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From the point of view of statistical thermodynamics, a single particle doesn't have a phase (state of matter), nor temperature.

What would happen if heat is transported to this single particle via radiation? How is this related to the same thing occurring for a mass of particles?

In addition, how can a single-molecule transistor, for example, depend on temperature via $f_{\alpha} = \left[ 1+\exp \left( \frac{\epsilon-\mu_{\alpha}}{k_B T} \right) \right]^{-1}$, if temperature is not even defined for a single molecule?

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Sparkler
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A single particle can be a 'system' within itself having modes depending on the particle ' s structure. These modes may be in 'tune' with the incident radiation and thus capture the energy which can increase the particle ' s momentum and therefore its velocity. We never say the particle's temperature has increased but rather it's momentum. When a system of particles absorb radiation, the increased momenta and higher mean velocity lead to the statistical measure of temperature.

docscience
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Can a single particle be “heated” by radiation?

Not really, because heat is an emergent macroscopic property of an ensemble of particles. But since you put the word "heated" in quotes, we can allow a yes of sorts. Take a look at the Wikipedia temperature page, where we can see this picture:

enter image description here CCASA image by Greg L, see Wikipedia

It's to do with the kinetic theory of gases. A hot gas is one where the molecules are moving fast. And radiation will heat this gas. It will makes the molecules move faster. The simplest version of this is Compton scattering, see Rod Nave's hyperphysics:

enter image description here

The single particle is "heated" by radiation because a fast electron is a "hot" electron. Google it.

From the point of view of statistical thermodynamics, a single particle doesn't have a phase (state of matter), nor temperature.

Yes, but thermodynamics isn't just statistical dynamics. Look at the name and its etymology. "It was first spelled in a hyphenated form as an adjective (thermo-dynamic) in 1849 and from 1854 to 1859 as the hyphenated noun thermo-dynamics to represent the science of heat and motive power". Motive power. Motion.

What would happen if heat is transported to this single particle via radiation?

Its energy is increased. But this is kinetic energy rather than rest-mass energy.

How is this related to the same thing occurring for a mass of particles?

If the particles are confined inside a box, their kinetic energy is effectively rest-mass energy. They aren't actually at rest, but the average position of the particles with respect to you doesn't change. The mass of the system increases because of the added energy. A box full of fast-moving particles is harder to accelerate than a box full of slow-moving particles.

John Duffield
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Phase and random kinetic energy are macro. If you don't call the kinetic energy of a single molecule temperature then fine but it is still kinetic energy.

In a single molecule you have rotation and vibration that can be effected by radiation. You can have an exited state where an electron is temporarily bumped to a higher orbit. Some reactions only take place when the molecule is in an excited state.

You can hit it with enough energy that it breaks up.

paparazzo
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