How to Conceptually Understand Long Wavelength Fluctuations? I have been trying to conceptually understand long-wavelength fluctuations of degrees of freedom, and I have been reading this (RG) to do so. I understand was it means for a degree of freedom to fluctuate, but I do not understand what it means for a degree of freedom to undergo long-wavelength fluctuations. How do I conceptually understand long-wavelength fluctuations of degrees of freedom and what causes a degree of freedom to undergo long-wavelength fluctuations? Examples in physical models will help.
 A: Consider a string, say on a guitar. By plucking the string, you can get the string to vibrate with different wavelengths. Each oscillation, with its associated wavelength, should be understood as a degree of freedom. Even when you are not plucking the string, it is vibrating, due to thermal fluctuations (i.e. the molecules in the air are hitting the string and lead to vibrations.) These thermal fluctuations are exciting your degrees of freedom (the different oscillation modes of the string), and are largest at long wavelengths. This is because long-wavelength oscillations are easiest to excite, energetically.
Hopefully, that provides you some intuition. This discussion is actually pretty general. For example, take a system of spins. Here, the degree of freedom is no longer the amplitude of the string, but the magnetization $\mathbf{m}(\mathbf{x})$. $\mathbf{m}$ could be a three-dimensional quantity  if the magnetic moments are allowed to point in any of the three directions (see the Heisenberg model), a two-dimensional quantity if the spins are constrained to exist within a plane (see the XY model), or a one-dimensional quantity if the spins are constrained to point along a single axis (see the Ising model.) As in the string example, the magnetization undergoes thermal fluctuations, with most of the excitations existing at long wavelengths, since this is what is most energetically favorable.
In other cases, the order parameter could be a complex number (see superconductivity), or even a tensor (see liquid crystals.)
Why are long-wavelength fluctuations energetically favorable? That's related to Goldstone's theorem, and what your order parameter looks like. Also relevant is how the dimensionality of your system changes the effect of thermal fluctuations (see the Mermin-Wagner theorem.) These are things you will want to look into.
