Bose-Einstein condensation in a Hyperbolic model for the Kompaneets Equation
In low-density or high-temperature plasmas, Compton scattering is the dominant process responsible for energy transport. Kompaneets in 1957 derived a non-linear degenerate parabolic equation for the photon energy distribution. In this paper we consider a simplified model obtained by neglecting diffusion of the photon number density in a particular way. We obtain a non-linear hyperbolic PDE with a position-dependent flux, which permits a one-parameter family of stationary entropy solutions to exist. We completely describe the long-time dynamics of each non-zero solution, showing that it approaches some non-zero stationary solution. While the total number of photons is formally conserved, if initially large enough it necessarily decreases after finite time through an out-flux of photons with zero energy. This corresponds to formation of a Bose-Einstein condensate, whose mass we show can only increase with time.
Key words and phrases:Bose-Einstein condensation, Kompaneets equation
2010 Mathematics Subject Classification:35Q85, 35L04, 35L60.
In low-density (or high-temperature) plasmas, Compton scattering is the dominant process responsible for energy transport. In 1957, Kompaneets  derived an equation modeling this and was allowed to publish his work because it was considered useless for weapons research. Today Kompaneets’ work has applications studying the interaction between matter and radiation in the early universe, the radiation spectra for the accretion disk around black holes, and various other fundamental phenomena in modern cosmology and high energy astrophysics [1, 15, 14].
In his work (see also ), Kompaneets derived a Fokker–Planck approximation for the Boltzmann–Compton equation in the setting of a spatially uniform, isotropic, non-relativistic plasma at a constant temperature assuming the heat capacity of photons is negligible, and the dominant energy exchange mechanism is Compton scattering. In this setting Kompaneets showed that the evolution of the photon density is given by
Here and are non-dimensionalized energy and time coordinates respectively. While the exact normalization in these coordinates is not important for the subsequent analysis, we remark that is proportional to the magnitude of the three dimensional photon wave vector. Consequently, is a radial variable, and the total number and total (non-dimensionalized) energy of the photons are given by
The boundary conditions associated to (1.1) are a little delicate. First, near it is natural to assume the incoming photon flux vanishes:
Near , the diffusion is degenerate and it is not clear a priori whether a boundary condition can be imposed. We will revisit the boundary condition at later.
Equation (1.1) formally possesses an entropy structure and dissipates the quantum entropy
This suggests that as , solutions to (1.1) approach an equilibrium for which
All non-negative solutions of this are given by , where
This leads to an interesting conundrum. Multiplying (1.1) by and integrating shows that the total photon number is formally a conserved quantity. Since solutions to (1.1) are expected to approach one of the stationary solutions above, the total photon number should equal the total photon number of some equilibrium solution . However,
and hence the total photon number in equilibrium is bounded above. Thus, if we start with more than photons, the total photon number will not be conserved.
Previous works by a number of authors suggest that a concentration of photons at low energy can develop, and may cause an “out-flux” of photons near the boundary [3, 5, 6, 7, 13, 16]. This is often interpreted as forming a Bose-Einstein condensate: a collection of zero-energy photons occupying the same quantum state. While the existence of such condensates was predicted in 1924 by Bose and Einstein, they were only exhibited experimentally for photons in 2010 by Klaers et al. , in circumstances dominated by physics different from Compton scattering. Actually, the Kompaneets equation (1.1) neglects physical effects, such as Bremsstrahlung radiation, which may act to damp the low-energy spectrum and suppress any out-flux at . Yet it remains interesting to investigate the behavior mathematically obtained from the dynamics of the pure Kompaneets equation (1.1) in order to understand how Compton scattering acts to create a photon flux toward low energy. Following terminology developed in earlier works, we will refer to any out-flux at as a contribution to a Bose-Einstein condensate at zero energy.
Mathematically, it was demonstrated by Escobedo et al.  that there do exist solutions for which no-flux boundary conditions at break down at some positive time, and an out-flux develops at this time. Moreover, a unique global solution continues to exist subject to a boundedness condition for on . However, a complete mathematical understanding of the behavior of the Bose-Einstein condensate and the long-time dynamics of solutions of (1.1) is still unresolved.
In an attempt to understand this problem better, many authors have studied simplified versions of (1.1). In , the authors considered a hyperbolic model obtained by dropping the term in (1.1). This makes an outflow boundary and the total photon number becomes a non-increasing function of time. This model, however, has no non-trivial stationary solutions, making the dynamics unphysical.
In  (see  for a published summary) the authors considered a linear model obtained by dropping the term in (1.1). In this case, solutions dissipate an associated entropy and the stationary solutions correspond to the classical statics. However, the no flux boundary condition at is automatically satisfied, without being imposed. Thus the total photon number is always conserved in time and no condensation can occur.
Finally, in  the authors consider the non-linear Fokker-Planck equation obtained by dropping the linear term in (1.1). This leads to dynamical behavior that is more like that which one expects for (1.1). They show that solutions are uniquely determined without imposing a boundary condition near , and obtain a complete description of the long-time behavior. In particular, the authors show that the total photon number is non-increasing in time, and as the solution converges to an equilibrium state of the form , for . However, because is unbounded, they work on the finite interval and impose a no-flux boundary condition at .
In this paper, we consider a purely hyperbolic model obtained by rewriting (1.1) in terms of the number density , and then neglecting a diffusive term which was found in  to have a negligible contribution to flux in the limit of small . This results in a system that is quite attractive from a dynamical point of view, and does not have many of the deficiencies described above. Indeed, the system we obtain has an infinite family of localized stationary solutions, the largest of which asymptotically agrees with the classical Bose-Einstein statistics near . Further, for this system, every solution converges to some equilibrium solution as , and the total photon number is a decreasing function of time. Being hyperbolic, this system naturally allows a non-zero out-flux of photons at corresponding to the formation of a Bose-Einstein condensate. We believe that the methods that we develop for our analysis may prove useful for study of the full Kompaneets equation and other models with related behavior.
To derive the model we study, let be the photon number density. Equation (1.1) now becomes
Neglecting the dissipation term in the flux gives us the model equation
on the domain . From physical considerations we impose the boundary condition
As we will see shortly, no boundary condition is required at . For convenience, we will work with solutions initially having compact support in . (This property is preserved for all time .)
The system (1.4)–(1.5) is a nonlinear hyperbolic problem with a position dependent flux. Following , it is natural to restrict our study to entropy solutions of this system. For clarity of presentation we postpone the definition of entropy solutions to Section 3 and present our main results below.
Our first result shows that (1.4)–(1.5) admits a unique entropy solution, without imposing a boundary condition at . This solution approaches a stationary solution as , and the total photon number is non-increasing as a function of time.
Additionally, this solution (denoted by ), satisfies the following properties:
There exists a unique such that
Here are (all) the equilibrium entropy solutions, and are defined by
The total photon number
is a non-increasing function of time.
Observe no boundary condition is imposed (or required) at the left endpoint , and we will directly prove uniqueness of non-negative entropy solutions without any flux condition at . As we will see a possible out-flux can occur at leading to a concentration of photons at zero energy (i.e. energy that is negligible on the scales described by the Kompaneets model). As remarked earlier, we interpret this out-flux as a contribution to a Bose-Einstein condensate. Our result above shows that if the Bose-Einstein condensate forms, it can only increase in mass.
We remark that the classical Bose-Einstein statistics postulate that the equilibrium photon energy distribution is
for . Near the origin, is linear for and quadratic for . All these solutions decay exponentially as . Because we neglect the diffusion term in (1.3), our equilibrium solutions no longer take this classical Bose-Einstein form. But they have similar asymptotic behavior for small : is linear near the origin for and vanishes in the interval for . All our equilibrium solutions are compactly supported, and are identically for .
Note Theorem 1.1 only guarantees the total photon number is decreasing. We can, however, obtain a more precise description of this phenomenon.
Physically, this means that photons can only be “lost” to the Bose-Einstein condensate, and not to infinity. Deferring the proof of Proposition 1.2 to Section 3, we now exhibit situations where the Bose-Einstein condensate must necessarily form in finite time. This is our next result, the proof of which is presented in Section 2.4.
In general, even though the system approaches one of the equilibria , we have no way of determining which one. We can, however, establish a non-zero lower bound on the total photon number in equilibrium. Below, we use the notation and .
Plan of this paper
This paper is organized as follows. In Section 2 we prove Theorem 1.1 and Corollaries 1.3 and 1.4. Our proof relies on several lemmas and uses the notion of entropy solutions á la . Even though this is now standard, it involves a number of technicalities to adapt the results to the present situation. Thus for clarity of presentation, we define entropy solutions and prove Proposition 1.2 (and the comparison and contraction lemmas) in Section 3. Finally in Section 4 we construct the appropriate “sub” and “super”-solutions required to control the long time behavior of the system.
2. Proof of the main theorem
Our goal in this section is to prove the main theorem. The proof consists of several ingredients, some of which are technical. For clarity of presentation we briefly explain each part in a subsection below, and then prove Theorem 1.1. Due to its technical nature we postpone the definition and proof of existence of entropy solutions to Section 3.
2.1. Stationary solutions
We begin by computing the stationary solutions.
Clearly if is a stationary solution to (1.4), then we must have for some constant . Since our boundary condition requires the incoming flux to vanish as , we must have . Thus looking for non-negative solutions to yields
We show in Lemma 2.3 that at points of discontinuity, entropy solutions (with compactly supported initial data) can only have upward jumps. Combined with (2.1) this immediately proves the lemma as desired. ∎
2.2. Regularity of Entropy Solutions and Compactness
In the proof of Lemma 2.1 we used the fact that entropy solutions can only have upward jumps. In fact, a much stronger result holds: the derivative of an entropy solution is bounded below, which leads to a BV estimate. Since this stronger fact will be used later, we state the lemmas leading to this result next.
Let be an entropy solution to (1.4) with non-negative initial data which is supported on for some . Then for any , a one-sided Lipschitz bound holds: whenever ,
where depends only on and and is increasing as a function of .
2.3. An contraction estimate, and uniqueness
The next step in proving (1.7) is to show that the -distance between and every cannot increase with time. We do this by proving a contraction estimate which also takes into account inflow and outflow flux.
Let and be two non-negative bounded entropy solutions of (1.4) with initial data. For any , we have
Namely, the distance from to between two solutions, as well as between the outgoing fluxes at , is controlled by the initial distance between the data and between the incoming fluxes at . We will later show that the incoming flux at vanishes, provided is greater than any value in the support of .
We also remark that -contraction estimates are well-known in the case when the flux is independent of position (see for instance ). However, for a position dependent flux, this estimate is not easily found in the literature, and for completeness we present a proof in Section 3.2.
As remarked above, in order to use Lemma 2.5, we need to show that the incoming flux from the right vanishes at any point beyond the support of .
If and is a non-negative entropy solution to (1.4) with initial data that is supported in , then for all the function is also supported in .
In fact, support of shrinks to the interval as , as shown at the end of the proof of Lemma 2.6.
Proposition 2.7 (Uniqueness).
If is compactly supported and non-negative, there is at most one non-negative entropy solution to (1.4) with initial data .
2.4. Proofs of the main results
Using the above, we now prove the results stated in Section 1. We begin with the main theorem.
Proof of Theorem 1.1.
We next prove part (1) of the theorem. First by Lemma 2.4 we know is relatively compact in . Thus, to show that converges in as , it is enough to show that subsequential limits are unique. For this, let be a sequence of times and be such that in . We claim that is independent of the sequence . Indeed, let
Thus must converge as , and we define
By assumption, since in we must also have
Of course, is independent of the sequence , and so if we show that can be uniquely determined from the constants we will obtain uniqueness of subsequential limits.
To recover from , note that Lemma 2.2 implies and hence
and hence, for a.e. ,
showing can be uniquely recovered from . This shows that in as . We note that is uniformly bounded for for each given as shown in the proof of Lemma 2.2 in Section 4.1. Using the convergence and standard dominated convergence arguments, we can pass to the limit through the integrals in (3.1) and (3.2) to show that is a stationary entropy solution to (1.4). Hence, there is some such that . This proves (1.7).
Part (2) of the theorem asserts that the total photon number is a non-increasing function of time. To prove this observe that the total photon number is given by
since and . By the contraction principle (Lemma 2.5) the right hand side is a non-increasing function of time, and hence the same is true for the total photon number, finishing the proof. ∎
The loss formula (1.9) uses techniques developed in the proof of Lemma 2.5 and we defer the proof to Section 3.2. Instead, we turn to Corollary 1.3 and show that if we start with a total photon number larger than , then the Bose-Einstein condensate must form in finite time.
Proof of Corollary 1.3.
Using (1.7) we see that
Consequently if , then at some finite time we must have
as desired. ∎
Finally, we prove Corollary 1.4 and show that for any non-zero initial data, the equilibrium solution approached is not identically . For this we need a comparison principle.
Lemma 2.8 (Comparison principle).
Let and be two non-negative entropy solutions to (1.4) with compactly supported initial data and respectively. Then if on , then on for any , .
Proof of Corollary 1.4.
Because , as a consequence of Proposition 1.2 the solution conserves total photon number, therefore
It remains to show that the equilibrium solution is not identically , provided the initial data isn’t either. For this, observe that if for all , then , showing is not identically . Alternately, if at some finite time , then by (1.9) we must have for some . Since the spatial discontinuities of can only be upward jumps (see Lemma 2.3), this forces
and (1.10) now implies is not identically . ∎
3. Entropy solutions
In this section, we define the notion of entropy solutions to (1.4)-(1.5) and prove existence as claimed in Theorem 1.1. We use the entropy introduced by Kruzkov  which takes the family of convex functionals as the entropies.
The function and for each test function , we have the weak formulation
For any and non-negative test function , we have the Kruzkov entropy inequality
The boundary condition (1.5) is satisfied in the sense, that is
for any .
3.1. Contraction and Comparison
In this subsection, we prove a contraction and comparison principle for non-negative, compactly supported entropy solutions to (1.4). Lemmas 2.6 and 2.8 are proved by controlling and , or more generally for some and . Our first lemma is the key step used to establish this.
Let and define . Then for any two bounded entropy solutions to (1.4) and ,
for any non-negative test function .
To begin, we let and be entropy solutions to (1.4). Take a smooth, non-negative function from and consider the weak entropy inequality (3.2) for . Fixing and , we substitute for in the generalization of (3.2) and integrate over and to obtain
By repeating the procedure with the entropy solution with serving the role of , integrating over and , and adding the result to (3.5) and multiplying by , we obtain
Using as the test function in the weak formulations (3.1) for and , we integrate the weak formulation for over and and integrate the weak formulation for over and , multiply each by , and add them together to obtain
We now take an arbitrary, non-negative test function and define a sequence of non-negative test functions in terms of by
Here is the approximate identity defined by
where is such that for and
We note that
Next, we refine Lemma 3.3 to control the difference between two solutions on a finite spatial domain.
Let and define . Let and be entropy solutions of (1.4). Then for any curve ,
In particular, if , then
Here, we generalize the work in [12, Section 3], in which a Lipschitz condition is assumed for the flux . However, for our model (1.4), we have no such condition. Thus, we must retain some terms involving the flux, but will use properties of our particular flux to control these terms for non-negative entropy solutions when we go to prove the contraction and the comparison principle.
For this proof, we use the result of Lemma 3.3 with an appropriate test function. The test function we choose approximates the characteristic function of the time-space domain of integration to mimic the use of the divergence theorem in and (see [2, Chapter 6]). To this end, we define and such that . Let be small, and define the test function by
and is the curve denoting in the right side of the domain in the -plane. Thus,
Note that as , we have
where is the Dirac delta distribution and is the indicator function on the set .
Proof of Lemma 2.5.
We now turn to showing Lemma 2.6, that if a non-negative entropy solution has compact support initially, then it will have compact support for all time.
Proof of Lemma 2.6.
where we have used (3.3) to eliminate the integral with the right-side flux terms. Setting , where is an upper bound for the support of , in (3.16) and using the fact that is a non-negative entropy solution complete the proof.
We can see that on a characteristic, if starts with a value of , it will remain on the characteristic. This observation allows us to strengthen our result from Lemma 2.6. For characteristics starting at values outside the support of , we will have
with along the characteristics given by (3.17). We also note that the equation for the characteristic is a logistic equation, so if , then as , decreases to along the characteristic starting at . Define such that and , where is the support of . Substituting this into (3.16) we obtain