The logarithm of the quantum Boltzmann density &‖X〉, where $H_V$=$p^2$/2m+V, is expressed as a cumulant expansion in powers of v=V-W, where W(x)=V(X)+V’(X)(x-X)+(1/2)V’’(X )(x-X${)}^2$ is the local quadratic approximant to V(x) at the point X. Where V’’(X)>0, this expansion behaves ‘‘nonsecularly’’ as β→∞ (all its terms ∼β), and thus remains a useful approximation scheme even as the temperature ${\beta }^{\mathrm{-}1}$→0 (in that limit, it yields Rayleigh-Schrödinger perturbation expansions of the ground state of $H_V$). By Taylor expanding v(x) about X in the cumulant expansion, we obtain an expansion which is a resummation over powers of V’’(X) of the Wigner-Kirkwood (WK) expansion of ln${\rho }_V$; this ‘‘renormalized’’ WK expansion, whose coefficients are simple functions of V’’(X), is as simple to use as the ordinary WK expansion, yet more accurate where V’’(X)≠0, and usable down to zero temperature where V’’(X)>0 (yielding, in that limit, WK-type expansions for the ground state of $H_V$). In lowest order, it yields an approximation initially proposed by Miller.

A new type of noise-induced precursor of a period-doubling bifurcation is described. The usual noise rise in the power spectrum at half the fundamental frequency, typical just before the onset of period doubling, may be preceded by a pair of symmetrically located noise bumps characteristic of a Hopf bifurcation precursor. The continuous evolution from the Hopf precursor to the period-doubling precursor may be understood in terms of the behavior of the Floquet multipliers as a control parameter is varied. The necessary appearance of the virtual Hopf phenomenon may be proven for a large class of low-dimensional systems, including the driven damped pendulum and the Lorenz equations, between each successive period doubling in the infinite cascade leading to chaos.

In order to elucidate the molecular mechanism of muscle contraction, we propose a theory of nonlinear lattice solitons in muscle proteins. In a superstructure of myosin molecules, there are two α-helical polypeptides. In each helix there are three one-dimensional chains of peptide groups joined together by H bonds which stabilize the α-helical structure. Since the H-bonding interaction between peptide groups has a remarkable nonlinearity due to its charge-transfer interaction, we can regard a H-bonded chain of peptide groups as a one-dimensional nonlinear lattice. By using the earlier results of theoretical studies for nonlinear lattice solitons and for the initial-value problems, we can describe the molecular mechanism in the processes of the contraction of muscle.

A systematic perturbation theory to deal with stationary, homogeneous turbulence in a dispersive dissipative system is developed and is shown to be renormalizable. General properties of the renormalized equations are discussed, and for the specific case of Vlasov-Poisson turbulence, it is shown that the present theory reduces to the conventional weak turbulence theory. Kramers-Kronig kind of dispersion relations are derived for the nonlinear dielectric.

This is the first in a series of papers in which we carry out a systematic study of multiparticle correlation effects in the dense, high-temperature plasmas, appropriate to the inertially-confined-fusion experiments and the interior of the main-sequence stars. In this paper (paper I), we develop a general density-response formalism with inclusion of varied degrees of the electron degeneracy and the local-field corrections (LFC’s) describing the strong Coulomb-coupling effects. An explicit theoretical scheme of calculating the static LFC’s is developed on the basis of the hypernetted-chain approximation; useful expressions for the dynamic LFC’s are proposed.

On the basis of the general theoretical framework constructed in the preceding paper (paper I in the present series), we calculate in this paper various correlation functions in dense plasmas, taking account of local-field corrections and varied degrees of electron degeneracy; physical implications of the results are investigated.

On the basis of the general formalism and the analysis of the correlation functions presented in the preceding two papers (I and II), we calculate in this paper (III) various thermodynamic functions with explicit inclusion of the varied degrees of the electron degeneracy and the local-field corrections describing the strong Coulomb-coupling effects. The physical implications of the computed results are investigated through comparison with other model calculations. Mindful of practical applications of the present theory, we derive the analytic interpolation formulas parametrizing the computed results accurately for the thermodynamic functions.

The stopping power of a dense, two-component plasma is calculated in the dielectric formulation, where the static and dynamic local-field corrections (LFC’s) describing strong Coulomb-coupling effects beyond the random-phase approximation are explicitly taken into account. We thus clarify the extent to which the LFC’s and the presence of ions act to modify the rate of inelastic scattering. It is found that when the velocity v of the injected particle is smaller than the Fermi velocity $v_F$ of the electrons, LFC’s enhance the stopping power; when v≪$v_F$, the ionic contribution may predominate over the electronic contribution, which takes on a much reduced magnitude.

On the basis of the quantum-statistical formulation of electronic transport, we calculate the electric and thermal conductivities of plasmas in a wide range of densities and temperatures where it is essential to take into account the varied degrees of electron degeneracy and local-field corrections describing the strong Coulomb-coupling effects. The physical implications of the results are investigated through comparison with other theories and experiments. For utility in the practical applications, we derive the analytic formulas parametrizing the computed results accurately for the generalized Coulomb logarithms appearing in those conductivities.

Theoretical predictions are given of the development of the current and the distributions of charge and electric field in negative corona, or Trichel current pulses [G. W. Trichel, Phys. Rev. 54, 1078 (1938)], in oxygen at a pressure of 6.67 kPa (50 Torr). For a 10-mm-diam negative sphere located 20 mm from a positive plane, the calculated current pulse has a rise time of 11 ns, a pulse width of 50 ns, and a peak amplitude of 13 mA. These results agree satisfactorily with experimental values. The predicted velocity of the cathode-directed light pulse also agrees well with observations. The theory is based on the accurate numerical solution of Poisson’s equation in conjunction with the continuity equations for electrons, positive ions, and negative ions. The effects of ionization, attachment, recombination, electron diffusion, and photoemission and ion secondary-electron emission from the cathode are all included. The initial steep rise of the current pulse is largely due to rapid ionization and electron motion in the high Laplacian field near the cathode. As the discharge develops, a dense plasma forms near the cathode, leading to strong space-charge distortion of the field. A prominent cathode fall region is formed immediately adjacent to the cathode, an almost zero field is formed within the plasma and the field is enhanced over the region to the anode. The current pulse is quenched because the low electric field in the plasma immobilizes the majority of the electrons which then undergo three-body attachment; furthermore, the cathode fall region becomes reduced to such a short distance that insignificant current is produced from this region. Because of the low mobility of the negative ions, the current remains low and the structure of the space-charge fields changes only slowly with time between pulses.

A previous model, designed so as to simulate the coupling between a one-dimensional, quantal harmonic excitation in a fermion heat bath, is extended to consider three-dimensional vibrations with axial symmetry. This investigation addresses future studies on the relaxation of collective modes in finite Fermi systems such as deformed nuclei. For this application, the equations of irreversible dynamics in the proposed model are extracted and their solutions are examined in a range of parameters appropriate for the nuclear fluid. The formulation, techniques, and discussion are far more general and can be applied to a variety of problems arising in statistical mechanics and many-body physics.

It is possible to adjust the density of cyclohexane by adding a small amount of deuterated cyclohexane. Then the mixture deuterated cyclohexane-cyclohexane-methanol can be made isopycnic. We have determined the critical properties of this system and of the systems cyclohexane-methanol and deuterated cyclohexane-methanol: coexistence curve, correlation length, and osmotic compressibility via refractive index and turbidity measurements. For that purpose, a detailed discussion of the validity of the volume additivity and of the Lorentz-Lorenz formula has been made. The main result is that the isopycnic system acts as a binary mixture for the phase transition properties as shown by the β exponent and the amplitude combinations ($R_{\xi }^{+}$$R_c^{\mathrm{-}1/3}$) which exhibit the universal values. Some aspects of microgravity conditions can then be created. We have studied in the isopycnic system the phase separation at critical concentration; new features are found after a thermal quench: macroscopic spinodal decomposition structures during the phase separation process and macroscopic wetting layers in the final equilibrium state.

We prove that the hypernetted-chain and soft-mean-spherical integral-equation theories, together with the variational perturbation theory (based on the Percus-Yevick approximation for the hard-sphere reference system) are all identical in the asymptotic high-density limit: They share the same Madelung energy that constitutes an exact lower bound to the true potential energy of the system. The Percus-Yevick equation for hard spheres is shown to diverge at total packing fraction equal to unity, with a direct correlation function c(r) given by the overlap volume of two spheres with separation r. We present the exact analytic solution of the mean-spherical approximation for any repulsive potential φ(r) satisfying φ̃(k)≥0 which is a Green’s function for an operator of the Sturm-Liouville-type. The corresponding direct correlation function is given by the interaction between the particles when they are ‘‘uniformly smeared’’ inside a sphere. Our results are general, applicable to any number of dimensions and to mixtures. The physically intuitive meaning for the direct correlation functions is offered as a starting point for the statistical thermodynamic analysis of nonspherical hard, and/or charged, objects.

Fréedericksz transition under twist deformation in a nematic layer is discussed when the magnetic field has a random component. A dynamical model which includes the thermal fluctuations of the system is presented. The randomness of the field produces a shift of the instability point. Beyond this instability point the time constant characteristic of the approach to the stationary stable state decreases because of the field fluctuations. The opposite happens for fields smaller than the critical one. The decay time of an unstable state, calculated as a mean first-passage time, is also decreased by the field fluctuations.

A large-amplitude longitudinal electrostatic electron plasma wave excited at the beat frequency of two laser beams is shown to be strongly unstable against the filamentation instability for arbitrary short-wavelength perturbations in the presence of an external transverse magnetic field. The Vlasov equation in terms of guiding-center coordinates has been employed to obtain the nonlinear response of electrons in the plasma. It is concluded that a strong static magnetic field efficiently reduces the growth rate of the filamentation instabilty of the excited electron plasma wave.

The Hasegawa-Wakatani nonlinear equations are used to study the propagation of localized interchange and drift-ballooning vortices in a nonuniform magnetic field.

The Ermakov problem of finding an exact invariant of the time-dependent harmonic oscillator is derived and solved within the framework of Nelson’s stochastic mechanics.

General formulas are derived for the two-center overlap and nuclear-attraction integrals with the arbitrary location and screening constants of Slater-type orbitals. The final results are expressed in terms of the spherical harmonics and the usual $A_k$ and $B_k$ functions. Useful formulas have also been established for the radial parts of overlap and nuclear attraction integrals in the form of polynomials in p=ζR.

Considerable interest has been generated in the asymptotic form of the singlet density for any atom or molecule. Using some earlier work on the second-order density matrix, it has been possible to obtain an analogous result for the pair correlation function in the same systems.

The ground-state energy of the two-dimensional (2D) hydrogen molecule has been calculated using a Heitler-London-Rosen-type variational calculation. The value of the dissociation energy is found to be 2.5 Ry. For the interatomic distance R=0, the variational parameter ${\alpha }^{2\mathrm{D}=1.7055\mathrm{}}$, corresponding to a 2D He atom. For the sake of comparison, the results of the 3D case are also recalculated. Special interest arises from the Slater integrals, which are rather difficult to obtain in the 2D case.