# Could time (or space) discretisation be needed in order to preserve the law of increase of entropy?

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In Introductory Statistical Mechanics, 2nd Edition, by R. Bowley and M. Sanchez, pages 81-83 is given a proof of the second law of thermodynamics by showing that the probability of a system being in a state described by an energy 20 standard deviations away from the mean (which is the energy corresponding to the maximum value of entropy) is so small that even considering that the system is in a different microstate every 10-50  seconds would yield approximately one such state every 1038 seconds (which is practically never).

My question is what stops us from considering that the system is sampling a different microstate every, say, 10-1000 seconds, provided that time and space are considered continuous variables? In this case the probabilities would have unreasonable values for the system being in a state in which the entropy would be much lower than its maximum possible value. One way of saving this argument could be that either time or space (or both) could be discrete so that no system could change its microstate faster than a given amount of time (which could depend on the system's microscopic parameters such as the number of particles). Is this a necessary assumption in order to keep the argument valid?

asked Aug 15, 2021 in Q&A

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