrostate, i.e., the macrostate that has a larger number of microstates [2] . The entropy definition given by Equation 4 is convenient for using entropy as a measure of disorder, since it deals with a finite (and, actually, integer) number of microstates. In the most ordered ideal state, i.e., at the absolute zero temperature, there is only one microstate and the entropy is zero. In the most disordered state of a particular system (e.g., the homogeneous mixing of two substances), the number of microstates reaches its maximum and thus the entropy is at maximum. The definition of Equation 3 can also be easily generalized for the theory of information, since the discrete microstates can be seen as bits of information required to uniquely characterize the macrostate, and thus the so-called Shannon entropy serves a measure of uncertainty in the information science. The concept of microstate is, however, a bit obscure. It is noted, that the statistical definition of entropy given by Equation 4 implies a finite number of microstates and thus a discrete spectrum of entropy, whereas the thermodynamic definition in Equations 1-3 apparently implies a continuum spectrum of entropy and an infinite number of microstates. According to Boltzmann, the microstates should be grouped together to obtain a countable set. Two states of an atom are counted as the same state if their positions, x, and momenta, p, are within δx and δp of each other. Since the values of δx and δp can be chosen arbitrarily, the entropy is defined only up to an additive constant. However, an apparent contradiction between the continuum and discrete approach remains, so the question is often asked, whether the entropy given by Equation 1 is the same quantity as the entropy given by Equation 4? The question -what is a microstate‖ is not completely clarified even if we take into consideration the fact that any measurement of any parameter is conducted with a finite accuracy and, therefore, a measuring device provides a discrete rather than continuum output. Thermodynamic parameters should be independent of the resolution of our measurement devices. Some authors prefer to use the concept of -quantum states‖ instead of the microstates [3]; however, the classical (non-quantum) description should use classical concepts. Fortunately, Equation 4 works independently of what is microstate. The only property of microstates that is of importance is their multiplicativity, that is, for a system consisting of two non-interacting subsystems, the total number of microstates is equal to the product of the numbers of microstates of the subsystems. This makes entropy, defined as a logarithm of the number of microstates, an additive function, that is, the entropy of a system is equal to the sum of entropies of the sub-systems. In a sense, Equation 4 serves as a definition of the microstate: the number of microstates is just an exponent of S/k. The arbitrary constant, which can be added to S, corresponds to the arbitrarily choice of δx and δp during the grouping of the microstates. The additive constant is usually chosen in such a manner that S = 0 at T = 0. In other words, there is only one microstate of any system at the absolute zero temperature when no thermal motion occurs. Entropy 2010, 12