We define entropy to be log of multiplicity. Is it just to reduce a large number like multiplicity or does it have any physical significance? Further, only if entropy is defined so, we get the correct expression for energy and temperature. Please clarify.
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2$\begingroup$ Please note that in English, questions must end with "?" and every sentence must begin with a capital letter. $\endgroup$– DanielSankCommented Dec 28, 2016 at 1:17
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$\begingroup$ If you don't take the log ,then entropy of a single microstate system is not zero ,but we expect it to be zero ,as entropy measures how much we don't know about the system.when there is a single microstate we know everything about the system $\endgroup$– PaulCommented Dec 28, 2016 at 1:56
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2 Answers
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It's because if I have two independent systems the multiplicity of the overall combined system is the product of the multiplicities of the systems: $$\Omega = \Omega_1 \Omega_2.$$
For entropy we need something that adds: $$S = S_1 + S_2.$$ The reason for requiring additivity is that it is the defining property for extensive quantities like mass ($M = M_1 + M_2$), number of particles ($N = N_1 + N_2$), volume ($V = V_1 + V_2$), and energy (for weakly interacting systems, $E = E_1 + E_2$). In general, extensive quantities scale with the size of a system. The opposite is intensive quantities that don't scale with system size, like temperature, pressure, chemical potential, etc.
Add in the requirement that entropy increases monotonically with multiplicity and it leads to: $$S = k \ln \Omega,$$ uniquely.
It is also important that the entropy reproduces the thermodynamic definition of changes in entropy: $$\operatorname{d} S = \frac{\delta Q}{T},$$ but I'll leave linking to or providing a proof of that to others.
answered Dec 28, 2016 at 1:17
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$\begingroup$ Leaning on entropy of two systems adding just pushes the question farther away, in my opinion. $\endgroup$ Commented Dec 28, 2016 at 1:18
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$\begingroup$ Are there any other mathematical functions that satisfy the given requirements? It's my understanding that this leads, uniquely, to a logarithm. $\endgroup$ Commented Dec 28, 2016 at 1:19
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1$\begingroup$ It does lead to a logarithm, but why are we requiring $S_{1+2} = S_1 + S_2$? $\endgroup$ Commented Dec 28, 2016 at 1:20
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1$\begingroup$ @DanielSank One wants an extensive quantity. Like the Clausius entropy. But large parts of the theory can be more easily derived using just Ω. $\endgroup$– user137289Commented Dec 28, 2016 at 1:25
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$\begingroup$ @Pieter "One wants an extensive quantity" <-- Why? $\endgroup$ Commented Dec 28, 2016 at 1:34
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Intuitively, this makes entropies add when we calculate the number of microstates for the combination of two systems: if the two systems are independent (uncorrelated), we simply multiply these numbers together.
Secondly, this is the unique definition that makes the quantity equal to the Clausius definition of entropy, namely as the function of state for which $t^{-1} = \frac{\partial S}{\partial Q}$.See my answer here for a sketch of a proof of this assertion.
Thirdly, and related to my first point above, the logarithm definition gives the quantity an information theoretic meaning as the Shannon information (entropy), i.e. the minimum number of bits (or proportional thereto) needed to describe the microscopic state of a thermalized system given knowledge of its macrostate. See my other answer linked above for further information. This state of affairs is proven by Shannon's Noiseless Coding Theorem if a multiparticle's constituent particles can be taken to be statistically independent.
For example, in the appendix of the seminal paper by E. T. Jaynes:
E. T. Jaynes, "Information Theory and Statistical Mechanics".
sketches Shannon's proof that the entropy, as you;ve defined it, is the unique function, modulo a positive scaling constant, with the three properties that (1) it is always positive, (2) it is a monotonically increasing function of uncertainty about a system's microstate (where the notion of "uncertainty" here is extremely broad and not needfully the multiplicty) and (3) linearly additive for independent sources of uncertainty. The proof is also given in the early chapters of the book Shannon and Weaver, "A Mathemetical Theory of Communication".
answered Dec 28, 2016 at 1:33