# Chapter 1. Condensate growth term: existence, uniqueness, propagations of moments

This post is based on my paper.

If a dilute gas of bosons, about one-hundred-thousandth the density of normal air, is cooled to a temperature very close to absolute zero (0 K or -273.15C), the gas will be changed into a new state of matter, called Bose-Einstein condensate (BEC). This state of matter was predicted by Satyendra Nath Bose and Albert Einstein in 1924-25.

In the pioneering MIT-BEC experiment (1998), which later led to the 2001 Nobel Prize in physics, one can observe the growth of the condensate after fast evaporative cooling, which cools the gas below the BEC transition temperature. The condensate growth is an interesting dynamical process.

In order to understand the dynamics of BECs and their excited atoms by the kinetic theory point of view , let first us look at the an example of a glass of water (Figure 1). The glass is divided into two parts: water and vapor. The energy of the whole system waver-vapor is increasing from low (water) to high (vapor).

The same thing happens for a BEC (Figure 2). We can see 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 in this Figure:

• excited atoms – excited atoms,
• condensate atoms – excited atoms.

Now, let us assume the following assumption, which facilitates the study of the BEC dynamics: Suppose the BEC is very stable and contains a sizable number of atoms, the interaction between excited atoms is small. As a consequence, the dominant interaction is the one between excited atoms and the BEC. We can see this in Figure 1: $C_ {12}$ is the operator that describes the interaction between excited atoms and the condensate atoms.

The dynamics of the BEC in this case is governed by the following equation, where $f(t,p)$ is the density distribution of excited atoms, at time $t$ with momentum $p$ (we will come back to the derivation of this equation in my joint work with Linda Reichl in Chapter 6):

$\frac{df}{dt}=C_{12}[f], f(0,\cdot)=f_0, (1)$

where the interaction operator is defined as

$C_{12}[f]:=\int _{ \mathbb{R}^3}\int _{ \mathbb{R}^3}dp_1dp_2 [R(p, p_1, p_2)-R(p_1, p, p_2)-R(p_2, p_1, p) ],$

$R(p, p_1, p_2):= |\mathcal{M}(p, p_1, p_2)|^2 [\delta (\mathcal{E}_p-\mathcal{E}_{p_1}-\mathcal{E}_{p_2}) \delta (p-p_1-p_2)][ f(p_1)f(p_2)(1+f(p))-(1+f(p_1)(1+f(p_2))f(p)].$

In the quantum kinetic master equation of Crispin W. Gardinier and Peter Zoller (2000), the condensate growth term – the principal term which gives rise to growth of the condensate – at the limit, has the form of (1).

Let me try to explain why the condensate growth term $C_{12}$ has the form of (1). In order to understand this, we start with the collision operator for classical particles. Suppose that we need to study the density of a spacially homogeneous gas, whose distribution function is $f(t,p)$, which indicates the probability of finding a particle with momentum $p$ and time $t$. Now, 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 that we lose two particles with momenta $p_1$ and $p_2$, and gain two new particles with momenta $p_3$ and $p_4$ (see Figure 3). 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 $f(t,p_3)f(t,p_3)$. 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 $p$. Finally the collision operator has to contain

$\delta(p_1+p_2-p_3-p_4)\delta(\mathcal{E}_{p_1} + \mathcal{E}_{p_2}-\mathcal{E}_{p_3} - \mathcal{E}_{p_4})[f(t,p_3)f(t,p_4)-f(t,p_1)f(t,p_2)].$

The classical Boltzmann equation then reads

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

where $K(p_1,p_2,p_3,p_4)$ is the collision kernel, which depends on the type of gas under consideration.

Figure 3: The collisions of classical particles and fermions

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 microscopic boxes with momenta $p_1$ and $p_2$ collide and after the collision some particles will move to the new boxes $p_3$ and $p_4$. As a result, we will have something similar 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 boxes $p_3$ 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 the fullness of the box $p_i$. The quantum Boltzmann equation for fermions can be written as follows

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

Figure 4: The collision operator $C_{12}$

The question now is how to adapt the above arguments to the case of $C_{12}$ (see Figure 4). Suppose again that two microscopic boxes with momenta $p_1$ and $p_4$ collide but the box $p_4$ 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 boxes $p_2$ and $p_3$. Similar as above, in the collision operator, we will have something similar 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: the bosons in $p_i$ are very friendly and they are trying to attract their friends to their home (Note that the fermions in $p_i$ are selfish and do not welcome their friends to share their home). 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

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

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 $p_3\to p_1+p_2$, that gives

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

and

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

Due to the symmetry of $p_2$ and $p_3$, we can combine them into

$-2\delta(p_2-p_1-p_3)\delta(\mathcal{E}_{p_2}-\mathcal{E}_{p_1}-\mathcal{E}_{p_3})[f(t,p_1)f(t,p_3)(1+f(t,p_2))-f(t,p_2)(1+f(t,p_2))(1+f(t,p_1))]. (3)$

Putting (2) and (3) together, (1) then follows.

The above explication explains why the operator has the name $C_{12}$. The interaction is a $3$-wave interaction of the type $1\leftrightarrow 2$. The case where $p_1+p_2\to p_3+p_4$ is ignored by our assumption, but will be considered in details in Chapter 3.

In this work, we prove that under some assumptions, Equation (1) has a unique, strong, global, positive, radial solution.

In this work, the transition probability $\mathcal M(p, p_1, p_2)$ and the particle energy $\mathcal{E}_p$ are approximated as:

$|\mathcal M|^2=\kappa |p||p_1||p_2|, \mathcal{E}_p=\kappa_2 |p|.$

Here $\kappa_1, \kappa_2>0$ are explicit constants.

Moreover, interestingly, we can also prove that the properties of propagation and creation of polynomial and exponential moments also hold true. For instance, we prove that: Let $f$ be a positive solution the equation (1). Then, there exists a constant $\alpha>0$ such that

$\int_{\mathbb{R}^3}\,f(t,p)|p|e^{\alpha \min\{1,t^{\frac{1}{6}}\}|p|} dp \leq \frac{1}{2\alpha},\quad\forall\, t\geq 0.$

A natural question one can ask is the following: Will all excited atoms fall into the BEC? In other words, could the pretty girl (BEC) make all the boys (excited atoms) fall in love with her? This is the topic of the next chapter:

Could the beautiful, sexy Milady de Winter seduce all the men?

Figure 5: Cover of the journal Scientific American

Reference for the picture:

[1]   http://bec.nist.gov/