Bose-Einstein Research Articles

Exact Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics

The exact Maxwell–Boltzmann (MB), Bose–Einstein (BE) and Fermi–Dirac (FD) entropies and probabilistic distributions are derived by the combinatorial method of Boltzmann, without Stirling's approximation. The new entropy measures are explicit functions of the probability and degeneracy of each state, and the total number of entities, N. By analysis of the cost of a “binary decision”, exact BE and FD statistics are shown to have profound consequences for the behaviour of quantum mechanical systems.

Wakefield of Bose–Einstein condensates in a background thermal gas

Wakefield excitation of sound waves by a Bose–Einstein (BE) condensate moving in a non-condensate gas is considered here. The BE condensate is described by a kinetic equation, which is valid in the quasi-classical limit. It describes the evolution of the Wigner function associated with its macroscopic wave-function. The non-condensate gas is described by fluid equations that are coupled to the kinetic equation by the mean field potential of the condensate. Wavefield solutions are established. Modulation instabilities occurring when the BE condensate beam velocity equals the sound velocity of the non-condensate gas are also considered. Cold and warm beam solutions for the instability growth rates are derived.

The BCS–BE crossover phase diagram atT = 0 K for a d-wave superconductor: the importance of the Debye frequency and the tightbinding band structure

We consider the phase diagram of the BCS (Bardeen–Cooper–Schrieffer)–BE (Bose–Einstein) crossover in theground state (T = 0 K) of a dx2−y2-wave superconductor, with a nearest neighbour tight binding structure, when wetake into account the Debye (phononic) frequency around the chemical potential,μ. This approach is a continuation of the work of den Hertog (1999 Phys. Rev. B60 559) and that of Soares et al (2002 Phys. Rev. B 65 174506). The latterauthors considered the influence of the second-nearest neighbours, butneither set of authors took into account the effect of the Debye frequency,ωD, or the influence of the next nearest neighbour matrix hopping element. We have found thefollowing results: (1) there is not a metallic phase—that is, when , , , and , where n is the carrierdensity per site, V is theattractive interaction, t is the nearest neighbour hopping integral, and is the next nearest neighbour hopping ratio; (2) the BCS–BE crossover line is strongly affected by thepresence of ωD/4t and that of —actually, the values of V/4t needed to achieve the Bose–Einstein regime become extremely large for small values ofωD/4t; and(3) both Δ/4t andμ/4t strongly dependon the values of ωD/4t and . The results (1) are in agreement with the ones found by Perali et al (2003 Phys. Rev. B 68066501 (Preprint cond-mat/0211132)) and Rodríguez-Núñez et al (2003 Phys. Rev. B 68066502), and in disagreement with those of den Hertog and Soares et al.

Bose–Einstein statistics behaviors of exciton–biexciton photoluminescence decay processes in a GaAs/AlAs type-II superlattice

We have investigated the Bose–Einstein (BE) statistics behavior of the exciton–biexciton (EX–BEX) system during the photoluminescence decay processes at 5 K in a (GaAs) 12/(AlAs) 12 type-II superlattice. Owing to the long lifetime of the type-II exciton, which is of the order of microseconds due to the indirect transition nature, we have precisely evaluated the density relation between the EX and BEX from the line-shape analysis of time-resolved photoluminescence spectra. At an EX density around 1×10 10/cm 2, the BEX density suddenly increases with a threshold-like profile. This behavior is quantitatively explained by the framework of BE statistics of the EX–BEX system. It is demonstrated that the threshold-like increase of the BEX density is recurred by the incidence of a second-excitation pulse with a time delay, which leads to the dynamical control of the BE statistics behavior.

Cooper pairing with finite angular momentum via a central attraction: From the BCS to the Bose limits

In the context of a simple model featuring an explicit, central interaction potential, and using a standard functional-integral technique, we study superconductivity with angular momentum quantum number l=2 as an emergent property of the many-body system. Our interaction potential is attractive at a finite distance r_0, and the breaking of the rotational symmetry is the result of an interplay between r_0 and the inter-particle distance r_s which we deem generic to interactions of this type. However such interplay, responsible for the preference of a d-wave state for a range of intermediate densities, takes place only in the BCS limit. In contrast, as the Bose-Einstein (BE) limit is approached the internal energy of the "preformed pairs" becomes the dominant contribution and there is a quantum phase transition in which the s-wave symmetry is restored. We also find that the limiting value of the critical temperature is k_B T_c --> 3.315(\hbar^2/2m*)[n/2(2l+1)]^{2/3}, which coincides with the usual result only for l=0; for l>0, it differs in the degeneracy factor 1/(2l+1), which lowers T_c. Our results thus place constraints on exotic pairing in the BE limit, while at the same time indicating a particularly interesting route to pairing with l>0 in a BCS superconductor.

The Equilibrium States for a Model with Two Kinds of Bose Condensation

We study the equilibrium Gibbs states for a Boson gas model, defined by Bru and Zagrebnov, which has two phase transitions of the Bose condensation type. The two phase transitions correspond to two distinct mechanisms by which these condensations can occur. The first (non-conventional) Bose condensation is mediated by a zero-mode interaction term in the Hamiltonian. The second is a transition due to saturation quite similar to the conventional Bose–Einstein (BE) condensation in the ideal Bose gas. Due to repulsive interaction in non-zero modes the model manifests a generalized type III; i.e., non-extensive BE condensation. Our main result is that, as in the ideal Bose gas, the conventional condensation is accompanied by a loss of strong equivalence of the canonical and grand canonical ensembles whereas the non-conventional one, due to the interaction, does not break the equivalence of ensembles, at least not on the level of the gauge invariant states. It is also interesting to note that the type of (generalized) condensate, I, II, or III (in the terminology of van den Berg, Lewis, and Pule), has no effect on the equivalence of ensembles. These results are proved by computing the generating functional of the cyclic representation of the Canonical Commutation Relation (CCR) for the corresponding equilibrium Gibbs states.

Bose–Einstein condensation of nonzero-center-of-mass-momentum Cooper pairs

Cooper pair (CP) binding with both zero and nonzero center-of-mass momenta (CMM) is studied with a set of renormalized equations assuming a short-ranged (attractive) pairwise interfermion interaction. Expanding the associated dispersion relation in 2D in powers of the CMM, in weak-to-moderate coupling a term linear in the CMM dominates the pair excitation energy, while the quadratic behavior usually assumed in Bose–Einstein (BE) condensation studies prevails for any coupling only in the limit of zero Fermi velocity when the Fermi sea disappears, i.e., in vacuum. In 3D this same behavior is observed numerically. The linear term, moreover, exhibits CP breakup beyond a threshold CMM value which vanishes with coupling. This makes all the excited (nonzero-CMM) BE levels with preformed CPs collapse into a single ground level so that a BCS condensate (where only zero CMM CPs are usually allowed) appears in zero coupling to be a special case in either 2D or 3D of the BE condensate of linear-dispersion-relation CPs.

Temperature variation of magnetic aftereffect

The paper analyzes the anomalous behavior of the magnetic aftereffect decay coefficient with temperature, which has been established experimentally from a thermodynamic point of view. The Arrhenius law, which has often been used to model magnetic aftereffect, is based on the Maxwell-Boltzmann (MB) statistics. At low temperatures, however, magnons obey Bose-Einstein (BE) statistics, which have a different energy spectrum. MB statistics are a good approximation to BE statistics in the temperature range in which most devices operate. The variation in temperature, governed by chemical potential that is essentially linear, has a negative slope and is zero below a low but finite temperature. Other explanations for this anomalous behavior would predict an incorrect type of temperature dependence. This new interpretation of magnetic aftereffect may have profound implications on accelerated testing to determine life expectancy of recordings.

Fluidity aspects of Bose-Einstein condensed systems

We analyze some aspects of the fluidity of Bose-Einstein (BE) condensed systems. We show that a two-fluid picture of a condensed system holds true. The condensate is a quantum fluid component of the system. This component of the fluid satisfies the continuity equation and a generalized Bernoulli equation. Under certain conditions, the condensate component becomes a classical superfluid. We suggest that condensed systems might exhibit mechanochalorical and thermomechanical effects, in close analogy with superfluid helium-4. Quantized vortices are also expected to appear in a BE condensed system. We discuss the structure of quantized vortices.

Non-standard peak values of the Bose-Einstein correlations and their possible interpretation by a metric description of strong interactions

We critically reanalyze some recent experimental data on the Bose-Einstein (BE) correlations in pion production. We show that there is, in some cases, an experimental evidence for a peak height of the correlation function greater than two, contrarily to the predictions of the “canonical” theory of BE correlations. Although an explanation of such an “anomalous” value can be given by means of suitable phenomenological models, we show that this result is a straightforward consequence of the treatment of BE correlations within the framework of a description of strong interactions in terms of a deformed Minkowski metric.

Generalization of a Fermi Liquid to a Liquid with Fractional Exclusion Statistics in Arbitrary Dimensions: Theory of a Haldane Liquid

In 1957 Landau [1] introduced the famous concept of quasiparticles with Fermi-Dirac statistics (FDS) in a Fermi liquid in order to describe electrons in a metal where the interaction between electrons was believed to be very significant, as a generalization of the Sommerfeld theory for a degenerate Fermi gas [2]. This description was later justified by Luttinger [3]. Recently Haldane [4] introduced the concept of fractional exclusion statistics (FES) which quasiparticles in strongly interacting systems in arbitrary dimensions might satisfy as a generalization of Pauli’s exclusion principle. Wu [5] first formulated quantum statistical mechanics (QSM) in the state representation and derived the distribution function for an ideal gas with FES as a generalization of the FD and the Bose-Einstein (BE) distribution functions. [Throughout this Letter the FES is referred to as the Haldane-Wu statistics (HWS) such as FDS and BES, while a gas or liquid with HWS is called a Haldane gas or liquid such as a gas or liquid with FDS (BES) was called a Fermi (Bose) gas or liquid.] This concept has played a very important role to understand strongly interacting system such as the TomonagaLuttinger model (TLM) [6], the Calogero-Sutherland model (CSM) [7], and the Haldane-Shastry model (HSM) [8] in one dimension and the fractional quantum Hall effect in two dimensions [9]. Especially, notable is that Dasnieres de Veigy and Ouvry [10] revealed a deep connection between the FES and the fractional quantum Hall system, while Bernard and Wu [11] and Isakov [12] found the one between the FES and the CSM. More recently, QSM formulation has been developed further by Nayak and Wiczek [13], Isakov, Arovas, Myrheim, and Polychronakos [14], and the author [15]. Here, the QSM formulation allows us to evaluate the equation of state for an ideal gas with HWS in arbitrary dimensions with obtaining all the exact cluster coefficients in the cluster expansion [14,15]. However, interacting quasiparticles with HWS in arbitrary dimensions have never been investigated yet, and therefore the concept of a Haldane liquid is still missing. In this Letter I will explore this concept as a generalization of the Landau’s Fermi liquid theory and generalize the Luttinger’s theorem for a Fermi liquid to that for a Haldane liquid.

INTERMITTENCY AND BOSE-EINSTEIN CORRELATIONS

The role of Bose-Einstein (BE) correlations in a widely discussed intermittency phenomenon is reviewed. In particular, it is shown that particle correlations of different origins are better displayed when analyzed as functions of appropriately chosen variables. Correspondingly, if the shape of the BE contribution is chosen to be Gaussian in three-momentum transferred, it provides the power-like law in four-momentum squared and is smeared out in (pseudo)rapidity. The increase in factorial moments in small cells of the phase space looks also different in different variables, which is ascribed to varying shares of various mechanisms of both dynamical (e.g. parton jets) and symmetry (BE correlations) origin.

Bosonic high-Tc superconductivity in two dimensions.

The mixed boson-fermion model of superconductivity of Friedberg and Lee is adapted to two dimensions. Owing to the finite correlation length $l(T)$, Bose-Einstein (BE) condensation can prevail only for a finite, but still macroscopic system. It is shown that for $Tl{T}_{c}\ensuremath{\sim}40$ K, BE condensation of charged bosons that are converted from fermion (electron or hole) pairs leads not only to a perfect Meissner effect but also an energy gap in the fermion excitation spectrum. For the temperature range ${T}_{c}lTl{T}_{\mathrm{May}}$, where ${T}_{\mathrm{May}}$ depends, in part, on the thickness of the two-dimensional layer, although the system shows no vestige of BE condensation, a near-perfect Meissner effect would yet persist until $T\ensuremath{\simeq}{T}_{\mathrm{May}}\ensuremath{\sim}150$ K, based on the adaptation of the theory of May to the present model.

Classical model of bosons and fermions.

In a previous work [Phys. Rev. E 48, 4263 (1993)] we have derived a nonlinear one-dimensional kinetic equation for the distribution function of particles obeying an exclusion principle. In the present work, on the same grounds, we extend this kinetics to D-dimensional continuous or discrete space, in order to study the distribution function of particles obeying a generalized exclusion-inclusion Pauli principle (EIP). This exclusion or inclusion principle is introduced into the classical transition rates by means of an inhibition or an enhancement factor, which contains a parameter \ensuremath{\kappa}, whose values range between -1 and +1 and can balance the effect of the full or partial validity of EIP. After deriving the kinetic equation we obtain a general expression of the stationary distribution function, depending on the value we give to the parameter \ensuremath{\kappa}. When we limit ourselves to Brownian particles, we derive exactly for \ensuremath{\kappa}=-1 the Fermi-Dirac (FD) distribution, for \ensuremath{\kappa}=0 the Maxwell-Boltzmann distribution, and for \ensuremath{\kappa}=1 the Bose-Einstein (BE) distribution. When \ensuremath{\kappa} assumes an intermediate value, except zero, between the extreme values -1 and +1, we obtain statistical distributions different from the FD and BE ones. We attribute to the parameter \ensuremath{\kappa} the meaning of the degree of indistinguishability of identical particles, the degree of antisymmetrization, or the symmetrization of the wave function of the particle system.

Classical model of intermediate statistics.

In this work we present a classical kinetic model of intermediate statistics. In the case of Brownian particles we show that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions can be obtained, just as the Maxwell-Boltzmann (MB) distribution, as steady states of a classical kinetic equation that intrinsically takes into account an exclusion-inclusion principle. In our model the intermediate statistics are obtained as steady states of a system of coupled nonlinear kinetic equations, where the coupling constants are the transmutational potentials ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\kappa}}\mathrm{\ensuremath{\kappa}}\ensuremath{'}}$. We show that, besides the FD-BE intermediate statistics extensively studied from the quantum point of view, we can also study the MB-FD and MB-BE ones. Moreover, our model allows us to treat the three-state mixing FD-MB-BE intermediate statistics. For boson and fermion mixing in a D-dimensional space, we obtain a family of FD-BE intermediate statistics by varying the transmutational potential ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. This family contains, as a particular case, when ${\mathrm{\ensuremath{\eta}}}_{\mathrm{B}\mathrm{F}=0}$, the quantum statistics recently proposed by L. Wu, Z. Wu, and J. Sun [Phys. Lett. A 170, 280 (1992)]. When we consider the two-dimensional FD-BE statistics, we derive an analytic expression of the fraction of fermions. When the temperature T\ensuremath{\rightarrow}\ensuremath{\infty}, the system is composed by an equal number of bosons and fermions, regardless of the value of ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. On the contrary, when T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$ becomes important and, according to its value, the system can be completely bosonic or fermionic, or composed both by bosons and fermions.

Two pion Bose-Einstein correlations in [formula omitted]p annihilations at rest

Bose-Einstein (BE) correlations between like-sign charged pions were studied in pp annihilations at rest into four-prong events, using data taken with the CPLEAR detector at LEAR (CERN). A strong enhancement was found in the production of pairs of like-sign pions of similar momenta, with respect to the pairs of unlike-sign pions. The observed BE- enhancement was used to extract the values for the strength λ of the effect and the radius r of the pion emitting source. The extracted value of λ > l is of relevant importance and clearly does not depend on the assumed parametrization of the correlation function. The influence of the normalization and fitting procedure, the detector resolution, the resonances production and decay and the neutral-pion multiplicity cuts, on the size of the pion source and the strength of the effect was investigated.

Stress Relaxation In Metals And Polymers: Theory, Experiment And Computer Simulations

ABSTRACTThere exist large amounts of experimental evidence on stress relaxation for metals and their alloys, synthetic and natural polymers, glasses and frozen non-polymeric organic liquids. The results, typically presented as curves a (log t) of relaxation of stress aas a function of logarithmic time t, exhibit common features, apparently independent of the type of Material. All curves consist of three regions: initial, nearly horizontal, starting at σ0; central, descending approximately linearly; and final, corresponding to the internal stress σi = σ(>). We discuss briefly the experimental evidence as well as the main features of the cooperative theory which does not involve specific features of different classes of Materials. The bulk of the paper deals with computer simulations. Simulation results obtained with the method of molecular dynamics are reported for ideal metal lattices, Metal lattices with defects, and for polymeric systems. In agreement with both experiments and the cooperative theory, the simulated σ (log t) curves exhibit the same three regions. In agreement with the theory, the slope of the simulated central part is proportional to the initial effective stress σ0* = σ0 - σi. The time range taken by the central part is strongly dependent on the defect concentration: the lower the defect concentration, the shorter the range. IMposition in the beginning of a high strain ε destroys largely the resistance of a material to deformation, resulting in low values of the internal stress σo. Since the cooperative theory assumes for particles (atoms, polymer chain segments) the existence of two states, unrelaxed and relaxed, and has a formal connection to the Bose-Einstein (B-E) distribution, we first simulate B-E systems, recording the formation of relaxed clusters of particles of different sizes. Differences in cluster sizes predicted from a B-E Model and those obtained from the simulations are recorded and analyzed. On the joint basis of experimental, theoretical

The nonstatistical fluctuations and low pt phenomena caused by Bose-Einstein correlations

For a study of the influence of known quantum mechanical effects as well as of detector effects on the intermittency patterns observed in nucleus-nucleus collisions, the Fritiof Monte Carlo program hasbeen modified. Bose-Einstein (BE) correlations were introduced as a final state interaction between two particles. For a realistic strength of the BE effect as measured in heavy ion collisions, a good description of the bin size dependence of the normalized factorial moments in the distributions of secondary particles in OAu NA 35 data is observed. The p t dependence of the normalized factorial moments, and the low p t enhancement in the single particle p t spectra, caused by BE correlation, are discussed.

Spatiotemporal intermittency in ultrarelativistic nuclear collisions

The effect of nonstatistical fluctuations in ultrarelativistic nuclear collisions is analyzed. We argue that it is a nonlinear effect, which cannot be accounted for by the simple nucleon-nucleon superposition models and may have strong implication on our understanding of the collision dynamics. The presence of spatiotemporal intermittency in the interaction region could explain the essential features of the experimentally observed intermittency patterns (impact parameter and projectile dependence). Assuming that the onset of the spatiotemporal intermittency is due to a higher-order phase transition, detailed calculations are made with the inclusion of the nonideal, inside-outside dynamics and the resonance decay. Also the behavior of the fluctuations during the hydrodynamical evolution is studied. The implication of the fractal structures in space-time for Bose-Einstein (BE) correlations is shown to be very small.

Temperature dependence of the low- and high-frequency Raman scattering from liquid water

Low frequency Δν̄=0–350 cm−1, Raman intensity data were obtained from liquid water between 3.5 and 89.3 °C using holographic grating double and triple monochromators. The spectra were Bose–Einstein (BE) corrected, I/(1+n), and the total integrated (absolute) contour intensities were treated by an elaboration of the Young–Westerdahl (YW) thermodynamic method, assuming conservation of hydrogen-bonded (HB) and nonhydrogen-bonded (NHB=bent and/or stretched, O–H O) nearest-neighbor O–O pairs. A ΔH°1 value of 2.6±0.1 kcal/mol O–H ⋅⋅⋅ O or 5.2±0.2 kcal/mol H2O (11 kJ/mol O–H ⋅⋅⋅ O, or 22 kJ/mol H2O) resulted for the HB→NHB process. This intermolecular value agrees quantitatively with Raman and infrared ΔH° values from the one- and two-phonon OH-stretching regions, and from molecular dynamics, depolarized light scattering, neutron scattering, and ultrasonic absorption, thus indicating a common process. A population involving partial covalency of, i.e., charge transfer into, the H ⋅⋅⋅ O units of linear and/or weakly bent hydrogen bonds, O–H ⋅⋅⋅ O; is transformed into a second high energy population involving bent, e.g., 150° or less, and/or stretched, e.g., 3.2 Å, but otherwise strongly cohesive O–H O interactions. All difference spectra from the fundamental OH-stretching contours cross at the X(Z,X+Z)Y isobestic frequency of 3425 cm−1. Also, total integrated Raman intensity decreases occurring below 3425 cm−1 with temperature rise were found to be proportional to the total integrated intensity increases above 3425 cm−1, indicating conservation among the HB and NHB OH-stretching classes. From the enthalpy of vaporization of water at 0 °C, and the ΔH°1 of 2.6 kcal/mol O–H ⋅⋅⋅ O, the additional enthalpy, ΔH°2, needed for the complete separation of the NHB O–O nearest neighbors is ∼3.2 kcal/mol O–H ⋅⋅⋅ O or ∼6.4 kcal/mol H2O (13 kJ/mol O–H ⋅⋅⋅ O or 27 kJ/mol H2O). The NHB O–O nearest neighbors are held by forces other than those involving H ⋅⋅⋅ O partial covalency, i.e., electrostatic (multipole), induction, and dispersion forces. The NHB O–O pairs do not appear to produce significant intermolecular Raman intensity because they lack H ⋅⋅⋅O bond polarizability, but the corresponding NHB OH oscillators do contribute weakened Raman intensity above 3425 cm−1. An ideal solution thermodynamic treatment employing ΔH°1 =2.6 kcal/mol O–H ⋅⋅⋅ O, the HB mole fraction, and the vapor heat capacity, yielded a very satisfactory specific heat value of 1.1 cal deg−1 g−1 H2O at 0 °C. The NHB mole fraction, fu, from the YW treatment is negligibly small, 0.06 or less, for t<−50 °C. However, fu increases to 0.16 at 0 °C, and fu≊1 at 1437 °C, where recent shock-wave Raman measurements indicate loss of all partially covalent, charge transfer hydrogen bonding.