Undergrad How Is Summation Changed to Integration in Reif's Statistical Mechanics?

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In Reif's Statistical Mechanics, the transition from summation to integration in Equation 1.5.17 is justified by the concept of approximating discrete values as continuous. This relationship is rooted in calculus, where limits are used to transition from finite sums to integrals as step sizes approach zero. However, not all mathematical functions are continuous and differentiable, which can complicate the application of these principles. Most physics problems are suitable for basic calculus methods, allowing for a straightforward application of these concepts. Understanding when to apply these rules requires deeper mathematical analysis, but for practical purposes, sticking to established methods is generally effective.
Abhishek11235
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This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?
 

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Abhishek11235 said:
This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?
Do you understand the relation between summation and integrals?

 
Abhishek11235 said:
Can anyone justify this change?
Basic ideas that you start with in Calculus take you from big steps to small steps and then you look at the limit as the step size approaches zero. That sort of relationship can be regarded as as 'continuous'. BUT that doesn't apply to all relationships. Not all relationships or mathematical functions are 'continuous and differentiable' over their whole range and you cannot do simple calculus in those cases. It is lucky (?) that most of the Physics we start off with is amenable to basic calculus methods (differentiation and integration). If you want to get deep into mathematical analysis methods then it will make you able to make choices about when you can and when you can't use basic calculus but, if you are like most of us, you just stick to the 'rules' that they use in book work and you won't go far wrong.
 

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