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

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SUMMARY

The discussion centers on the transition from summation to integration as presented in Reif's "Statistical Mechanics," specifically in Equation 1.5.17. The justification for this change lies in the approximation of discrete values as continuous, a fundamental concept in calculus. Participants emphasize that while basic calculus principles apply to many physical scenarios, not all mathematical functions are continuous and differentiable, which can limit the applicability of these methods. A deeper understanding of mathematical analysis is recommended for discerning when to apply calculus effectively.

PREREQUISITES
  • Understanding of basic calculus concepts, including limits and continuity.
  • Familiarity with the principles of integration and differentiation.
  • Knowledge of statistical mechanics as outlined in Reif's textbook.
  • Basic mathematical analysis to differentiate between continuous and discrete functions.
NEXT STEPS
  • Study the relationship between summation and integration in calculus.
  • Explore the concept of limits and their role in defining continuity.
  • Learn about mathematical analysis techniques relevant to statistical mechanics.
  • Review advanced calculus topics, focusing on differentiability and continuity of functions.
USEFUL FOR

Students and professionals in physics, particularly those studying statistical mechanics, as well as mathematicians interested in the application of calculus in physical theories.

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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