# Why is the entropy of most systems not infinite?

We can define a microstates as follows:

Definition 1:

A complete specification of the six coordinates of each molecule of a system, within the limits of the dimensions of the cell in which its representative points lie... Such a specification states where each molecule is, within the limits $dx, dy, dz,$ and how fast and in what direction it is moving, within the limits of the differentials of velocity. (Thermodynamics, Sears, 1953, pg 277)

Whilst the entropy of system can be defined as:

Definition 2:

$$S=k_B \ln(\Omega)$$ Where $\Omega$ is the number of microstates associated with the macrostates under consideration. (paraphrased from Concepts in Thermal Physics, Blundell & Blundell, 2010,pg 147)

Consider the case of a single particle in a box. Definition 1 implies that at every position the particle could be at with every possible different velocity represents a unique microstates, since there are (a near) infinite places in the box we could put the particle and (a near) infinite number of velocities we could give it then surly we have (a near) infinite number of microstates $\Omega$ and thus $S\approx \infty$ A simple argument could be made for all similar systems. But surly the entropy of these systems is not $\infty$, but using these two definitions I cannot see how this cannot be so?

• You have made a good observation. See e.g. Wikipedia – Danu Sep 1 '15 at 7:51

The key to the answer is this part of your quoted text:

...within the limits of the dimensions of the cell in which its representative points lie...

Thus in the book you quote, the number $\Omega$ is to be obtained by counting the number of cells (whose dimensions are arbitrary), not the number of all possible states.

Alternatively, one can replace the number of cells by phase space volume of the phase space $\Gamma$ compatible with the macrostate $\mathbf X$:

$$\Omega(\mathbf X) = \int_{\Gamma(\mathbf X)}\,dq_1...dq_{3N}dp_1...dp_{3N}$$

to get ln-entropy of the macrostate:

$$S_{ln}(\mathbf X) = k_B \ln \Omega(\mathbf X).$$