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Classical Physics
Thermodynamics
How Do Microstates, kT, and Entropy Interact in Statistical Mechanics?
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[QUOTE="Stephen Tashi, post: 5666535, member: 186655"] The situation in quantum mechanics is clearer than the situation in classical mechanics, but it may be useful to consider the situation in classical statistical mechanics - useful because it is (to me) murky. The "number of microstates" obviously depends on how you define the microstates. As far as I can tell, there is no standard definition for microstates that would specify each microstate as a unique subset of the (detailed) state space of a system. For example, suppose the detailed state of a system is given by a vector ##(x_1,x_2,...x_n)## where the ##x_i## take values in a continuous range of real numbers and specify the position and velocity of each particle in the system. I know of no standardized definition of "microstate" that would be so specific as to say things like "microstate number 37 consists of the set of all ##(x_1,x_2,...x_n)## such that the total kinetic energy of the particles is 28 J and ##(x_1,x_2,...x_n##) satisifes ##( 0.3m < x_1 \le 0.3002 m, 1.7 m < x_2 \le 1.7002 m ..., .0001 m/s < x_n < .0002 m/s##." If you are thinking of the system being in a specific state ##(x_1,x_2,...x_n)## at a given time then you are (conceptually) contradicting a basic assumption of statistical mechanics that says the system has a non-trivial [I]probability distribution [/I] of being in different microstates. If the system is conceptualized as being in a particular microstate at a given time, its probability of being there is 1 and its probability of being in a different microstate is 0. You might be able to force the concept of probability into the situation by saying that you sample the state of the system "at a random time" in some time interval. My impression of the concept of a microstate in classical physics is that it is analogous to the concept of "dx" in elementary calculus. We don't conceive of "dx" as have a particular numerical value. We conceive of it as something that could be given a particular value when we do an approximation. If my view is correct, then it would be interesting to discuss this further because the final goal of conceptualizing a finite number of microstates seems to be to derrive formulas by taking a limit as the number of microstates becomes infinite and the volume of each microstate becomes smaller. [/QUOTE]
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How Do Microstates, kT, and Entropy Interact in Statistical Mechanics?
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