# Huh? Condensates can exchange particles without effect?

• jshrager
In summary, the conversation discusses the exchange of particles in a condensate, specifically in the context of a fermion exchanging with a boson. The concept of the energy of the condensate not changing when a virtual particle is absorbed into it is also explored, and the idea of the emitting particle being part of the condensate is proposed to resolve this apparent impossibility. The conversation also touches on the topic of the thermodynamic limit and the calculation of energy density for a relativistic or non-relativistic ideal Bose gas. However, the meaning of the initial question about the exchange of particles in a condensate is unclear.
jshrager
Gold Member
I have now seen it repeated multiple times that a particle (a fermion, perhaps?) moving in a condensate can exchange particles (bosons, most probably) "without effect" -- the version of this that I run into usually goes something like that the energy of the condensate does not change AT ALL when, say, a virtual photon (or z-boson) is absorbed into a BEC. Now, first off, hunh? Does this really mean that the energy of the condensate doesn't change when something is absorbed by it? How can that be? And if so, what aspect of what equation(s) tell us this? One possible way that I can resolve this apparent impossibility is that to require that the emitting particle be a part of the condensate to begin with, so that the absorption is balanced by the emission, but this doesn't seem to be what is being said.

One usually assumes that the condensate is infinitely large, so its total energy is infinite or at least undefined anyhow.

If you go into the thermodynamical limit ##V \rightarrow \infty##, total energy, total particle number etc. get undefined, and only the corresponding densities make sense. You also must say, whether you look at relativistic particles, where energy is usually defined as including the rest-mass contribution, ##E=\sqrt{m^2+p^2}## (setting ##c=\hbar=1##), or a non-relativistic gas.

For an ideal relativistic Bose gas you have
$$f(\vec{p})=(2 \pi)^3 n_c \delta^{(3)}(\vec{p})+\frac{1}{\exp[(E(\vec{p})-m)/T]-1},$$
where ##n_c## is the particle-number density of the particles in the condensate, ##m## the mass of the particles and (in the thermodynamic limit) ##\mu=m##.

For the energy density, it follows
$$\epsilon=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} E(\vec{p}) f(\vec{p})=m n_c + \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} E(\vec{p}) \frac{1}{\exp[(E(\vec{p})-m)/T]-1}.$$
Thus, for a relativistic gas, the energy density for the particles in the condensate is ##n_c m##.

The calculation is the same for a non-relativistic ideal Bose gas. The only difference is that one doesn't include the rest-mass energy in the energy but uses kinetic energy ##E(\vec{p})=\vec{p}^2/(2m)##, and then (in the thermodynamic limit) you have ##\mu=0## and thus
$$f(\vec{p})=(2 \pi)^3 n_c \delta^{(3)}(\vec{p}) + \frac{1}{\exp[E(\vec{p})/T]-1},$$
and the energy density of the condensate particles is 0.

Concerning the question in the OP, I don't know, what it means. Of course interchanging a particle within the BEC doesn't change anything, because all these particles are completely indistinguishable. I don't get the idea about a fermion exchanged with a boson at all. Of course, you can consider mixtures of fermions with bosons, where the bosons can be in a condensate, but I can't make sense about the question also in this context.

## 1. What are condensates?

Condensates are states of matter that form when a large number of particles, such as atoms or molecules, come together and behave as a single entity. They can exist at extremely low temperatures, close to absolute zero.

## 2. How do condensates exchange particles without effect?

In a condensate, particles are in a state of quantum superposition, meaning they can exist in multiple places at once. This allows them to exchange particles without changing the overall state of the condensate.

## 3. What is the significance of this phenomenon?

This phenomenon, known as superfluidity, has important implications for our understanding of quantum mechanics and can potentially lead to advances in technologies such as superconductors and quantum computers.

## 4. Can this phenomenon be observed in everyday life?

No, this phenomenon can only be observed at extremely low temperatures and in highly controlled laboratory conditions. It is not something that occurs in our daily lives.

## 5. How does this relate to other states of matter?

Condensates are considered a fourth state of matter, alongside solid, liquid, and gas. They possess unique properties that distinguish them from the other states, including the ability to exchange particles without effect.

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