Bose-Einstein Particles Research Articles

An explicit coercivity estimate of the linearized quantum Boltzmann operator without angular cutoff

The quantum Boltzmann-Bose equation describes a large system of Bose-Einstein particles in the weak-coupling regime. If the particle interaction is governed by the inverse power law, the corresponding collision kernel has angular singularity. In this paper, we give a constructive proof of the coercivity estimate for the linearized quantum Boltzmann-Bose operator to capture the effects of the singularity and the fugacity. Precisely, the estimate explicitly reveals the dependence on the fugacity parameter before the Bose-Einstein condensation. With the coercivity estimate, the global in time well-posedness of the inhomogeneous quantum Boltzmann-Bose equation in the perturbative framework and stability of the Bose-Einstein equilibrium can be established.

Global well-posedness of the quantum Boltzmann equation for bosons interacting via inverse power law potentials

We consider the spatially inhomogeneous quantum Boltzmann equation for bosons with a singular collision kernel, the weak-coupling limit of a large system of Bose-Einstein particles interacting through inverse power law. Global well-posedness of the corresponding Cauchy problem is proved in a periodic box near equilibrium for initial data satisfying high temperature condition.

Singularities in $L^1$-supercritical Fokker–Planck equations: A qualitative analysis

A class of nonlinear Fokker–Planck equations with superlinear drift is investigated in the L^1 -supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis with main focus on the isotropic case, where solutions will be shown to converge to the unique minimiser of the free energy as time tends to infinity. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the three-dimensional Kaniadakis–Quarati model for Bose–Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.

Relativistic BGK model for massless particles in the FLRW spacetime

<p style='text-indent:20px;'>In this paper, we address the Cauchy problem for the relativistic BGK model proposed by Anderson and Witting for massless particles in the Friedmann-Lemaȋtre-Robertson-Walker (FLRW) spacetime. We first derive the explicit form of the Jüttner distribution in the FLRW spacetime, together with a set of nonlinear relations that leads to the conservation laws of particle number, momentum, and energy for both Maxwell-Boltzmann particles and Bose-Einstein particles. Then, we find sufficient conditions that guarantee the existence of equilibrium coefficients satisfying the nonlinear relations and we show that the condition is satisfied through all the induction steps once it is verified for the initial step. Using this observation, we construct explicit solutions of the relativistic BGK model of Anderson-Witting type for massless particles in the FLRW spacetime.</p>

Bose–Einstein Condensation as a Deposition Phase Transition of Quantum Hard Spheres and New Relations between Bosonic and Fermionic Pressures

We investigate the phase transition of Bose–Einstein particles with the hard-core repulsion in the grand canonical ensemble within the Van der Waals approximation. It is shown that the pressure of non-relativistic Bose–Einstein particles is mathematically equivalent to the pressure of simplified version of the statistical multifragmentation model of nuclei with the vanishing surface tension coefficient and the Fisher exponent тF = 5/2 , which for such parameters has the 1-st order phase transition. The found similarity of these equations of state allows us to show that within the present approach the high density phase of Bose-Einstein particles is a classical macro-cluster with vanishing entropy at any temperature which, similarly to the system of classical hard spheres, is a kind of solid state. To show this we establish new relations which allow us to identically represent the pressure of Fermi–Dirac particles in terms of pressures of Bose–Einstein particles of two sorts.

The Spatially Homogeneous Boltzmann Equation for Bose–Einstein Particles: Rate of Strong Convergence to Equilibrium

The paper is a continuation of our previous work on the spatially homogeneous Boltzmann equation for Bose-Einstein particles with quantum collision kernel that includes the hard sphere model. Solutions $F_t$ under consideration that conserve the mass, momentum, and energy and converge at least weakly to equilibrium $F_{{\rm be}}$ as $t\to\infty$ have been proven to exist at least for radially symmetric and non-singular initial data, and for the case of low temperature, $F_t$ have to be positive Borel measures. The new progress is as follows: we prove that the long time convergence of $F_t(\{0\})$ to the Bose-Einstein condensation $F_{{\rm be}}(\{0\})$ for low temperature holds for all radially symmetric and non-singular initial data $F_0$. This immediately implies the long time strong convergence to equilibrium. We also obtain an algebraic rate of the strong convergence for arbitrary temperature. Our proofs are based on the entropy control, Villani's inequality for the entropy dissipation, a suitable time-dependent convex combination between the solution and a fixed positive function (in order to overcome the lack of positive lower bound), the convex-positivity of the cubic collision integral, and an iteration technique for obtaining a positive lower bound of condensation.

Global existence of solutions of the Boltzmann equation for Bose–Einstein particles with anisotropic initial data

In this paper we prove the global in time existence and uniqueness of solutions of the spatially homogeneous Boltzmann equation for Bose–Einstein particles for the hard sphere model for bounded anisotropic initial data. The main idea of our proof is as follows: we first establish an intermediate equation which is closely related to the original equation and is relatively easily proven to have global in time and unique solutions, then we use the multi-step iterations of the collision gain operator to obtain a desired uniform L∞-bound for solutions of the intermediate equation so that if an initial datum is sufficiently small relative to the inverse of the Planck constant (which belongs to the case of very high temperature), then the corresponding solution of the intermediate equation becomes the solution of the original equation.

Coefficient of performance at maximum [formula omitted]-criterion of thermochemical refrigerators with near-independent particles

Thermochemical refrigerators, constituted by a cold/hot reservoir with an energy filter between them, are established by utilizing Maxwell–Boltzmann, Fermi–Dirac, and Bose–Einstein particles as working substances. With the help of chemical potential gradient, heat coupled with the particle flux can be transferred against the temperature gradient, thus the cold reservoir is cooled. The coupling between the heat and particle fluxes, determined by the width Γ of the energy filter, will be weakened when Γ increases. As the coupling gets weak gradually, the εC-dependent coefficient of performances (COPs) of such refrigerators are optimized under the χ-criterion, where εC is the Carnot COP and χ is defined as the product of the COP and cooling rate. It is found three different regions of COP at maximum χ in three different kinds of near-independent systems respectively. The corresponding three upper bounds, obtained from the case of strong coupling (i.e., Γ→0), are also bounded by 9+8εC−3∕2 from above. When Γ increases from 0 to infinity, the COPs at maximum χ decreases monotonously to three lower bounds. In addition, the upper bound of Bose–Einstein system can be approximated by the Curzon–Ahlborn COP: εCA=1+εC−1.

Long time strong convergence to Bose-Einstein distribution for low temperature

We study the long time behavior of measure-valued isotropic solutions \begin{document}$F_t$\end{document} of the Boltzmann equation for Bose-Einstein particles for low temperature. The global in time existence of such solutions \begin{document}$F_t$\end{document} that converge at least semi-strongly to equilibrium (the Bose-Einstein distribution) has been proven in previous work and it has been known that the long time strong convergence to equilibrium is equivalent to the long time convergence to the Bose-Einstein condensation. Here we show that if such a solution \begin{document}$F_t$\end{document} as a family of Borel measures satisfies a uniform double-size condition (which is also necessary for the strong convergence), then \begin{document}$F_t$\end{document} converges strongly to equilibrium as \begin{document}$t$\end{document} tends to infinity. We also propose a new condition on the initial datum \begin{document}$F_0$\end{document} such that a corresponding solution \begin{document}$F_t$\end{document} converges strongly to equilibrium.

Efficiency at maximum power of thermochemical engines with near-independent particles.

Two-reservoir thermochemical engines are established by using near-independent particles (including Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein particles) as the working substance. Particle and heat fluxes can be formed based on the temperature and chemical potential gradients between two different reservoirs. A rectangular-type energy filter with width Γ is introduced for each engine to weaken the coupling between the particle and heat fluxes. The efficiency at maximum power of each particle system decreases monotonously from an upper bound η(+) to a lower bound η(-) when Γ increases from 0 to ∞. It is found that the η(+) values for all three systems are bounded by η(C)/2 ≤ η(+) ≤ η(C)/(2-η(C)) due to strong coupling, where η(C) is the Carnot efficiency. For the Bose-Einstein system, it is found that the upper bound is approximated by the Curzon-Ahlborn efficiency: η(CA)=1-sqrt[1-η(C)]. When Γ → ∞, the intrinsic maximum powers are proportional to the square of the temperature difference of the two reservoirs for all three systems, and the corresponding lower bounds of efficiency at maximum power can be simplified in the same form of η(-)=η(C)/[1+a(0)(2-η(C))].

Condensation phenomena in nonlinear drift equations

We study non-negative, measure-valued solutions to nonlinear drifttype equations modelling concentration phenomena related to Bose-Einstein particles. In one spatial dimension, we prove existence and uniqueness for measure solutions. Moreover, we prove that all solutions blow up in finite time leading to a concentration of mass only at the origin, and the concentrated mass absorbs increasingly the mass converging to the total mass as t → ∞. Our analysis makes a substantial use of independent variable scalings and pseudo-inverse functions techniques.

Long Time Convergence of the Bose–Einstein Condensation

We study long time behavior of the Bose–Einstein condensation of measure-valued solutions \(F_t\) of the space homogeneous and velocity isotropic Boltzmann equation for Bose–Einstein particles at low temperature. We prove that if \(F_0\ge 0\) is a non-singular Borel measure on \({{\mathbb R}}_{\ge 0}\) satisfying a very low temperature condition and that the ratio \(F_0([0,\varepsilon ])/\varepsilon ^{\alpha }\) is sufficiently large for all \(\varepsilon \in (0, R]\) for some constants \(0 0\), then there exists a solution \(F_t\) of the equation on \([0,+\infty )\) with the initial datum \(F_0\) such that \(F_t(\{0\})\) converges to the expected Bose–Einstein condensation as \(t\rightarrow +\infty \). We also show that such initial data \(F_0\) exist extensively.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L 1 norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.

Coincidence problem within dark energy as a coupled self-interacting Bose–Einstein gas

A late accelerated expansion of the Universe is obtained from non-relativistic Bose–Einstein particles with a short-range attractive interaction, with a temperature of the same order as the critical one, and an energy interchange with a dark-matter source. We show that for a sizable range of parameters, these particles evolve as a coupled dark-energy source, solving the coincidence problem, and in agreement with the luminosity distance versus redshift derived from supernova data.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose–Einstein Condensation. Part I: Local Existence of Solutions

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of a local-in-time nonnegative continuous weak solution is proven. If the solution is not global, it blows up with respect to the $L^\infty$ norm in finite time. The proof is based on approximation arguments, interpolation inequalities in weighted Sobolev spaces, and suitable a priori estimates for a weighted gradient $L^2$ norm.

The Boltzmann Equation for Bose–Einstein Particles: Regularity and Condensation

We study regularity and finite time condensation of distributional solutions of the space-homogeneous and velocity-isotropic Boltzmann equation for Bose–Einstein particles for the hard sphere model. Global in time existence of distributional solutions had been proven before. Here we prove that the equation is locally and can be globally (in time) well-posed for the class of distributional solutions having finite moment of the negative order $$-1/2$$ , and solutions in this class with regular initial data are mild solutions in their regularity time-intervals. By observing a necessary condition on the initial data for the absence of condensation at some finite time, we also propose a sufficient condition on the initial data for the occurrence of condensation at all large time, and then using a positivity of a partial collision integral we prove further that the critical time of condensation can be strictly positive.

The Boltzmann Equation for Bose-Einstein Particles: Condensation in Finite Time

The paper considers the problem of the Bose-Einstein condensation in finite time for isotropic distributional solutions of the spatially homogeneous Boltzmann equation for Bose-Einstein particles with the hard sphere model. We prove that if the initial datum of a solution is a function which is singular enough near the origin (the zero-point of particle energy) but still Lebesgue integrable (so that there is no condensation at the initial time), then the condensation continuously starts to occur from the initial time to every later time. The proof is based on a convex positivity of the cubic collision integral and some properties of a certain Lebesgue derivatives of distributional solutions at the origin. As applications we also study a special type of solutions and present a relation between the conservation of mass and the condensation.

Finite Time Blow Up in Kaniadakis–Quarati Model of Bose–Einstein Particles

We study a Fokker–Planck equation with linear diffusion and super-linear drift introduced by Kaniadakis and Quarati [12, 13] to describe the evolution of a gas of Bose–Einstein particles. For kinetic equation of this type it is well-known that, in the physical space ℝ3, the structure of the equilibrium Bose–Einstein distribution depends upon a parameter m*, the critical mass. We are able to describe the time-evolution of the solution in two different situations, which correspond to m ≪ m* and m ≫ m* respectively. In the former case, it is shown that the solution remains regular, while in the latter we prove that the solution starts to blow up at some finite time t c , for which we give an upper bound in terms of the initial mass. The results are in favor of the validation of the model, which, in the supercritical regime, could produce in finite time a transition from a normal fluid to one with a condensate component.

On the Boltzmann Equation for 2D Bose-Einstein Particles

This paper considers the spatially homogeneous Boltzmann equation for 2D Bose-Einstein particles. Suppose the collision kernel satisfies some assumptions that include the hard disk model and other possible physical models. We prove the existence of global in time conservative measure solutions of the equation for isotropic initial data, and that for any initial datum which is not totally singular and has positive energy, the solution always converges strongly to the Bose-Einstein distribution as time goes to infinity. This implies that for the present 2D model there is no Bose-Einstein condensation in the sense of long-time limit.

The grazing collision limit of Kac caricature of Bose–Einstein particles

We discuss the grazing collision limit of certain kinetic models of Bose-Einstein particles obtained from a suitable modification of the one- dimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a nonlinear Fokker-Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in [13 ]. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation.