I've seen order parameter used in two different ways. One is to distinguish between an ordered and an unordered phase, like whether the net magnetization is stable or not. The second way is to distinguish what the magnetization is, up or down.

More broadly, does it just mean a macroscopic observable? For us to see it at large scales, doesn't it have to be relatively stable?

EDIT: by macroscopic observable I mean like the net magnetization, which is an average of a local operator.

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    $\begingroup$ Yes but a particular local observable may not match the average value as it will fluctuate. So for instance, the net magnetization is a macroscopic observable even though it depends on microscopic observables. $\endgroup$ – sebastianspiegel Oct 1 '14 at 21:48
  • $\begingroup$ I would like to see an answerto this question as well. Is there a general and precise definition? $\endgroup$ – becko Nov 29 '17 at 13:00

An order parameter distinguishes two different phases (or orders). In one phase the order parameter is zero and in another phase it is non-zero. It does not have to be macroscopic.

For example, in the BCS theory of superconductivity the order parameter is called the gap $\Delta$. It can be interpreted as the binding energy of a Cooper pair, namely two electrons that become correlated over long ranges due to an attractive interaction. The order parameter, or gap, shows up as a minimum energy that would be needed to excite a single-particle in the system.

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    $\begingroup$ Is there a general definition of order parameter? $\endgroup$ – becko Nov 29 '17 at 16:49

In a wide variety of phase transitions (both for first order and second order transitions) there exists a quantity which is zero below the transition temperature and becomes nonzero above the transition temperature. For example, in a gas-liquid phase transition as T goes to below the coexistence line at constant pressure, the liquid phase that appears has a much higher density than the gas phase. In this case, we can define $(\rho_l - \rho_g)$ as the order parameter!

Please see this Book:

Statistical Mechanics for Chemistry and Material Science - Biman Bagchi (2018)


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