Why is the logarithm of the number of all possible states of

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Discussion Overview

The discussion revolves around the mathematical interpretation of the temperature of a system as defined by the derivative of the logarithm of the number of accessible states (Ω) with respect to energy. Participants explore the implications of treating Ω as a discrete function and the conditions under which it can be differentiated, as well as alternative definitions of temperature.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the mathematical implications of differentiating ln(Ω), noting that Ω can only take discrete values, which raises ambiguity regarding its derivative.
  • Another participant asserts that there is no better definition of temperature and explains that energy flows from higher to lower temperatures based on the increase of accessible states in the combined system.
  • A different viewpoint suggests that there are multiple ways to approximate a function defined on discrete integers, leading to different derivatives, but these approximations may be empirically indistinguishable.
  • One participant humorously remarks on the tension between physicists' approaches and mathematicians' expectations, implying that practical outcomes often take precedence over rigorous mathematical definitions.

Areas of Agreement / Disagreement

Participants express differing views on the mathematical treatment of Ω and its implications for defining temperature. There is no consensus on the best approach to handle the discrete nature of Ω or on the necessity of rigorous definitions in this context.

Contextual Notes

The discussion highlights limitations in the treatment of discrete functions and the assumptions underlying the approximation methods used in defining temperature. The ambiguity in defining "reasonable" approximations is also noted.

Who May Find This Useful

This discussion may be of interest to those studying statistical mechanics, thermodynamics, or the mathematical foundations of physics, particularly in relation to temperature definitions and the treatment of discrete systems.

Kiarash
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Temperature of a system is defined as
$$\left( \frac{\partial \ln(\Omega)}{ \partial E} \right)_{N, X_i} = \frac{1}{kT}$$Where Ω is the number of all accessible states (ways) for the system. Ω can only take discrete values. What does this mean from a mathematical perspective? Many people say we have 10^23 particles so Ω is almost continuous function of energy. Why is 10^23 a nice number but 1000 is not? When can one be sure they can differentiate ln(Omega)?

If you agree with me, do you know an alternative accurate definition for temperature?
 
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There's no better definition of temperature. Ω is only approximately equal to the number of accessible states. I don't see why that bothers you.

When two systems are in contact with each other, energy will flow in the direction that increases the number of accessible states of the combined system. A fairly simple argument shows that if we define temperature this way, then energy is flowing from higher temperature to lower. We define Ω and T this way to be able to make this argument.
 
@Fredrik Thank you for your reply. log[Ω] can only take discrete values; so it is ambiguous what is its derivative. I mean there is infinitely many functions that have the same values as Ω has, but they have different derivative. What does it mean when we differentiate log[Ω]; it is not trivial.
 
OK, so the main issue is that there's more than one way to approximate a function defined on a set of integers by a function defined on an interval. I suppose the answer has to be that all reasonable ways to define what we mean by a "best" approximation (e.g. for some n, the nth-degree polynomial that's the best approximation in the least squares sense) give us functions that are empirically indistinguishable from each other.

I haven't tried to verify this, but I also don't think it's necessary. There's obviously some ambiguity in a word like "reasonable", but this is to expected in arguments that are used to find the theory that we're going to use. Such arguments don't have to be rigorous, since the goal is to guess what definitions will be useful, not to prove that the theory is right.
 
Physicists are well-known for doing things that make mathematicians tear their hair out, but that somehow "work", anyway. :-p
 
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