I've been told white dwarfs don't produce stellar wind like the sun produces solar winds. I don't quite understand why that is and can't really find any evidence that confirms or denies that.
My understanding is that solar wind is the product of the sun's variable magnetic field and consists of charged particles. Does nuclear fusion have to be occurring at the star's core for these charged particles to exist?
The strong gravity of the white dwarf star makes it difficult for stellar wind to escape. Unglaub (2008) shows that for surface gravity $\log g > 7.0$ hydrogen-helium winds are not possible for solar or sub-solar metallicity white dwarfs and only a weak metallic wind would be possible (note that in astronomy, "metal" tends to refer to any element heavier than helium). There is a group of hot white dwarfs with temperatures above 60000 K called "hot wind white dwarfs" showing ultra-high excitation absorption lines. According to Reindl et al. (2019), these features arise from wind material trapped in the magnetosphere of the white dwarfs.
Stellar winds tend to be either "thermal" in nature, driven by high temperatures in the gas surrounding the star (e.g. the solar wind); or they are "radiatively driven", the wind is accelerated by radiation pressure from the stellar photosphere (e.g. the winds from most hot stars).
White dwarfs are not surrounded by hot coronae - they have comparatively weak X-ray emission, suggesting very little hot plasma surrounding them. They are incapable of supporting thermally driven winds.
White dwarfs can be hot though, especially when young, and emit intense radiation from their surfaces. Whether a radiatively driven wind can be setup depends on a competition between the outward acceleration provided by radiation pressure and the inward acceleration due to gravity.
The outward radiative acceleration depends to first order on luminosity divided by distance from the star squared. But the luminosity depends on the radius of the star squared times its surface temperature to the fourth power. That means if we express distance as a multiple of the stellar radius, the radiative driving at a set number of stellar radii only depends on $T^4$.
Gravity on the other hand depends on the mass of the star divided by the square of the distance from it. So the gravitational force at a set number of stellar radii increases as the inverse square of the absolute stellar radius.
Thus we can compare a hot white dwarf with a B star. The B star might have a mass of $\sim 10 M_{\odot}$ and a radius of $\sim 10 R_{\odot}$ and $T\sim 20000$ K; it will have a strong, radiatively driven wind.
A young white dwarf at the same temperature has a mass of $\sim 1M_{\odot}$ and radius $\sim 0.01R_{\odot}$.
If we consider a point 1 stellar radius above the surface of each star, then material at this point will be outwardly accelerated by approximately the same radiative flux. However, the inward gravitational force on that material will be a hundred thousand times higher for the white dwarf.
That is why most white dwarfs do not have strong winds.
Why most? Well white dwarfs can be hotter than this. There are a fraction of much hotter white dwarfs ($T>60000$ K) that do show high excitation UV emission lines with asymmetries suggesting weak mass loss via a wind. Unglaub & Bues (2000) discuss theoretical calculations of mass loss rates for very young and hot white dwarfs. At $T\sim 80000$ K, the radiative accelerations are increased by a factor of 250 over the case considered above. Simultaneously, hot white dwarfs are not completely degenerate and are larger (by factors of a few) than cool white dwarfs of the same mass (see Do white dwarfs lose mass as they fade to black dwarfs? Is there a correlation between temperature, mass, and radius?) and thus the gravitational accelerations are smaller by an order of magnitude. This allows the possibility (for white dwarfs with helium envelopes) of weak winds (of order $<10^{-11}\ M_{\odot}$/yr, which fade drastically on timescales of a few hundred thousand to a few million years as the white dwarf cools, the radiative flux decreases and the surface gravity increases. This level of mass loss is insufficient to affect the white dwarf's subsequent evolution.
One thing to be aware of is that prior to becoming white dwarfs, low-mass stars shed their envelopes as planetary nebulae. Then the star itself is called the central star of a planetary nebula (CSPN), which is a star on its way to becoming a white dwarf. These stars are very luminous so they have strong winds, and indeed that's the tail end of the creation of the planetary nebula. Also, the cores of these stars are degenerate, so they are basically white dwarfs that are wearing a "coat", and they are in the process of shedding that coat. The act of shedding that coat sounds a lot like a white dwarf core that has a very strong wind-- the wind is the envelope being shed. So you could say that white dwarfs have a history of very strong winds, but if you don't count it as a white dwarf until it has finished shedding that envelope, then you are basically saying you are defining the white dwarf to be the star that no longer has much of a wind. So that's why white dwarfs don't have much in the way of winds, they've already lost everything that isn't "nailed down." But since there is not necessarily clear demarcations between a CSPN, a hot white dwarf, and a cold white dwarf, whether you count it as a white dwarf with a wind, or a bare white dwarf that has already lost everything it will lose, kind of depends on where you draw those lines.
