What is the significance of d(lnW) in physical problems?

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Discussion Overview

The discussion revolves around the mathematical expression d(lnW) and its significance in physical problems, particularly in the context of thermodynamics and statistical physics. Participants explore the meaning of the variables involved and the nature of the derivatives used in the equation.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents the equation d(lnW)=Ʃ_{i}(dlnW_{i}/dN_{i})*dN_{i} but expresses confusion regarding the mathematical interpretation.
  • Another participant requests clarification on the meanings of W and Wi and their relationship.
  • A different participant suggests that W is related to thermodynamic concepts, mentioning Maxwell-Boltzmann and Bose-Einstein statistics, but expresses uncertainty about their connection to the original equation.
  • One participant clarifies that the derivatives in the equation are partial derivatives, except for the last one, which is a total derivative, and apologizes for the initial confusion.
  • Another participant agrees with the clarification about total and partial derivatives and relates it to the definition of total differentials in functions of multiple variables.
  • A later reply indicates that the original poster's confusion was primarily due to notation rather than the underlying concepts.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the notation and the mathematical relationships involved. While some clarifications are made, there is no consensus on the broader implications of the equation or its application in physical problems.

Contextual Notes

There are unresolved aspects regarding the definitions of W and Wi, as well as the specific context in which the equation is applied. The discussion also highlights the potential for confusion between total and partial derivatives.

Who May Find This Useful

This discussion may be of interest to those studying thermodynamics, statistical physics, or anyone dealing with mathematical expressions involving differentials in physical contexts.

M. next
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The equation says: d(lnW)=Ʃ[itex]_{i}[/itex](dlnW[itex]_{i}[/itex]/dN[itex]_{i}[/itex])*dN[itex]_{i}[/itex]

I chose the constants as so since I ran across this in a Physical problem, but it doesn't matter. It is the maths that I don't understand.
 
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It would help if you explained what W and Wi meant and how they are related.
 
I guess it is only a mathematical relation. Some rule that am not getting.. But if that would help, summation of Wi's will give W. These are related to thermodynamics particularly statistical physics, "W" can be for Maxwell Boltzmann, Bose-Einstein, and so on.. But I am pretty sure that these that I just defined have nothing to do with the expansion above.
Thanks!
 
Please note that IN THE FRACTION above in my first post (in the question) the derivatives are not total derivatives but instead are partial derivatives, ONLY the derivatives included in the fraction - the last derivative is excluded from this correction, i.e it is a total derivative. Sorry about that.
 
M. next said:
Please note that IN THE FRACTION above in my first post (in the question) the derivatives are not total derivatives but instead are partial derivatives, ONLY the derivatives included in the fraction - the last derivative is excluded from this correction, i.e it is a total derivative. Sorry about that.
Well, that makes more sense. It's just the definition of a total differential. For a function f of n variables

[tex]\mathrm{d}f(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1}^{n}\frac{\partial{f}}{\partial{x_{i}}}\mathrm{d}x_{i}[/tex]

They are often encountered in thermodynamics. Do you have problems actually calculating them, or was it just the notation that was the problem?
 
Thank you for the reply. No, actually it was just the notation. Your reply is just the answer I wanted.
 

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