Log simplification question

  • #1

Main Question or Discussion Point

This is from a worked example involving using Stirlings approximations.

I have the log equation, where c<<1:

S=K(Nln(N)-cNln(cN)-N(1-c)ln(1-c))


In the next line of the example, this is simplified to

S=-kNcln(c)



I've tried a few ways of getting to this, but have had no success. I've tried making the last term 0, but still don't get the right answer.

Any help would be great :) Thanks
 

Answers and Replies

  • #2
Mute
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First off, I think you're missing a factor of N inside the last logarithm. I.e., your expression (aside from the Boltzmann constant) is

[tex]N\ln N - cN\ln(cN) - N(1-c)\ln(N(1-c))[/tex]

From there a bunch of stuff should cancel, leaving you with

[tex]-Nc\ln c - N(1-c)\ln(1-c)[/tex].

Using the fact that c is small, you can show that the first term is much larger than the other term, hence you can neglect the other term.
 
  • #3
Redbelly98
Staff Emeritus
Science Advisor
Homework Helper
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It looks like they are using

ln(cN) = ln(c) + ln(N)​

and then neglecting the ln(N) terms relative to -ln(c), i.e. N « 1/c.

UniPhysics90, from the context of what you are reading, does that seem a reasonable assumption?
 

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