Chapter 6: Derivation of quantum kinetic equations

This post is based on my work

Let us recall from the previous chapters that the BEC occupies the lowest quantum state, at that point macroscopic quantum phenomena become apparent. Above the BEC are excited atoms, which occupy higher quantum states. The are two types of interactions:

excited atoms – excited atoms: $C_{22}$ (Uehling-Ulenbeck term),
condensate atoms – excited atoms: $C_{12}$ (condensate growth term).

The above model has been remarkably successful in describing several Bose-Einstein Condensate (BEC) dynamical problems, in both the hydrodynamic and collisionless regimes, with extremely broad temperature ranges, and even the description of the condensate growth. It also provides a clear picture of what physical processes are taking place in a partially condensed Bose gas and gives information about the superfluid properties of the system. However, as pointed out by Linda Reichl et. al. (Phys. Rev. A, 2013 & 2014), the description of the thermal component of the BEC is simplified in this system due to the use an approximate particlelike Hartree-Fock excitation spectrum; that makes predictions for transport properties at low temperature inaccurate, for instance, sound speeds at $T = 0$ . As a consequence, according to the breakthrough idea of Linda Reichl, a collision operator is missing in the model!

Figure 1: The three collision operators $C_{12}, C_{22}, C_{31}$

In order to understand the intuition of the derivation of $C_{31}$ . We start with the collision operator for classical particles. Suppose that we need to study the density of a spatially homogeneous gas, whose distribution function is $f(t,p)$ , which indicates the probability of finding a particle with momentum $p$ and time $t$ . Suppose the two particles with momenta $p_1$ and $p_2$ collide and they change their momenta into $p_3$ and $p_4$ . That means we lose two particles with momenta $p_1$ and $p_2$ , and gain two new particles with momenta $p_3$ and $p_4$ . The probability of losing two particles with momenta $p_1$ and $p_2$ is $-f(t,p_1)f(t,p_2)$ and of gaining two particles with momenta $p_3$ and $p_4$ is $f(t,p_3)f(t,p_4)$ . As a result, the collision operator has to contain $f(t,p_3)f(t,p_4)-f(t,p_1)f(t,p_2)$ . Since the collision conserves mass and momentum, we also have $p_1+p_2=p_3+p_4$ and $\mathcal{E}_{p_1} + \mathcal{E}_{p_2}= \mathcal{E}_{p_3} + \mathcal{E}_{p_4}$ , in which $\mathcal{E}_p$ is the energy of the particle with momentum $p$ . As a consequence, the collision operator should be proportional to $\delta(p_1+p_2-p_3-p_4)\delta(\mathcal{E}_{p_1} + \mathcal{E}_{p_2}- \mathcal{E}_{p_3} - \mathcal{E}_{p_4})[f(t,p_3)f(t,p_4)-f(t,p_1)f(t,p_2)]$ . The classical Boltzmann equation then reads

$\partial_t f(t,p_1) = \int_{\mathbb{R}^9}\delta(p_1+p_2-p_3-p_4)\delta(\mathcal{E}_{p_1} + \mathcal{E}_{p_2}- \mathcal{E}_{p_3} - \mathcal{E}_{p_4})[f(t,p_3)f(t,p_4)-f(t,p_1)f(t,p_2)] dp_2dp_3dp_4$

where the collision kernel is omitted for the sake of simplicity.

We will try to apply the above argument for a gas of fermions. For fermions, we cannot consider the collisions of each quantum particle as in the case of classical gas. We then modify the argument as follows: Suppose that two momentum states with momenta $p_1$ and $p_2$ collide and after the collision some particles will move to the new momentum states $p_3$ and $p_4$ . As a result, we will have a term proportional to $f(t,p_3)f(t,p_4)-f(t,p_1)f(t,p_2)$ in the collision operator. However, fermions follow the Pauli Exclusion Principle, which says that if the momentum states $p_3$ and $p_4$ are already full, then particles cannot move from $p_1$ and $p_2$ to $p_3$ and $p_4$ . We therefore modify the collision as

$f(t,p_3)f(t,p_4)(1-f(t,p_1))(1-f(t,p_2))-f(t,p_1)f(t,p_2)(1-f(t,p_3))(1-f(t,p_4)),$

where the factor $1-f(t,p_i)$ indicates that momentum state $p_i$ is occupied. The quantum Boltzmann equation for fermions can be written as follows

$\partial_t f(t,p_1) =\int_{\mathbb{R}^9}\delta(p_1+p_2-p_3-p_4)\delta(\mathcal{E}_{p_1} + \mathcal{E}_{p_2}- \mathcal{E}_{p_3} - \mathcal{E}_{p_4})[f(t,p_3)f(t,p_4)(1-f(t,p_1))(1-f(t,p_2))-f(t,p_1)f(t,p_2)(1-f(t,p_3))(1-f(t,p_4))] dp_2dp_3dp_4.$

The question now is how to adapt the above arguments for fermions to the case of the operator $C_{12}$ for bosons. Suppose again that two momentum states with momenta $p_1$ and $p_4$ collide but the momentum state $p_4$ is hidden inside the condensate: we have a collision between $p_1$ and the condensate. After the collision some particles will move to the new momentum states $p_2$ and $p_3$ . Similar as above, in the collision operator, we will have a term proportional to $f(t,p_2)f(t,p_3)-f(t,p_1)$ . If we apply the same argument as above, the collision operator is proportional to

$f(t,p_2)f(t,p_3)(1-f(t,p_1))-f(t,p_1)(1-f(t,p_2))(1-f(t,p_3)).$

However, different from fermions, bosons do not follow the Pauli Exclusion Principle. Therefore, instead of multiplying with $1-f(t,p_i)$ , we use the factor $1+f(t,p_i)$ and modify the collision as

$f(t,p_2)f(t,p_3)(1+f(t,p_1))-f(t,p_1)(1+f(t,p_2))(1+f(t,p_3)).$

The conservation of momentum and energy in this case are $p_1=p_2+p_3$ and $\mathcal{E}_{p_1}=\mathcal{E}_{p_2} +\mathcal{E}_{p_3}$ . The collision operator should contain

$\delta(p_1-p_2-p_3)\delta(\mathcal{E}_{p_1}-\mathcal{E}_{p_2}-\mathcal{E}_{p_3})[f(t,p_2)f(t,p_3)(1+f(t,p_1))-f(t,p_1)(1+f(t,p_2))(1+f(t,p_3))]$

Denoting this case as $p_1\to p_2+p_3$ , we observe that there are still other two cases $p_2\to p_1+p_3$ and $p_3\to p_1+p_2$ . Due to the symmetry of $p_2$ and $p_3$ , we can combine them into

$-2\delta(p_2-p_1-p_3)\delta(\mathcal{E}_{p_2}-\mathcal{E}_{p_1}-\mathcal{E}_{p_3})[f(t,p_1)f(t,p_3)(1+f(t,p_2))-f(t,p_2)(1+f(t,p_1))(1+f(t,p_3))]$

The form of $C_{12}$ then follows. The interaction is of the type $1\leftrightarrow 2$ . The operator $C_{22}$ describes the process that two momentum states with momenta $p_1$ and $p_2$ collide and after the collision some bosons will move to the new momentum states $p_3$ and $p_4$ . The same argument as above also gives the form of $C_{22}$ . The $C_{22}$ operator describes the $2\leftrightarrow2$ interaction. Let us look carefully into the structure of $C_{22}$ : We call this a four-excitation process due to the fact that there are four particles in each collision. In a four-excitation process, besides the $2\leftrightarrow2$ interaction, there should be another $3\leftrightarrow1$ one. The missing collision operator is, therefore, of the type $C_{31}$ , which describes the process of three momentum states $p_1$ , $p_2$ and $p_3$ collide and particles are moving into a new momentum state $p_4$ .

The new kinetic equation should be (see Figure 3)

$\partial_t f=C_{22}[f]+C_{31}[f]+C_{12}[f].$

Figure 2: The Bose-Einstein Condensate (BEC) and the excited atoms: $C_{22}+C_{31}+C_{12}$

The new model has been tested with several experiments. For example, the speed and lifetime of the first and second sound modes in mono-atomic BECs computed by the new model, with the new collision operator $C_{31}$ , are in agreement with many experimental results. So, $C_{31}$ should be the missing piece of the picture.

In this work, we revisit a method developed by Peletmiskii and Yatsenko to derive kinetic equations for quantum gases.

Mathematical properties of Boltzmann equations in Wave Turbulence and Quantum Kinetics

Minh-Binh Tran's Blog

Chapter 6: Derivation of quantum kinetic equations

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