Bose-Einstein condensation in relativistic plasma

TL;DR

This paper predicts that relativistic plasma at extremely high temperatures can undergo Bose-Einstein condensation, extending the phenomenon beyond low-temperature atomic systems to plasma conditions.

Contribution

It demonstrates from first principles that relativistic plasma can undergo Bose-Einstein condensation under certain initial conditions, a novel extension of the phenomenon.

Findings

Relativistic plasma can condense at billions of Kelvin.

Necessary conditions for condensation are identified.

Potential laboratory and astrophysical observations discussed.

Abstract

The phenomenon of Bose-Einstein condensation is traditionally associated with and experimentally verified for low temperatures: either of nano-Kelvin scale for alkali atoms [1-3] or room temperatures for quasi-particles [4,5] or photons in two dimensions [6]. Here we demonstrate out of first principles that for certain initial conditions non-equilibrium plasma at relativistic temperatures of billions of Kelvin undergoes condensation, predicted by Zeldovich and Levich in their seminal work [7]. We determine the necessary conditions for the onset of condensation and discuss the possibilities to observe such a phenomenon in laboratory and astrophysical conditions.

Equations

$$f=\frac{2}{{\left(2\pi \hslash \right)}^3}\frac{1}{\exp \left(\frac{E-{\mu }_i}{k{T}_k}\right)\pm 1},$$

$$t_k\simeq A{\left({\sigma }_T{n}_{e}c\right)}^{-1},$$

$$n_{\gamma }>\frac{2\zeta (3)}{{\pi }^2}{\left(\frac{\hslash }{mc}\right)}^{-3}{\left(\frac{k{T}_k}{m_e{c}^2}\right)}^3,$$

$$\frac{1}{c}\frac{\partial f_i}{\partial t}=\sum _q\left({\eta }_i^q-{\chi }_i^q{f}_i\right),$$

$$n_i=\int f_i(\mathbf{p},t)d^{3}p;\frac{d{\rho }_i}{dϵ}=\frac{4\pi }{c^2}|\mathbf{p}|{ϵ}^2{f}_i.$$

$${\eta }_I^{2p}$$

$$\times f_{III}{f}_{IV}\left(1+\xi f_I\right)\left(1+\xi f_{II}\right),$$

$${\chi }_I^{2p}{f}_I$$

$$\times f_I{f}_{II}\left(1+\xi f_{III}\right)\left(1+\xi f_{IV}\right),$$

$${\eta }_I^{3p}$$

$$\times f_{III}{f}_{IV}{f}_V\left(1+\xi f_I\right)\left(1+\xi f_{II}\right),$$

$${\chi }_I^{3p}{f}_I$$

$$\times f_I{f}_{II}\left(1+\xi f_{III}\right)\left(1+\xi f_{IV}\right)\left(1+\xi f_V\right),$$

$$dw$$

$$\times \left(\prod _{in}\frac{\hslash c}{2{ϵ}_{in}V}\right)\left(\prod _{fin}\frac{d^3{p}_{fin}}{(2\pi \hslash {)}^3}\frac{\hslash c}{2{ϵ}_{fin}}\right),$$

Full text

1 Introduction

The phenomenon of quantum condensation of bosons was predicted by Bose and Einstein [8, 9, 10]. In physics textbooks it is associated with cooling to low temperatures [11, 12], and it was indeed observed for ultracold atoms [1, 2, 3] and for quasi-particles [4, 5]. It was also observed for photons in a microcavity [6], where special boundary conditions ensured necessary pre-requisites for Bose-Einstein condensation: (a) photon number conservation or (b) generation of effective mass via spatial confinement [13].

Following the original prediction, Bose-Einstein condensation (BEC) is understood as a quantum phenomenon, occurring in ideal Bose gas of massive particles when the temperature decreases at constant density, or, alternatively, when the density of particles increases at fixed temperature, leading to condensation of a fraction of particles in the lowest energy state. It consists of appearance of a separate phase in a gas whose particles occupy the lowest quantum state. This phenomenon is traditionally associated with low temperatures, as well demonstrated by cooling alkali atoms to nanokelvin degrees [1, 2, 3].

In a pure photon gas such phenomenon cannot occur, because photons are massless particles and cooling leads to disappearance of photons. Nevertheless, it was predicted in opaque plasma in a pioneering work [7]. There it was conjectured that, in absence of photon absorption, the dominant interaction process in a rarefied hot plasma is Compton scattering, which conserves photon number, fulfilling the condition (a). Considering the properties of nonlinear Kompaneets equation [14], which describes time evolution of photon spectrum due to Compton scattering on nonrelativistic electrons with Maxwellian distribution, the mechanism of condensation was illustrated. It was shown that, unlike ideal Bose gases, BEC manifests itself as an excess of photons over the Planck distribution [15], which is only possible at intermediate energies: between the spectral peak and the critical energy, below which absorption dominates. It was proposed that such phenomenon may occur in astrophysical conditions, when hot radiation passes through cold plasma. A related phenomenon called comptonization is indeed observed, albeit in an almost transparent plasma, and is known as Sunyaev-Zeldovich effect [16, 17]. The photon condensation in plasma is a bulk phenomenon, arising in homogeneous and unbounded system; moreover, photons in plasma acquire effective mass [18] ensuring condition (b). The mechanism of BEC of photons in thermodynamic equilibrium with the atoms of diluted gases has been also discussed in [19], while in [20] statistical theory of photon condensation is developed.

The purpose of the present work is to demonstrate out of first principles that for certain initial conditions photons in non-equilibrium optically thick electron-positron plasma undergo BEC. Thus, in contrast with the traditional belief that BEC is related to cooling to low temperatures, we provide an example for photon condensation at very high, relativistic temperatures. We show, that although the condition (a) is violated, as the number of particles changes due to triple interactions, BEC is present as a transient phenomenon both at nonrelativistic and relativistic temperatures.

2 Kinetic versus thermal equilibrium

Considering relaxation of non-equilibrium electron-positron-photon plasma [21], as well as relativistic (average energy per particle exceeds its rest mass energy) plasma with proton load [22] with arbitrary initial conditions it was found that this process occurs in two steps. First, detailed balance is established in two-particle (binary) interactions, such as Compton and Coulomb scattering, pair creation and annihilation in two photons. This balance corresponds to a metastable state called kinetic equilibrium, which is characterized by the same temperature $T_{k}$ of all particles, and nonzero chemical potentials $\mu_{i}$. It is important to stress that condition (a) is satisfied in kinetic equilibrium. The distribution function of particles with energy $E$ in this state has the form

[TABLE]

where $\hbar$ is reduced Planck’s constant $k$ is Boltzmann’s constant, and signs ”$+$” and ”$-$” correspond to Fermi-Dirac and Bose-Einstein statistics, respectively. Kinetic equilibrium is established if the rates of binary interactions exceed the ones of three-particle (triple) interactions, namely relativistic bremsstrahlung, double Compton scattering, radiative pair production and three-photon annihilation. The characteristic timescale of kinetic equilibrium can be estimated as

[TABLE]

where $\sigma_{T}$ is the Thomson cross section, $n_{e}$ is electron number density, $c$ is the speed of light, and the coefficient $A\simeq 20$ [21]. When triple interactions finally come into detailed balance, thermal equilibrium is established and the chemical potential of photons vanishes.

In a recent work [23] quantum statistics of particle was accounted for, and in addition the rates of triple interactions were evaluated directly from the QED matrix elements. It was found that kinetic equilibrium is established in the relaxation process prior to the thermal one only for nonrelativistic plasma, with final thermal equilibrium temperatures $T_{th}<0.3m_{e}c^{2}/k$.

The possibility of condensation of photons in kinetic equilibrium state occurs if the initial number density of photons $n_{\gamma}$ exceeds the one given by eq. (1) with zero chemical potential of photons $\mu_{\gamma}=0$, namely

[TABLE]

where $\zeta\left(s\right)$ is the Riemann $\zeta$-function. Provided that binary interactions do not change the number of photons, these particles tend to accumulate in lowest energy states and form an excess over the Planck distribution as long as kinetic equilibrium is maintained. When the rates of triple interactions become significant, the number of photons reduces and the excess over the Planck distribution tends to disappear and plasma relaxes towards thermal equilibrium described by the distribution (1) with zero chemical potential of photons. The characteristic timescale of thermal equilibrium can be estimated as $t_{th}\sim\left(\alpha\sigma_{T}n_{e}c\right)^{-1}$, where $\alpha$ is the fine structure constant. Detailed calculations show that both kinetic and thermal equilibrium timescales are functions of total energy density [24, 23] and at high temperatures $T_{th}>0.3m_{e}c^{2}/k$ they nearly coincide.

We have found that BEC of photons may occur for a broad class of initial distribution functions, including Gaussian and Wien distributions. In what follows we give several examples.

3 Boltzmann equations

We solve relativistic Boltzmann equations [25] for one-particle distribution functions of electrons $e^{-}$, positrons $e^{+}$ and photons $\gamma$ with quantum corrections described by Uehling-Uhlenbeck collision integrals

[TABLE]

where $f_{i}(\epsilon,t)$ are their distribution functions, index $i$ denotes the sort of particles, $\epsilon$ is their energy, $\eta_{i}^{q}$ and $\chi_{i}^{q}$ are the emission and the absorption coefficients of a particle of type ”$i$” via the physical process labelled by $q$. All binary and triple interactions between particles are taken into account.  The number density and spectral energy density are defined as follows

[TABLE]

The emission and absorption coefficients for the particle $I$ in a binary process $I+II\leftrightarrow III+IV$ have the following form

[TABLE]

where transition rates are $W_{(3,4|1,2)}d^{3}p_{3}d^{3}p_{4}=Vdw_{(3,4|1,2)}$ and $W_{(1,2|3,4)}d^{3}p_{1}d^{3}p_{2}=Vdw_{(1,2|3,4)}$, $V$ is normalization volume, $dw$ is differential reaction probability per unit time, $\xi=\psi h^{3}/2$ and $\psi$ is +1,-1 for Bose-Einstein and Fermi-Dirac statistic, respectively, $h=2\pi\hbar$. The emission and absorption coefficients for the particle $I$ in a triple process $I+II\leftrightarrow III+IV+V$ have the following forms

[TABLE]

where $W_{(3,4,5|1,2)}d^{3}p_{3}d^{3}p_{4}d^{3}p_{5}=Vdw_{(3,4,5|1,2)}$ and $W_{(1,2|3,4,5)}d^{3}p_{1}d^{3}p_{2}=V^{2}dw_{(1,2|3,4,5)}$. The expression for $dw$ is given in quantum electrodynamics as

[TABLE]

where $p_{f\!in}$ and $\epsilon_{f\!in}$ are respectively momenta and energies of outgoing particles, $p_{in}$ and $\epsilon_{in}$ are momenta and energies of incoming particles, $M_{fi}$ is the corresponding matrix element, $\delta$-functions stand for energy-momentum conservation. Therefore, collision integrals, i.e. right-hand side of equations (4), are integrals over the phase space of interacting particles, which include the quantum electrodynamics matrix elements, see e.g. [25, 26] for binary reactions and [27] for double Compton scattering, [28] for relativistic bremsstrahlung and [29] for substitution rules in computation of remaining matrix elements for triple reactions.

4 Numerical results

The coupled system of integro-differential equations (4) is solved numerically using a finite difference method by introducing a computational grid in the phase space to represent the distribution functions and to compute collisional integrals [21, 30, 23]. Thus the system of integro-differential equations is reduced to a system of stiff ordinary differential equations which are solved by the Gear method [25, 31]. Due to a finite numerical resolution, the effective photon mass in plasma [18] always turns out to be outside our grid, therefore we consider photons as massless particles.

Evidently, the possibility of the development of photon condensation strongly depends on the initial state of plasma. An example considered in [7] is the initial state with hot photons and cold electrons, where photons have the Planck spectrum with temperature $T_{\gamma}$, and electrons have the Maxwellian distribution with temperature $T_{e}<T_{\gamma}$. It is expected that Compton scattering redistributes energy between these components so that photons would cool down. Given that the number of particles is conserved, photons are expected to undergo condensation.

It turns out that these initial conditions are not suitable to obtain BEC. In our simulations with cold degenerate electrons and high temperature photons during the relaxation process the cooling of photons is accompanied with the decrease of their number density, due to triple reactions not properly taken into account in [7] in such a way that their condensation does not appear [32]. Besides, at relativistic temperatures annihilation of photons is efficient enough and prevents accumulation of photons at low energies. Therefore, the necessary condition for BEC of photons cannot be met with such initial conditions.

We found that in order to favour BEC, initial distribution of photons should not be broader than Wien spectrum, and the peak of the distribution should be above the critical energy, below which triple interactions dominate over the binary ones. Initial state for pairs can be arbitrary, and for high temperature plasma pairs initially can be even absent: they are quickly produced from photons. As Coulomb interactions are much faster than Compton scattering and two-photon creation/annihilation [22], initial distribution function of pairs acquires the Fermi-Dirac form well before balance is achieved for Compton scattering and two-photon creation/annihilation, so photons interact essentially with Maxwellian electrons.

Our results show that photon condensation appears as a transient state both in non-relativistic and relativistic plasma, yet each case has its own peculiarity. In the non-relativistic case, we present a particular result with total energy density $\rho_{tot}=8.7\times 10^{20}\,\text{erg cm}^{-3}$ corresponding to a final equilibrium temperature $\theta=k_{B}T/m_{e}c^{2}=0.098$. Total initial particle number density is $n_{tot}^{in}=5n_{tot}^{f\!in}$, where $n_{tot}^{f\!in}=3.5\times 10^{27}\,\text{cm}^{-3}$ is the final total particle number density in thermal equilibrium. The initial energy density spectrum is a Wien distribution centered at the energy $\varepsilon=0.06m_{e}c^{2}$ (the 27th energy grid node) for photons and a single peak (delta function) for pairs located at the energy $\varepsilon=0.001m_{e}c^{2}$ (the 1st energy grid node). Both energy density and particle number density of pairs are much smaller that that of photons. The time evolution of energy density and particle number density is presented on Fig. 1, while Fig. 2 shows spectral energy density and emission and absorption coefficients (reaction rates) for selected time moments.

The energy conservation ensures that the total energy density does not change with time. The total particle number density changes only due to the imbalance in triple processes, e.g. bremsstrahlung. Before the time moment $t\simeq 10^{-12}$ sec the triple processes play no significant role. Both photon number and pair number are almost unchanged; photon annihilation is also not essential (most photons have energy less than $m_{e}c^{2}$). The photon energy density slightly decreased and the pair energy density slightly increased due to Compton scattering. After the time moment $t\simeq 10^{-12}$ sec the total particle number starts to decrease. As a result at the time moment $t\simeq 10^{-11}$ sec the total particle number density is $n_{tot}\simeq n_{\gamma}\simeq 6.5\times 10^{27}\text{cm}^{-3}\simeq 1.85\ n_{tot}^{f\!in}$. Binary processes are balanced in a broad energy region implying kinetic equilibrium is established and photons are described by the Bose-Einstein distribution function (1). The maximum number of photons supported by the equilibrium with pairs given by eq. (3) is $3.8\times 10^{27}\text{cm}^{-3}$. The photon number density at this moment is as large as $6.5\times 10^{27}$ cm*-3* implying that the excess of photons with the density $2.7\times 10^{27}\text{cm}^{-3}$, comparable to the number density of non-condensed photons, should form a condensate. This excess is indeed visible in the middle panel of Fig. 2. At low energies bremsstrahlung and double Compton scattering are more efficient than single Compton scattering, which results in a steep decrease of the spectrum. Small deviations from the Planck spectrum seen in the right panel of Fig. 2 in the final state are within the numerical accuracy obtained on the grid (logarithmic and homogeneous, correspondingly) with 60 intervals in energy and 24 intervals in angles used for the computation. As long as triple interactions are not balanced, thermal equilibrium is not achieved until about $\sim 10^{-8}$ sec when the condensation disappears completely. It implies the condensate is sufficiently long lived state, in comparison with the kinetic equilibrium time $t_{k}$. Similar results were obtained with different initial conditions, in particular for Gaussian distribution of photons $d\rho/d\epsilon=a\epsilon^{3}exp\{-(\epsilon-b)^{2}/c^{2}\}$ with $a=10^{48}~{}\text{erg}^{-3}\text{cm}^{-3}$, $b=3\times 10^{-7}~{}\text{erg}$, $c=6\times 10^{-8}~{}\text{erg}$, implying that the phenomenon is quite generic.

We also studied the development of BEC for different degree of initial degeneracy, by increasing the initial photon number density, which exceeded equilibrium value by a factor 3, 5, 7 and 10. We found that the system loose memory of initial distribution at the moment when number density starts to change due to triple interactions, at $10^{-11},10^{-12},5\times 10^{-13},2\times 10^{-13}$ sec, correspondingly. At this moment, which we call pre-condensation, the photon distribution functions still have non-equilibrium shape, but power law excess over Planck spectrum starts to appear. In all cases the condensation (with Planck spectrum and the corresponding excess at intermediate energies) occurs at $10^{-11}$ sec, and thermalization finishes at $10^{-8}$ sec, independent on the degree of degeneracy.

Finally, we present the results for the relativistic case with a total energy density $\rho=2.1\times 10^{27}$ erg/cm3, corresponding to the final equilibrium temperature $\theta=3$. The initial state is pairless and photons are placed at the energy node with $\varepsilon=1.06m_{e}c^{2}$.

Like in nonrelativistic case, BEC develops as a transient state before the final thermal equilibrium is established. However, unlike the nonrelativistic case, the photon spectrum does not show the characteristic power law, but has an excess over the associated Planck spectrum in a form of a bump. This case is interesting, since it is known [23] that triple interactions in plasma with relativistic temperature are faster than binary ones, see Fig. 3. Yet, condensation occurs, as triple interactions are not fast enough to remove photons in excess over the Planck spectrum by the time when spectrum acquires an equilibrium shape.

Note that BEC in a microcavity [6] represents an optical analogy of photon condensation discussed in this work. Indeed, photon number in a microcavity is conserved and after photon thermalization via interaction with a solution, photons in excess over the thermal number density undergo condensation. We also point out that in experiments with microcavity injected photons initially have a narrow energy distribution, similar to the conditions found in our work. Moreover, there is an absorption of photons by the walls of the cavity, so that photon condensate also represents a transient phenomenon.

5 Conclusions

In this work we presented the results of the first principles calculations demonstrating BEC of photons in relativistic plasma. This phenomenon was predicted in 1969 and still awaits confirmation in the laboratory.

It is found that condensation of photons may occur on a timescale given by eq. (2) both in nonrelativistic and in relativistic cases and it manifests in photon spectra described by the Planck law with an excess formed in the energy range above the critical energy for the dominance of triple interactions and below the peak of the spectrum. At nonrelativistic temperatures it is well described by a power law, while at relativistic temperatures it represents a bump. In our nonrelativistic example with $k_{B}T\simeq 0.1m_{e}c^{2}$ the condensation persist until about $\sim 10^{-8}$ sec, when complete thermal equilibrium is established.

It is found that necessary condition for the development of BEC is an excess of photon number over the equilibrium number, see eq. (3), as well as initial distribution of photons not broader than Wien spectrum with the peak of the distribution located above the critical energy below which triple interactions dominate over the binary ones. Broader initial distributions, even the Planck spectrum, contain too many photons at low energies, and triple interactions such as bremsstrahlung quickly eliminate excess photons, preventing the condensation. This is the reason why the cooling of photons by electrons proposed by Zeldovich and Levich does not lead to photon condensation.

Likewise fermion degeneracy manifests itself in relativistic systems such as white dwarfs and neutron stars, it is possible that BEC of photons might be as well observed in astrophysical conditions. Given a transient character of condensation and short time of its existence, the necessary condition is a supply of soft nonthermal photons, which might provide support for condensation on longer timescales. A possible candidate could be gamma-ray bursts, whose time resolved spectra are often fit with a cut-off power law function, which, as we have shown, is expected when BEC occurs.

Regarding the possibility to observe this phenomenon in the laboratory, we propose to study the interaction of X-ray lasers with dense plasma targets. Our example in nonrelativistic case refers to initial conditions with isotropic distribution of photons with energy about $10$ keV, and number density about few $10^{28}$ $cm^{-3}$ interacting with Maxwellian electrons with temperature $50$ keV. The discovery of photon condensation at relativistic temperatures $k_{B}T_{\gamma}>m_{e}c^{2}$, reported in this work, makes it possible to think about initial conditions, when dense non-equilibrium photon gas at MeV energies itself generates electron-positron pairs and undergoes such a condensation.

Acknowledgement

This work is supported within the joint BRFFR-ICRANet-2018 funding programme.