In statistical mechanics, entropy (usual symbol $S$) is related to the number of microscopic configurations Ω that a thermodynamic system can have when in a state as specified by some macroscopic variables. Specifically, assuming for simplicity that each of the microscopic configurations is equally probable, the entropy of the system is the natural logarithm of that number of configurations, multiplied by the Boltzmann constant $k_b$

For my argument to be much more imaginable by the reader let us assume that the "thermodynamic system" that we are talking about is just a single molecule named (let's say) $z$. Also we (again for the simplicity) let us assume that the molecule $z$ can only move in one dimension, it can go back and forth, that's all.

The Wikipedia article on entropy also states that entropy is linked to the microscopic configurations of a system in the manner described by the equation below.

$S = k_b\lnΩ $ (assuming equiprobable states)

What I don't understand is, if entropy is related to the number of macroscopic configurations of a thermodynamical system why isn't the entropy infinitely large? Because in our cases the molecule $z$ can move infinitely small distances to back or forth. Is there a limit to the small distances that a molecule or any other particle can move?


1 Answer 1


Actually, the "counting" of states should be replaced by something well-defined if there infinitely many possibilities: A suitable measure on phase-space.

In classical mechanics, phase space is indeed continuous and you would always count $\infty$ many points as possible states of a classical system. Therefore, the correct notion is that $\Omega$ is the phase space volume of the given macrostate.

(One might add that this changes in quantum mechanics where some kind of "quantization" allows a discrete counting and avoids this infinity.)


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