Kinetic equations and time correlation functions of critical fluctuations.

Pattern dynamics in spatiotemporal chaos: Pattern selection, diffusion of defect and pattern competition intermettency

The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from ${T}_{c}$, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near ${T}_{c}$, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.

Scaling Laws for Dynamic Critical Phenomena

The chemistry and magnetic properties of metal nitronyl nitroxide complexes

An introduction is given to the methods and results of some recent researches into statistical thermodynamics bearing upon the correlation functions of magnetic moments in Heisenberg-coupled spin-only magnets, and their intimate connection with neutron-scattering theory and practice is brought out. The interrelationships between the correlation function, the relaxation function, the generalized susceptibility, the power spectrum of the fluctuations and the neutron scattering are explained, and it is shown that insights into any one of these aspects can serve to illuminate the others. Different forms of approximate theory are seen to be suitable in approaching the topic through these different avenues; the method of moments for analysing the power spectrum and the connection with Green function theory are described in particular. Some examples are calculated in molecular-field theory and with various other simple approximations; expressions for the frequency spectrum of the susceptibility are obtained in various temperature ranges. The extreme forms of the frequency spectrum in appropriate conditions as a set of delta functions, as a pseudo-Gaussian and as a pseudo-Lorentzian are derived, and the transitions between these cases are considered. The approximation of spin diffusion and its limitations are analysed; the problem of the dynamical slowing-down of magnetic fluctuations near the phase-transition point is examined and the current position in this enquiry is set out.

https://iopscience.iop.org/article/10.1088/0034-4885/31/2/305/pdf

Magnetic correlations and neutron scattering

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Dynamic Critical Phenomena in Magnetic Systems. I

The temperature dependence of EPR frequencies is studied theoretically for pure and pseudo-one dimensional Heisenberg magnets by means of Mori's theory of generalized Brownian motions. In a system with uniaxial symmetry, simple expressions for resonance frequencies are obtained. It is seen that the resonance frequencies are shifted from the Zeeman frequency when the magnetic susceptibility is anisotropic. Fisher's classical spin model is used to calculate the spin correlation functions. The anisotropy of the magnetic susceptibility is found to be enhanced by an effect of the short range order. The pseudo-one dimensional systems are treated with the help of the mean field approximation to interchain interactions. The result is reduced to that of a pure-one dimensional system, when the interchain interactions are absent. The effects of interchain interactions are negligible for antiferromagnets even near the Neel point, whereas those for ferromagnets are divergent near the Curie point. Nagata and Tazuke's e...

Temperature Dependence of EPR Frequencies in Pure-and Pseudo-One Dimensional Heisenberg Magnets

Most nonlinear ordinary differential equations exhibit chaotic attractors which have singular local structures at their bifurcation points. By taking the driven damped pendulum and the Duffing equation, such chaotic attractors are studied in terms of the q-phase transitions of a q-weighted average A(q), (-oo<q<oo) of the coarse-grained expansion rates A of nearby orbits along the unstable manifolds. We take their Poincare maps in order to obtain the expansion rates A and their spectrum g\(A) explicitly. It is shown that q-phase transitions occur at crises in the differential equations. Just before the crises, qp-phase transitions occur due to the collisions of the attractors with unstable periodic orbits. Numerical values of the transition points qp thus obtained agree fairly well with theoretical predictions. q.-phase transitions occur just after the crises where the chaotic . attractors are suddenly spread over chaotic repellers. Thus it turns out that the q-phase transitions of A(q) are useful for characterizing the chaotic attractors of differential equations at their bifurcation points.

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q-Phase Transitions in Chaotic Attractors of Differential Equations at Bifurcation Points

A rectangular micro-electrophoresis cell is commonly used for measuring the zeta-potential of mineral particles in suspension. It is thus important to analyze the electro-osmotic circulation of the dispersion liquid in the cell for determing the electrophoretic velocities of suspended solids.A theoretical calculation of this circulation was presented by Smoluchowski (1921) and Komagata (1933). According to their theories, the velocity of liquid changes with depth such that the velocity distribution may be represented by a symmetrical parabola in the cell. However, we have obtained results which conflict with their theories. In our experiments, the velocity distribution curve is represented by a parabola which is asymmetric with respect to the center line of the cell. A similar phenomenon had been suggested by Lane and White (1937) based on the Iatter's theory which seemingly was incomplete. By using methods of measuring electrophoretic velocities based on the above three theories, serious errors are obvious.We are thus proposing a unified theory for determining the electrophoretic velocity of mineral particles in a rectangular cell (equation (31)). The equations of Smoluchowski, Komagata and White are derivable from our theory as special cases. By our theory, we propose an improved method of determining the precise electrophoretic velocity of mineral particles (equation (41)) based solely on the coefficients of the velocity distribution curve which is represented by an asymmetrical parabola (equation (38)). In our experiments, this method gives satisfactory results.

A Unified Theory of Determining the Electrophoretic Velocity of Mineral Particles in the Rectangular Micro-Electrophoresis Cell

A new lattice theory of liquid-solid transitions is developed to clarify the statistical­ mechanical mechanism of freezing and melting processes. This is done by employing the expandable-lattice model in which the cell size is determined as a function of temperature and density by minimizing the free energy, and by introducing a simple approximation which takes into account the short-range correlation between molecules locating within the force range. In the pressure versus density diagram, the fluid and solid phase are represented by different, separate branches which have different cell sizes and molecular fractions. The branch which yields the fluid phase is disordered all over, while another consists of disor­ dered and ordered subbranches, the ordered subbranch leading to the solid phase. This result differs from those of Lennard-Jones and Devonshire's theory and the rigid-lattice theo­ ries, in which the fluid and solid phase are described by one branch which consists of dis­ ordered and ordered subbranches. Thus the fluid-solid transition turris out to have no critical point above which the transition ceases to be of the first order. It also turns out that although the solid state meets a thermodynamic instability below the melting density, the liquid metastable state remains stable far beyond the freezing density. We assume the Lennard-Jones potential and take up to the second-nearest-neighbor interaction. Then we obtain the usual shape of the phase diagram with the liquid range Tcr1t!Ttrip=l.38. Various constants of the critical and triple point are in fairly good agree­ ment with experiments except the pressure at the triple point. It is shown that the liquid states near the critical and triple point have adequate numbers of holes.

Hisao Okamoto

Papers

Dynamic Critical Phenomena in Magnetic Systems. I

Temperature Dependence of EPR Frequencies in Pure-and Pseudo-One Dimensional Heisenberg Magnets

q-Phase Transitions in Chaotic Attractors of Differential Equations at Bifurcation Points

A Unified Theory of Determining the Electrophoretic Velocity of Mineral Particles in the Rectangular Micro-Electrophoresis Cell

A Simplified Theory of Liquid-Solid Transitions. I