# Fluctuation-Dissipation Theorem and QED

Staff Emeritus

## Main Question or Discussion Point

Using classical (non-quantum, that is) electrodynamics, one can predict that a charged particle accelerated by a nonuniform electric field will radiate. This can be modeled (although not without problems, such as unphysical runaway solutions) by the Abraham–Lorentz force, which is a dissipative force proportional to the rate of change of the acceleration.

Now, a general argument known as the fluctuation-dissipation theorem says that if a system has a dissipative force, then there must be a corresponding fluctuation. If one applies the fluctuation-dissipation theorem to the Abraham-Lorentz force, one would get a fluctuation in the electromagnetic field, I assume.

If this is true, it's sort of amazing, because the Abraham-Lorentz force is derived from classical electrodynamics. But it seems to point to a result from quantum field theory, that the electromagnetic field inherently has fluctuations.

Does anyone know of a paper working out the fluctuations predicted by the fluctuation-dissipation theorem applied to the electromagnetic field? How does the predicted fluctuations compare with the predictions made by QED?

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Bill_K
Here's the closest thing I can find.

Jano L.
Gold Member
Now, a general argument known as the fluctuation-dissipation theorem says that if a system has a dissipative force, then there must be a corresponding fluctuation.
I do not think that is what the fluctuation-dissipation theorem says. The FD theorem states that the properties of thermal fluctuations are related to damping constant and temperature of the system, provided the damped motion does not get the system far from thermal equilibrium.

For such systems, both damping and fluctuations are present, but it is groundless to think that one is caused by the other. They are just two parts of one thing: the action of the environment on the system. In case the system is in thermal equilibrium, there is just some relation between these two parts.

The phenomenon of radiation damping in classical electromagnetism is a totally different phenomenon, having nothing to do with thermodynamics or fluctuations. The currents in the antenna are damped not because there is some friction due to fluctuations in the EM field, but because one part of the current distribution pulls another part with some retardation, which leads to damping. For example, in the antenna this can be described approximately by expanding the field due to current distribution into powers of ##v/c## and finding the additional component that acts on the current distribution. This leads in the first approximation to the radiation damping electric field proportional to ##\frac{d^3}{dt^3}\mathbf d##, where ##\mathbf d## is dipole moment of the oscillating charge distribution (see Landau&Lifgarbagez, sec. 75). The L-A equation was derived in a similar manner - the additional term was calculated by Lorentz as the part of the force due to the field due to accelerating charge itself.

Staff Emeritus
I do not think that is what the fluctuation-dissipation theorem says. The FD theorem states that the properties of thermal fluctuations are related to damping constant and temperature of the system, provided the damped motion does not get the system far from thermal equilibrium.

For such systems, both damping and fluctuations are present, but it is groundless to think that one is caused by the other. They are just two parts of one thing: the action of the environment on the system. In case the system is in thermal equilibrium, there is just some relation between these two parts.

The phenomenon of radiation damping in classical electromagnetism is a totally different phenomenon, having nothing to do with thermodynamics or fluctuations. The currents in the antenna are damped not because there is some friction due to fluctuations in the EM field, but because one part of the current distribution pulls another part with some retardation, which leads to damping. For example, in the antenna this can be described approximately by expanding the field due to current distribution into powers of ##v/c## and finding the additional component that acts on the current distribution. This leads in the first approximation to the radiation damping electric field proportional to ##\frac{d^3}{dt^3}\mathbf d##, where ##\mathbf d## is dipole moment of the oscillating charge distribution (see Landau&Lifgarbagez, sec. 75). The L-A equation was derived in a similar manner - the additional term was calculated by Lorentz as the part of the force due to the field due to accelerating charge itself.
I'm not saying that the derivation of the L-A force has anything to do with fluctuations. It certainly wasn't. The question is whether there is some necessary connection between dissipative forces and fluctuations that's more general than the setting of the original fluctuation-dissipation theorem.

If one considers a closed system of charged particles and electromagnetic radiation (imagine placing particles and radiation into an idealized perfectly rigid, perfectly reflecting box), the charged particles certainly would be subject to time-varying electric fields. If you modeled this as a stochastic system, then the question is whether the charged particles would experience an effective dissipative force due to the random action of the radiation. Would this dissipative force be the same as the L-A force?

Quantum mechanics is possibly involved here because to treat the radiation stochastically, one would need to have a thermodynamics of radiation, which would require solving the same sort of problem that Planck was faced in developing his theory of black-body radiation.

Jano L.
Gold Member
If you modeled this as a stochastic system, then the question is whether the charged particles would experience an effective dissipative force due to the random action of the radiation.
If such system can be in some kind of equilibrium, then I would expect effective dissipation force too.

Would this dissipative force be the same as the L-A force?
This is hard to say with certainty since solving such model is a hard task, but I think it is very unlikely. The LA force is due to the extended charge body itself - "self-force". It is proportional to ##\dot{\mathbf a}## of the body ; it does not depend on any other variable properties of anything else. Force of this kind can be derived if more charges move together coherently - consider currents in antenna, or accelerated motion of extended body, or oscillations of electrons in a molecule; here the charges move in a very correlated way.

The dissipation force encountered in the FD theorems is very different - it is due to the other bodies (environment), not the body itself and its magnitude depends on the properties of the fluctuations of the environment (temperature). For example, increasing the energy of the system would most probably affect the magnitude of such dissipation force.

Einstein thought of similar model in his paper on the quantum theory of radiation, but instead of charged particles he had neutral molecules. He assumed that the effective friction force due to EM radiation, in order to slow down fast molecules, have to be of the ordinary friction kind ##\mathbf F = - \gamma \mathbf v## with ##\gamma## dependent on the temperature.