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It is often stated that we work in thermodynamic limit at the beginning of courses on statistical physics

$$N \to \infty, V \to \infty, \quad\frac{N}{V}=n=\textrm {constant}$$

what is less often stated is why we do it.

Having studied some applied probability to statistical physics, I relate it to the Central Limit Theorem: we're interested in macroscopic extensive quantities made up by contributions of microstates, that we can model as independent, and we need $N \to \infty$ to have small fluctuations of these quantities (since the variance should go as $\frac{1}{\sqrt{N}}$ for CLT). Then the other two hypotheses are needed in order to have a quantity with physical meaning (density)?

Are there any other reasons? Can it be explained in other ways?

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  • $\begingroup$ Unfortunately, the central limit theorem doesn't work. The necessary assumption is called "ergodic hypothesis" and it is not quite as valid as one would wish. $\endgroup$
    – CuriousOne
    Feb 6 '16 at 9:57
  • $\begingroup$ Yes, that's true, I took it for granted, since I usually study systems in which ergodicity is assumed. But I know for example for a harmonic oscillator ergodicity is not true. But my real question is: why do we work in thermodynamic limit? $\endgroup$
    – iacolippo
    Feb 6 '16 at 10:03
  • $\begingroup$ If the goal is to recover thermodynamics as a limit of statistical mechanics, then the fluctuations have to be suppressed completely, which requires infinite system size, but I am not certain that's sufficient or that the central limit theorem applies in general. For the case of phase transitions in low dimensions where critical exponents have to be calculated with renormalization the long range fluctuations may not average out the way the central limit theorem suggests. I could be wrong... $\endgroup$
    – CuriousOne
    Feb 6 '16 at 11:00
  • $\begingroup$ Yes, the goal is the first one in my case. For the case of phase transition, I studied them with large deviations theory and they can be recovered as typical outcome with a trick, but CLT is not recovered in the neighborhood of it :) $\endgroup$
    – iacolippo
    Feb 6 '16 at 11:17
  • $\begingroup$ Sounds about right. Central limits may apply for systems with sufficiently weak correlations... but that's boring, isn't it? $\endgroup$
    – CuriousOne
    Feb 6 '16 at 11:43
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I see at least 3 reasons.

  1. It is only in this limit that macroscopic observables become deterministic.
  2. It is only in this limit that one has equivalence of the different statistical ensembles.
  3. It is only in this limit that one has sharp phase transitions (genuine singularities of thermodynamic potentials).

(The first two properties may fail to hold even in the thermodynamic limit at phase transitions.)

Note that all these properties are necessary if one wants to recover the thermodynamical description.

Of course, all three remain approximately valid (and the error can be sometimes quantified) in large finite systems.

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