Field-theory methods in coagulation theory

Coagulating systems are systems of chaotically moving particles that collide and coalesce, producing daughter particles of mass equal to the sum of the masses involved in the respective collision event. The present article puts forth basic ideas underlying the application of methods of quantum-field theory to the theory of coagulating systems. Instead of the generally accepted treatment based on the use of a standard kinetic equation that describes the time evolution of concentrations of particles consisting of a preset number of identical objects (monomers in the following), one introduces the probability W(Q, t) to find the system in some state Q at an instant t for a specific rate of transitions between various states. Each state Q is characterized by a set of occupation numbers Q = {n ₁, n ₂, ..., n _g , ...}, where n _g is the total number of particles containing precisely g monomers. Thereupon, one introduces the generating functional Ψ for the probability W(Q, t). The time evolution of Ψ is described by an equation that is similar to the Schrödinger equation for a one-dimensional Bose field. This equation is solved exactly for transition rates proportional to the product of the masses of colliding particles. It is shown that, within a finite time interval, which is independent of the total mass of the entire system, a giant particle of mass about the mass of the entire system may appear in this system. The particle in question is unobservable in the thermodynamic limit, and this explains the well-known paradox of mass-concentration nonconservation in classical kinetic theory. The theory described in the present article is successfully applied in studying the time evolution of random graphs.     

Screening of a moving charge in a nonequilibrium plasma

Based on the model of point sinks, we consider the problem on the screening of the charge of a moving macroparticle in a nonequilibrium plasma. The characteristic formation times of the polarization cloud around such a macroparticle have been determined by the method of a three-dimensional integral Fourier transformation in spatial variables and a Laplace transformation in time. The screening effect is shown to be enhanced with increasing macroparticle velocity. We consider the applicability conditions for the model of point sinks and establish that the domain of applicability of the results obtained expands with decreasing gas ionization rate and macroparticle size. We consider the problem of charge screening at low velocities and establish that the stationary potential of the moving charge has a dipole component that becomes dominant at large distances. We show that the direction of the force exerted on the dust particle by the induced charges generally depends on the relationship between the transport and loss coefficients of the plasma particles in a plasma. When the Langevin ion recombination coefficient β _iL = 4πeμ _i exceeds the electron-ion recombination coefficient β _ei , this force will accelerate the dust particles in the presence of sinks. In the absence of sinks or when β _ei > β _iL , this force will be opposite in direction to the dust particle velocity. We also consider the problem on the energy and force of interaction between a moving charged macroparticle and the induced charges.     

Forced snaking: Localized structures in the real Ginzburg-Landau equation with spatially periodic parametric forcing

We study spatial localization in the real subcritical Ginzburg-Landau equation u _t = m ₀ u + Q(x)u + u _xx + d|u|² u ?|u|⁴ u with spatially periodic forcing Q(x). When d>0 and Q ≡ 0 this equation exhibits bistability between the trivial state u = 0 and a homogeneous nontrivial state u = u ₀ with stationary localized structures which accumulate at the Maxwell point m ₀ = ?3d ²/16. When spatial forcing is included its wavelength is imprinted on u ₀ creating conditions favorable to front pinning and hence spatial localization. We use numerical continuation to show that under appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders structure centered on the Maxwell point, and refer to this phenomenon as forced snaking. We determine the stability properties of these states and show that longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly localized, weakly interacting pulses exhibiting foliated snaking.     

Derivation of the Time Dependent Gross-Pitaevskii Equation Without Positivity Condition on the Interaction

Using a new method (Pickl in A simple derivation of mean field limits for quantum systems, 2010) it is possible to derive mean field equations from the microscopic N body Schrödinger evolution of interacting particles without using BBGKY hierarchies.In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii equation which is usually derived assuming positivity of the interaction (Erdös et al. in Commun. Pure Appl. Math. 59(12):1659–1741, 2006; Invent. Math. 167:515–614, 2007). The new method for dealing with mean field limits presented in Pickl (2010) allows us to relax this condition. The price we have to pay for this relaxation is however that we have to restrict the scaling behavior of the interaction and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function \(\mu^{\Psi_{0}}\) to a pure state |φ ₀〉〈φ ₀|.     

Quantenstatistik eines Gases mit verschiedener Bahn- und Spintemperatur

A system of particles with spin in a magnetic field may possess an orbital temperatureT _o different from the spin temperatureT _s (?0), if it is possible to neglect the energetic interaction between the orbital and the spin system. The calculation of the quantum statistical most probable distribution of identical independent particles on the orbital and spin energy levels yields the introduction of three Lagrange multipliers—according to the fact that the orbital and the spin energy and the number of particles are fixed—representing the orbital and spin temperature and a generalizedPlanck's “characteristic function”. Apart from the Boltzmann-approximation being valid in the case of small spin values forT _o ?T _e (T _e =customary degeneration temperature) and arbitraryT _s ?0, the distributions and the orbital and the spin energy depend onboth the temperaturesT _o andT _s coming from the principle of exclusion forFermi resp.Bose particles. The equations of state are discussed. There are four heat capacities, which possess characteristic peaks. In stead of the well-known temperature independence of the paramagnetism of degenerated conducting electrons one obtains χ～T _o /T _s . The behaviour of the Einstein-condensation of aBose gas is considered.     

Determination of nuclear excitation energies from the number of evaporated particles

The mean number ?N_b〉 of particles evaporated in the interaction of ²²Ne, ³²S, and ⁵⁶Fe nuclei with photoemulsion nuclei was measured as a function of the number of alpha particles emitted within the fragmentation cone. It is found that ?N_b〉 decreases with increasing number of the alpha particles and increases with increasing number of projectile nucleons involved in the interaction with a target nucleus and that ? N_b〉 is a linear function of the excitation energy E_ex of the target-nucleus residue. The maximum experimental value of the mean number of evaporated particles is ?N_bmax〉 ? 12–13, which corresponds to E_exc ? 540 ± 60 MeV.     

Vortex-antivortex oscillation and tunneling in Bose-Einstein condensates

We study interacting condensates in anisotropic traps. Employing a two-level mean-field theory, which is valid provided the interaction energy is much smaller than ?ω_x and ?ω_y and the number of particles N is much larger than unity, we see that even a small interaction can drastically modify the dynamics of the system as predicted by García-Ripoll et al. [Phys. Rev. Lett. 87, 140403 (2001)]. In the present work, we supplement the discussion of the previous work and point out the important role of coupling between population difference and phase difference between two p states in the x and y directions. We also explore the stability of the vortex state for small systems with N ～ O(1), for which the mean-field theory is inapplicable. We performed the full quantum mechanical calculations using up to six single-particle states and showed that, when N is comparable to unity, quantum tunneling between the vortex and antivortex states can occur even though the interaction coefficient is so large that the vortex-antivortex oscillation is prohibited within the mean-field theory.     

Covalence-induced stabilization of an intermediate-spin state and the magnetic susceptibility of LaCoO<Subscript>3</Subscript>

The energies of terms with spins S = 0, 1, 2 have been found using exact diagnoalization of the multielectron Hamiltonian of a multiband pd model for the CoO₆ cluster. Co (e _g orbital)-O hops, which form the covalent σ bond, are shown to decrease the energy of the state (IS) with an intermediate spin (S = 1) as compared to the energy of the state (LS) with a low spin (S = 0). An analogue of the Tanabe-Sugano diagram that takes into account the covalence of the CoO₆ cluster is constructed. The state with S = 1 is shown to be a ground state at certain model parameters. An increase in temperature is established to decrease the crystal field and, thus, favors the transition of the ground state from LS to IS at T = 100 K and the transition of the IS ground state to a state (HS) with a high spin (S = 2) at T = 550 K. The magnetic susceptibility of LaCoO₃ is calculated with allowance for the LS, IS, and HS states and for the fact that the HS state exhibits threefold orbital degeneracy of the t _2g shell, which results in an effective orbital moment L = 1 and the importance of spin-orbit interaction. The behavior of this magnetic susceptibility agrees well with the experimental x(T) dependence of LaCoO₃.     

Ground State Energy of the Low Density Hubbard Model

We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani (J. Math. Phys. 48:023302, [2007]), our result proves that in the low density limit the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by 8π a ? _u ? _d , where ? _u(d) denotes the density of the spin-up (down) particles, and a is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.     

Distribution function and electron state density in nonequilibrium plasma created by dye laser

Ultracold nonequilibrium plasma created by a dye laser has been studied by the molecular dynamics method. Electrons and protons in this model of nonequilibrium plasma interacted according to the Coulomb law. In the case of electron-proton interaction and a distance between particles r < a ₀ (Bohr radius), the interaction energy is constant, e ²/a ₀ (e is the charge of electron). An initial proton kinetic energy is set randomly so that the average kinetic energy is 0.01–1 K. Initial full electron energy is also set randomly, but at the same time it is positive; i.e., all the electrons according to our task are located in the continuous spectrum. Average kinetic electron energy per one particle varies from 1 to 50 K. The motion equations in periodical boundary condition for this system have been solved by molecular dynamics method. We have calculated the distribution function in the region near the ionization threshold. The distribution function is being described using electron state density in the nearest neighbor approximation with activity correction.     

Exclusive electrodisintegration of the deuteron within the point form of relativistic quantum mechanics

The process of exclusive deuteron electrodisintegration is considered within the point form of relativistic quantum mechanics. Relevant calculations are performed on the basis of the Nijmegen and Moscow nucleon-nucleon potentials, the latter involving forbidden states. Final-state interaction is taken into account. For either type of the potentials used, the results of the calculations describe well experimental data at final-neutron momenta p _n of up to 500 MeV/c.     

An exact solution for uniformly accelerated particles in general relativity

We present an exact solution of the Einstein empty-space equations referring to four particles in relative motion. The particles move with different uniform accelerations relative to a co-ordinate system which is Minkowskian at infinity, except in certain directions. If positive and negative masses are allowed, the particles can move freely under their own gravitation; if all four masses are positive, stresses extending to infinity are needed to cause the motion, but two of the particles can move freely. There are three results of interest. First, the field can be described in terms of a classical potential which is the average of retarded and advanced potentials corresponding to the particles. Secondly, the field at spatial infinity is entirely different from that of a static mass, and theg _ik fall off like the inversesquare of the distance. Thirdly, the world-lines of free particles are geodesics of the space-time.     

Negativity and Quantum Phase Transition in the Spin Model Using the Quantum Renormalization-group Method

In this paper, we study the anisotropy parameter and Dzyaloshinskii-Moriya (DM) interaction on negativity and quantum phase transition (QPT) by using the quantum renormalization-group (QRG) method in the spin model. In our model, the anisotropy parameter and DM interaction can influence the phase diagrams. Negativity can develop two different values which separated two phases i.e. Spin-fluid phase and the Néel phase with the number of QRG iterations increased, and can obviously exhibit QPT at the critical point. Then, we find that negativity of particles 1, 3 throughout is less than negativity of particles 1, 2 or particles 2, 3. Because of information between the three particle distributions, please see the conclusion. We find that the negativity difference value (S) can also clearly detect QPT at the critical point. Most importantly, the maximum S _max become more and more close to the critical point. So S _max can be used as a criterion of the quantum phase transition occurrence when the spin chain is infinity (N → ∞).     

Causal propagation of signals in strangeon matter

The state equation for strangeon matter is very stiff due to the non-relativistic nature of its particles and their repulsive interaction,such that pulsar masses as high as～3 M_⊙ would be expected. However, an adiabatic sound speed, c_s=(?P/?ρ)～(1/2), is usually superluminal in strangeon matter, and the dynamic response of a strangeon star(e.g., binary merger) is not tractable in numerical simulations. In this study, we examined signal propagation in strangeon matter and calculate the actual propagation speed, c_(signal).We found that the causality condition, c_(signal) c, is satisfied and the signal speed is presented as a function of stellar radius.     

Screening of a moving charge in a nonequilibrium plasma

The problem of screening a moving charged dust particle is analyzed in the model of point sinks. Typical time scales for the formation of a polarization cloud around the moving macroscopic particle are found using the three-dimensional integral transform with respect to the spatial coordinates and the Laplace transform in time. It is shown that the stationary potential of a moving charge has a dipole component dominating at sufficiently large distances. The force exerted on a moving charged macroscopic particle by the electric field of induced charges is calculated. It is shown that, in general, the direction of this force depends on the ratio between the transfer coefficient and the decay rate of plasma particles in the plasma. In the presence of sinks, a dust particle is accelerated by this force if the Langevin recombination rate for ions, β _iL = 4πeμ _i , exceeds the electron-ion recombination rate β _ei . In the absence of sinks or if β _ei > β _iL this force is antiparallel to the dust-particle velocity.     

Structural Transitions in Elemental Tin at Ultra High Pressures up to 230 GPa

The quantum heat generation, interaction force, and friction torque for two rotating spherical nanoparticles with the radius R are calculated. In contrast to a static case where an upper bound in the radiative heat transfer between two particles exists, the quantum heat generation for two rotating particles diverges at distances between particles d < d ₀ = R(3/ε″(ω₀))^1/3 (where ε″(ω₀) is the imaginary part of the dielectric function for the material of a particle at the resonance frequency ω₀), when the rotation frequency coincides with poles in the excitation generation rate at Ω = 2ω₀. These poles are due to the anomalous Doppler effect and the mutual polarization of particles and exist even in the presence of dissipation in particles. The anomalous heat generation is associated with the conversion of mechanical rotation energy into heat mediated by quantum friction. Similar singularities also exist for the interaction force and friction torque. The results can be of significant importance for biomedical applications.     

X-Ray Spectral Studies of the Interface Interaction in CuO<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>/MWCNTs Nanocomposite

Results of a comprehensive study of the interface interaction of a nanostructured CuO_x and multiwalled carbon nanotubes (MWCNTs) in CuO_x/MWCNT nanocomposite by X-ray absorption spectroscopy (XANES, NEXAFS) and X-ray photoelectron spectroscopy (XPS) methods using a synchrotron radiation are presented. It is established that a nanostructured CuO_x in CuO_x/MWCNT nanocomposite is predominantly formed by CuO and has the form of flakelike particles 200–500 nm in size uniformly dispersed over an array of nanotubes. A chemical interaction of CuO_x and nanotubes with formation of covalent carbon–oxygen bonds, which does not lead to a significant destruction of the outer layers of carbon nanotubes, is observed at the interfaces of the nanocomposite.     

Existence and Regularity of Propagators for Multi-Particle Schrödinger Equations in External Fields

We consider Schrödinger equations for N number of particles in (classical) electro-magnetic fields that are interacting with each other via time dependent inter-particle potentials. We prove that they uniquely generate unitary propagators \({\{U(t,s), t,s \in \mathbb{R}\}}\) on the state space \({\mathcal{H}}\) under the conditions that fields are spatially smooth and do not grow too rapidly at infinity so that propagators for single particles satisfy Strichartz estimates locally in time, and that local singularities of inter-particle potentials are not too strong that time frozen Hamiltonians define natural selfadjoint realizations in \({\mathcal{H}}\). We also show, under very mild additional assumptions on the time derivative of inter-particle potentials, that propagators possess the domain of definition of the quantum harmonic oscillator \({\Sigma(2)}\) as an invariant subspace such that, for initial states in \({\Sigma(2)}\), solutions are C¹ functions of the time variable with values in \({\mathcal{H}}\). New estimates of Strichartz type for propagators for N independent particles in the field will be proved and used in the proof.     

Specific features of the states of cobalt fluoride in the vicinity of the critical field

The state of cobalt fluoride in the vicinity of the critical value H _c of a longitudinal magnetic field H, in which the magnetic subsystem of a CoF₂ crystal with a strong Dzyaloshinskii interaction is transformed from the antiferromagnetic phase into the canted phase, has been investigated taking into account the increasing number of experimental studies related to the use of cobalt fluoride. It has been found that, despite the unusually high magnetic anisotropy of the crystal, the state of the magnetic subsystem at H = H _c is extremely sensitive to a small deviation of the vector H from the C ₄ axis. Another feature is that the high sensitivity disappears with an increase or decrease in the magnetic field by only a few thousandths of H _c . The results of the investigations performed in this work are applicable to magnetically ordered crystals FeF₃ and Cu₂OSeO₃, which, as well as the CoF₂ crystals, are characterized by a strong Dzyaloshinskii interaction and a significant magnetic anisotropy. The revealed anomaly in the reduction of the effective magnetic anisotropy is of interest in connection with numerous attempts to decrease the magnetic anisotropy in crystals with giant magnetostriction, which are necessary for the use as sensors and vibrators.     

Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to \(\frac{1}{2} \sqrt{T_{L}T_{R}}\) where T _L and T _R are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier’s Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach.