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I was reading Nigel goldenfeld's Lectures on Phase Transition and Renormalisation Group and came across the following statement:

'If there were perfect instrumental resolution, a change in the physical properties in a finite system would not occur over an infinitesimal interval of the relevant coupling constant, but would occur over some range. This phenomenon is an example of a finite size effect,..'

Should it not be opposite like with perfect instrumental resolution, an infinitesimal change of relevant coupling constant(parameter) should cause slight changes in physical properties which should be detectable with instruments instead of detecting changes over some range of coupling constant?

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You are focusing on the wrong part of the sentence. The smearing out of the phase transition is not due to the perfect instruments, it is due to the system having a finite size, rather than being infinite. We expect this smearing out to be a real physical effect (I.e. a perfect instrument could detect it) but we typically don't observe this finite size effect due to finite instrumtal precision.

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  • $\begingroup$ By smearing out of phase transition do you mean that instead of the non analyticity of some nth order derivative of free energy at a fixed parameter we have a continuous approximation to the respective derivative $\endgroup$ Aug 27 '19 at 17:56
  • $\begingroup$ I would not say it is an approximation. The partition function of a finite system is completely analytic, so all its derivatives really are continuous. But yes as the system gets larger you will get some sort of continuous interpolation, which approaches the discontinuity in the thermodynamic limit $\endgroup$ Aug 28 '19 at 8:52

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