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.     

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.     

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.     

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.     

Core-softened potentials,multiple liquid–liquid critical points,and density anomaly regions: An exact solution

The pressure versus temperature phase diagram of a system of particles interacting through a multiscale shoulder-like potential is exactly computed in one dimension. The N-shoulder potential exhibits N density anomaly regions in the phase diagram if the length scales can be connected by a convex curve. The result is analyzed in terms of the convexity of the Gibbs free energy.     

Thermodynamische Stabilität makromolekularer Kristalle

The density of free energy of a chain crystal contains two terms of opposed sign dependent on the numberN of chain elements in the straight section of the macromolecules between the surfaces of the crystal perpendicular to thec-axis. The surface energy contributes a positive term decreasing withN. The amplitude Φ of the periodic lattice field opposing the chain translation in thec-axis yields the negative term. Due to the incoherent longitudinal thermal vibration of the four first-order neighbours of any chain the fieldΦ is smeared out. Its amplitude decreases the more the higherN and hence yields an increase in free energy density with increasingN. As a consequence of the opposite sign of surface energy and lattice field changes withN the free energy density shows a minimum at finiteN corresponding to the thermodynamically stable crystal thickness. With increasing temperature and lower interaction between adjacent chainsN increases in perfect qualitative agreement with experimental data.     

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.     

Dependence of the surface energy on the size and shape of a nanocrystal

An expression is derived for the surface energy σ as a function of the size and shape of a nanocrystal. It is shown that the wider the deviation of the shape parameter f from unity, the more pronounced the decrease in the surface energy σ with a decrease in the number N of atoms in the nanocrystal. The dependences of the average coordination number, the surface energy, and the melting temperature on the number N exhibit an oscillatory behavior with maxima at points corresponding to numbers of atoms forming a defect-free cube. The surface energy decreases with an increase in the temperature T. It is found that the smaller the nanocrystal size or the greater the deviation of the nanocrystal shape from the thermodynamically most stable shape (a cube), the larger the quantity-(dσ/dT). It is established that the nanocrystal undergoes melting when the surface energy decreases to a value at which it becomes independent of the nanocrystal size and shape. The conditions providing fragmentation and dendritization of the crystal are discussed. It is demonstrated that, at N>1000, the dependence σ(N) coincides, to a high accuracy, with the dependence of the surface tension of the nanocrystal on N. The inference is made that bimorphism is characteristic of nanocrystals. This implies that nanocrystals can have platelike and rodlike shapes with equal probability.     

Neutral Particle Motion around a Schwarzschild Black Hole in Modified Gravity

We investigate circular motion of neutral test particles on equatorial plane near a black hole in scalar-tensor-vector gravity. We consider three cases (i) α < G/G_N (ii) α = G/G_N and (iii) α > G/G_N to find the regions where motion can exist. The corresponding effective potential, energy, angular momentum and center of mass energy are evaluated. Further, we define four different cases for α > G/G_N and identify stable and unstable regions of circular orbits. It is found that circular orbits having zero angular momentum exist at r = αGNM due to repulsive gravity effects. We conclude that the structure of stable regions for α < G/G_N as well as α = G/G_N case is completely different from that of α > G/G_N.     

Nonperturbative quark dynamics in a baryon

The field-correlator method is used to calculate nonperturbative dynamics of quarks in a baryon. The general expression for the 3q Green’s function is obtained using the Fock-Feynman-Schwinger (world-line) path-integral formalism, where all dynamics is contained in the 3q Wilson loop with spin-field insertions. Using the lowest cumulant contribution for the Wilson loop, one obtains a Y-shaped string potential vanishing at the string-junction position. Using the einbein formalism for the quark kinetic terms, one automatically obtains constituent quark masses, calculable through the string tension. The resulting effective action for 3q plus Y-shaped strings is quantized in the path-integral formalism to produce two versions of Hamiltonian: one is in the c.m. and the other is in the light-cone system. The hyperspherical formalism is used to calculate masses and wave functions. Simple estimates in the lowest approximation yield baryon masses in good agreement with experiment without fitting parameters.     

Thermodynamics of the rupture in a Morse lattice

The rupture of a Morse lattice is considered in the presentpaper. The critical rupture force F _cr is found todecrease with the number of particles N as F _cr ~ 1/\(\sqrt{N}\). The partition function is obtained for two states ofthe lattice – with all equal bond lengths and one broken bond.In the first case an accurate expressions for thermodynamicparameters are obtained, and thermodynamic expressions arederived in the harmonic approximation in the latter case. Theanalytical predictions are confirmed by extensive MD simulations.Cis-trans isomerization is considered as an example. Volumefractions of trans- and cis-isomers versus number ofmonomer units N are found depending on the torsion stiffnesses.     

Energy dependence of effective nucleon-nucleon interaction and position of the nucleon drip line

A semimicroscopic version of the self-consistent theory of finite Fermi systems is proposed. In this approach, the standard theory of finite Fermi systems is supplemented with relations that involve the external values of the invariant components of the Landau-Migdal amplitude and which follow from microscopic theory. The Landau-Migdal amplitude at the nuclear surface is expressed in terms of the off-shell T matrix for free nucleon-nucleon scattering at the energy E equal to the doubled chemical potential of the nucleus being considered. The strong energy dependence of the free T matrix at low E changes the properties of nuclei in the vicinity of the nucleon drip line. It is shown that, upon taking into account the energy dependence of the effective interaction, the neutron drip line is shifted considerably toward greater neutron-excess values. This effect is illustrated by considering the example of the tin-isotope chain.     

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.     

Mittlere Energien und Driftgeschwindigkeiten von Elektronen in Stickstoff,Argon und Xenon,ermittelt aus Bildverstärkeraufnahmen von Elektronenlawinen

Local and temporal development of electron avalanches in a pulsed discharge gap (d=3,00 cm) are investigated in N₂, Ar, Xe and mixtures of N₂ and CH₄ by simultaneously applying high gain image intensifier- and photomultiplier techniques. Electron drift velocities are obtained from time-of-flight and way-of-flight measurements in these gases. The mean energy of agitation of the electrons is derived both from electron mobility and avalanche image trace profile (diffusion broadening). The results obtained (for 20°C), being in fair agreement with one another, read N₂: (4·6...5·0) eV forE/p=50...200 V/cm Torr; Ar: (9·0...9·5) eV forE/p=24... 45 V/cm Torr; Xe: (4·8...5·0) eV forE/p= 40... 90 V/cm Torr; CH₄(10% N₂): 6·3 eV forE/p= 89 V/cm Torr. The mean energy of agitation does not change very much withE/p in the ranges investigated. Some results concerning the radiation properties of these gases are included such as lifetime of the excited states, quenching pressure etc.     

Temperature dependence of the spectrum of electrons levitating above solid hydrogen

A theory of photoresonance transitions for the electrons localized above the solid H₂ surface is proposed on the basis of the experimental data obtained by V. V. Zav’yalov and I. I. Smol’yaninov [Pis’ma Zh. Éksp. Teor. Fiz. 44, 142 (1986); Zh. Éksp. Teor. Fiz. 92, 339 (1987); Zh. Éksp. Teor. Fiz. 94, 307 (1988)] and a simple model of condensed hydrogen, H₂. A new explanation is offered for the strong dependence of the transition frequency v on the hydrogen vapor pressure revealed in the above-cited works. In contrast to the notions that were proposed by V. V. Zav’yalov and I. I. Smol’yaninov [Zh. Éksp. Teor. Fiz. 92, 339 (1987); Zh. Éksp. Teor. Fiz. 94, 307 (1988)]; and V. B. Shikin and S. N. Nazin [Pis’ma Zh. Éksp. Teor. Fiz. 82, 752 (2005)] and based on the effects of the quantum refraction of electrons by hydrogen vapor atoms, the temperature dependence of the frequency ν = ν (T) is attributed here to the H₂ vapor inhomogeneity. The nonanalytic dependence of ν on the vapor density n _ν , ν(n _ν ) ? ν(0) ～ n _ν ^γ ～ e^?Δ/T , γ ? 0.56 has been revealed from the experimental data. The activation energy Δ corresponds to the “incomplete evaporation” of solid hydrogen [see J. I. Frenkel, Z. Phys. 26, 37 (1924); Kinetic Theory of Liquids (Nauka, Leningrad, 1945; Clarendon, Oxford, 1946)]. The parameter Δ is lower than the energy of the “complete evaporation” of solid H₂ equal to 92.6 K.     

Partial virial theorems for atomic-molecular systems

New virial relations for three-and four-particle atomic-molecular systems are proposed. Using operators of extension or squeezing of interparticle distances, it is shown that, for all pairs of j and k particles in S states of these systems, the following partial virial relations are valid: 〈2T _jk 〉+〈 V _jk 〉=0, where 〈V _jk 〉 is the average Coulomb interaction energy for a pair of particles and 〈T _jk 〉 is a part of the average kinetic energy of the system. There are three and six such relations for three-and four-particle systems, respectively. The conventional virial theorem (〈 2T〉+〈V〉=0) for the average total kinetic and potential energies of the system (〈 T〉 and 〈V〉, respectively) corresponds to the summation of partial virial relations over all pairs of particles. It is shown by an example of variational calculations of the helium atom ⁴He²⁺ e ^? e ^? and the helium muon-electron mesoatom ⁴He²⁺μ^? e ^? that partial virial relations are a highly sensitive indicator of the accuracy of wave functions.     

More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|~(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.