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In many (biological and living systems) physics courses I encountered the notion that systems operating far from equilibrium are related with more efficiency with regard to their specific tasks.

One example I vaguely remember is the comparison of the energy efficiency of ATP to ADP convertion reaction with the efficiency of combustion engines.

Unfortunately, I can not think of many specific examples and can only recall this abstract notion of efficiency of systems far from equilibrium.

Could someone break this notion down by using many specific examples of simple physical systems, and in each case clearly defining the notion of far from equilibrium and the notion of efficiency ? You could start with the example I mentioned.

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There are two factors that governs which pathway a reaction takes. The first is the thermodynamic factor, the second is kinetic. Thermodynamic factor deals with the efficiency of the process, more efficient the process the more likely it is to take place. However, the kinetic factor also has a major role to play. Say reaction A is thermodynamically more plausible than reaction B, but it is also slower than reaction B. Then more of the reactants are converte to products via reaction B than reaction A.

One example I vaguely remember is the comparison of the energy efficiency of ATP to ADP convertion reaction with the efficiency of combustion engines.

One must understand that reactions at equilibrium are extremely slow. However, the requirement for life is to have quick access to energy, which is provided by reactions that deviate far from equilibrium. It is not that these reactions are more efficient, rather they are more essential.

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There is the efficiency of the Carnot cycle that is very well defined for systems executing cycles between two fixed temperatures at which heat exchange can take place, and this efficiency is always higher than other thermodynamic cycles whose thermal energy exchanges do not exceed those two temperatures.

Since these are processes and not equilibrium states your question must be referring to reversible versus irreversible processes. Note too that Carnot's cycle efficiency needs an ideal reversible processes that are usually approximated quasi-statically in small equilibrium or near equilibrium steps, hence it is of low working rate (power). But high power does not mean high efficiency, the two concepts are unrelated. Biological processes are not thermodynamic cycles and comparing their efficiency to, say, that of an internal combustion engine or of a nuclear reactor, etc., is at best misleading.

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