Entropy question in classical physics

[Glenn G]

Hi I've been wondering about Boltzmann's equation
S = k ln W
Where W is the number of different distinguishable microscopic states of a system.
What I don't get is that if it's the position and velocity of a particle that describes a microstate doesn't it mean that W would be infinite classically since every infinitesimally small change in say a particles position represents a different microstate. Am I missing something?
Cheers,
Glenn.

 

[Stephen Tashi]

What I don't get is that if it's the position and velocity of a particle that describes a microstate doesn't it mean that W would be infinite classically since every infinitesimally small change in say a particles position represents a different microstate. Am I missing something?

You are correct in the sense that the style of reasoning using "microstates" in classical thermodynamics is similar to the way that "dx" is used in the intuitive presentation of integration. We imagine dividing up a interval into tiny sub-intervals of length dx. This type of intuitive thinking is an abbreviation for arguments whose precise statement would involve taking limits.

The most general definition of entropy when we have divided state space up into a finite number of microstates (each of which had finite volume instead of being a single point) is $S = -k \sum_i { p_i \ln(p_i) }$ The formula $S = k \ln W$ would be the special case where the system has the same probability $p_i = p$ of being in each of the microstates.