E. Gross

Hydrodynamics of a Superfluid Condensate

The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self‐consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self‐consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a bo...

Kinetic Models and the Linearized Boltzmann Equation

Attention is directed to some unsatisfactory features of kinetic theory treatments of problems for which the linearized Boltzmann equation is applicable. The main defects are in the region where nearly free molecular flow conditions prevail. They can be overcome when the problems are treated by simplified kinetic models. In this paper relations between the linearized Boltzmann equation and some models are established. The method is based on a comparison of the eigenvalue spectra of the respective collision operators. Particular attention is paid to inverse fifth molecules. This allows evaluation of the limitations of a given model and shows how more accurate models can be constructed. It is shown how one may overcome the chief shortcomings of approximate solutions of the linearized Boltzmann equation.

Boundary value problems in kinetic theory of gases

Abstract The state of a gas, flowing between two parallel plates, is analyzed from the viewpoint of kinetic theory. When the mean free path is infinitely greater than the distance between the plates, the exact solution shows that the distribution function is discontinuous in velocity. One must distinguish between molecules impinging on a plate and those leaving. We investigate the problem of finding a theory valid for arbitrary ratio of mean free path to plate distance, and of plate speed to sound speed. This is most easily achieved by splitting the distribution function into the above mentioned parts, and expanding each part in polynomials in velocity space, which are orthogonal over half the velocity range. In every approximation, exact account is given of (1) the microscopic boundary conditions, (2) the conservation laws, and (3) the behavior in the low pressure region. The method, which can be applied to the Boltzmann equation, is here developed for the kinetic model of Bhatnager, Gross, and Krook. Variational principles are stated by noting the similarity of the linearized version of this theory to the Milne equation of radiative transfer. For the nonlinear, high speed case, a new approach in the low pressure region is indicated. The relationship to alternative methods is discussed. When the distribution function is expanded in full-range orthogonal polynomials it is necessary to go to high order to obtain an adequate representation of the low pressure region, and of the boundary layer. Very simple half-range distribution functions yield an accurate description of the state of the gas.

The DELPHI Microvertex detector

The DELPHI Microvertex detector, which has been in operation since the start of the 1990 LEP run, consists of three layers of silicon microstrip detectors at average radii of 6.3, 9.0 and 11.0 cm. The 73728 readout strips, oriented along the beam, have a total active area of 0.42 m2. The strip pitch is 25 μm and every other strip is read out by low power charge amplifiers, giving a signal to noise ratio of 15:1 for minimum ionizing particles. On-line zero suppression results in an average data size of 4 kbyte for Z0 events. After a mechanical survey and an alignment with tracks, the impact parameter uncertainty as determined from hadronic Z0 decays is well described by (69pt)2 + 242 μm, with pt in GeV/c. For the 45 GeV/c tracks from Z0 → μ− decays we find an uncertainty of 21 μm for the impact parameter, which corresponds to a precision of 8 μm per point. The stability during the run is monitored using light spots and capacitive probes. An analysis of tracks through sector overlaps provides an additional check of the stability. The same analysis also results in a value of 6 μm for the intrinsic precision of the detector.

Inertial Effects and Dielectric Relaxation

The study of dielectric relaxation in compressed gases requires an extension of existing theory. As the gas pressure is lowered below several hundred atmospheres, the inertial response of the dipoles gives rise to large deviations from the Debye equations. The absorption and dispersion depend on collision time and frequency of applied field in a manner which is sensitive to the molecular constants and to the state of dynamical order of the compressed gas.These phenomena are investigated by studying the response to an alternating electric field of a dilute solution of dipolar molecules in a nonpolar compressed gas. The dipoles are described by a classical distribution function which is a function of angular velocity, orientation, and time. We assume that the duration of collision may be neglected compared to the time between collisions, the period of the applied field, and the mean thermal period. The distribution function satisfies a kinetic equation; the effects of collisions are described by a collision...

Kinetic Theory of Linear Shear Flow

The flow of a monatomic gas between two parallel plates kept at the same temperature and moving in opposite directions is studied. The relative velocity of the plates is much smaller than the speed of sound. The deviation from the equilibrium distribution, φ(c, x), satisfies the linearized Boltzmann equation. The customary boundary conditions are adopted in which a fraction of the molecules is specularly reflected and the rest emitted with a Maxwellian distribution characteristic of the plate. The method consists of setting φ = φ+ for cx > 0 and φ = φ− for cx < 0 so that positive and negative velocities are distinguished. We take φ± = a0±(x)cz + a1±(x)czcx. The space functions are determined by taking half‐range velocity moments of the Boltzmann equation. Explicit results for the distribution function, flow velocity and stress are given for a general law of force. Numerical results are worked out for hard sphere molecules. The method treats both microscopic boundary conditions and conservation laws exactl...

Heat Flow Between Parallel Plates

A study is made of the flow of heat between parallel plates of slightly different temperatures. The problem is described by the linearized Boltzmann equation which is subject to microscopic boundary conditions. We approximate the distribution function by half‐range polynomials in velocity space and determine the space‐dependent coefficients by forming half‐range moment equations. An approximation involving four pairs of space functions suffices to give an accurate treatment of the heat flow and of the density and temperature profiles for the entire range of conditions from free molecule to hydrodynamic. Detailed numerical results for the temperature slip and molecular boundary structure are obtained for hard‐sphere molecules. The accuracy of cruder half‐range approximations and other methods of fixing the coefficients is established.

Classical theory of boson wave fields

Abstract The subject of discussion is the Hamiltonian for a system of bosons interacting by two body forces, as expressed in the formalism of second quantization. In this paper, we examine properties of the classical wave field governed by the Hamiltonian. For a general potential there is always an exact solution representing a uniform density. Exact solutions are exhibited, which represent disturbances of a definite velocity and of arbitrary amplitude. For small amplitudes the disturbances obey Bogolyubovs dispersion relation. Corresponding solutions are found for disturbances when the system moves as a whole. For suitably attractive potentials we find a class of exact solutions, degenerate in energy, with spatially periodic density. These solutions have a lower energy than the uniform type. Small amplitude excitations are investigated for the periodic case. They are phonons for long wavelengths, but show a band character at shorter wavelengths. A theory of the motion of foreign atoms in the boson fluid is formulated.

Analytical methods in the theory of electron lattice interactions Part I

Abstract A study is made of the low-lying levels of an electron interacting with a quantized lattice vibration field. To treat the case of arbitrary coupling strength, a canonical transformation is performed. The new lattice variables describe motions carried out partly relative to the instantaneous electron position and partly about mean values depending on the average electron state. In the present initial work the wave functions are assumed to be a product of electron and lattice functions in the new variables, and are not translationally invariant. They contain functions describing the tie between particle and field and a function describing the tendency of the particle to be localized in space. The theory that results when these functions are determined in the optimum way is explored, and it is noted that the method points to a natural systematic perturbation theory. The ground state energy and effective mass of a particle is studied in detail. The approach can be directly generalized to treat more complicated Hamiltonians which involve periodic potentials, magnetic fields or many electrons.

Kinetic Theory of the Impulsive Motion of an Infinite Plane

The half‐range method of solution of the Boltzmann equation is used to give a kinetic theory treatment of the Rayleigh problem. In contrast to the Navier‐Stokes or Grad theories, the method yields exact results for the initial stress and slip velocity at the boundary. It also indicates corrections to the classical Rayleigh results, even at long times compared to the collision periods of the gas molecules. The corrections are shown to be related to the usual slip boundary conditions of hydrodynamics plus an additional part arising from boundary layer effects. The half‐range method predicts corrections to the standard methods of between 10%–25% for all values of the time.