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Rotational diffusion of a spheroidal Brownian particle
Abstract
The rotational Brownian motion of a large particle is considered in terms of mode-mode coupling theory. A modified Debye law for the rotational diffusion coefficient is obtained.
The structure and function of the influenza A M2 proton channel have been the subject of intensive investigations in recent years because of their critical role in the life cycle of the influenza virus. Using a truncated version of the M2 proton channel (i.e., M2TM) as a model, here we show that fluctuations in the fluorescence intensity of a dye reporter that arise from both fluorescence quenching via the mechanism of photoinduced electron transfer (PET) by an adjacent tryptophan (Trp) residue and local motions of the dye molecule can be used to probe the conformational dynamics of membrane proteins. Specifically, we find that the dynamics of the conformational transition between the N-terminal open and C-terminal open states of the M2TM channel occur on a timescale of about 500 μs and that the binding of either amantadine or rimantadine does not inhibit the pH-induced structural equilibrium of the channel. These results are consistent with the direct occluding mechanism of inhibition which suggests that the antiviral drugs act by sterically occluding the channel pore.
The microscopic origin of Stokes’ law for a hard sphere system is examined from two points of view. First, we derive a set of hydrodynamic equations with source terms, which account for the presence of a Brownian particle in the fluid, from the generalized Langevin equation. The friction coefficient can be expressed in terms of averages of the hydrodynamic fields over the Brownian particle. The solution of the source hydrodynamic equations for these averaged fields leads to the slip Stokes’ law result. Second, we consider the problem from kinetic theory and show that the same source hydrodynamic equations are obtained when projections onto hydrodynamic states are made.
There is a close relation between the Knudsen number dependence of the drag coefficient for a macroscopic body in a gas stream, and the density dependence of the transport coefficients for moderately dense gases in both two and three dimensions.
The analysis of the translational diffusion coefficient of a Brownian particle, given in the preceding paper, is extended to treat the rotational motion. Agreement with the hydrodynamic value for the frequency dependent friction coefficient is found.
The long-time behavior of autocorrelation functions of Q1, the lth rank irreducible product of components of a unit vector fixed in a single tagged member of a pure fluid, is studied. The Q1 couple to nonlinear products of the conserved hydrodynamic variables. The asymptotic time dependence is determined by the coupling to bilinear forms. The correlation function for 〈Q1(t)Q1〉 decays as t-7/2 for l = 1 and t-5/2-(1−2) for l ≥ 2.
A theory of many-particle systems is developed to formulate transport, collective motion, and Brownian motion from a unified, statistical-mechanical point of view. This is done by, first, rewriting the equation of motion in a generalized form of the Langevin equation in the stochastic theory of Brownian motion and then, either studying the average evolution of a non-equilibrium system or calculating the linear response function to a mechanical perturbation. (1) An expression is obtained for the damping function φ(t), the real part of whose Laplace transform gives the damping constnat of collective motion. (2) A general equation of motion for a set of dynamical variables At) is derived, which takes the form
where is a frequency matrix determining the collective oscillation of A(t). The quantity f(t) consists of those terms which are either non-linear in A(s), t ≧s ≧0, or dependent on the other degrees-of-freedom explicitly, and its time-correlation function is connected with the damping function φ(t) by (f(t1), f(t2)*) = φ(t1 − t2)·(A, A*). (3) An expression is obtained for the linear after-effect function to thermal disturbances such as temperature gradient and strain tensor. Both the conjugate fluxes and the time dependence differ from those of the mechanical response function. The conjugate fluxes are random parts of the fluxes of the state variables, thus depending on temperature. (4) The difference in the time dependence arises from a special property of the time evolution of f(t) and ensures that the damping function and the thermal after-effect function are determined by the microscopic processes in strong contrast to the mechanical response function. The difficulty of the plateau value problem in the previous theories of Brownian motion and transport coefficients is thus removed. (5) The theory is illustrated by dealing with the motion of inhomogeneous magnetization in ferromagnets and the Brownian motion of the collective coordinates of fluids. (6) Explicit expressions are derived for the thermal after-effect functions and the transport coefficients of multi-component systems
We construct a closed-form expression for the self-diffusion constant, D, for a hard-sphere particle whose mass and radius are large compared to the corresponding bath-particle quantities. The expression yields the Stokes-Einstein law at high bath-particle densities and the Boltzmann form for low densities. In addition, the first density correction to D is obtained and the higher-order density corrections are shown to diverge. The second density correction diverges as -log(k0R), where k0 is a cutoff wavevector and R is the radius of the particle.
The 13C nuclear spin-lattice relaxation time T1 was studied in liquid CS2 from −106°C to +35°C at resonance frequencies of 14, 30, and 62 MHz. The relaxation is caused by anisotropic chemical shift and spin-rotation interaction. It is shown that for linear molecules the spin-rotation constant C and the anisotropy of the chemical shift Δσ can be obtained from the relaxation rates without use of adjustable parameters. The analysis yields: C = -13.8 ± 1.4 kHz and Δσ = 438 ± 44 ppm.
Kawasaki has shown how to construct a nonequilibrium theory which relaxes to an equilibrium described by the standard Ising model. The main significance of Kawasaki's work is his proof that transport coefficients do not diverge near the critical point in his model. In this paper, his approach is generalized. Two master-equation models of transport are examined: one gives spin diffusion and thermal diffusion but no sound waves; the other gives thermal diffusion and sound waves. The first model involves a Hermitian transition matrix in the master equation. The Hermiticity enables one to prove a variational theorem which requires the transport coefficients to be finite. However, sound waves appear as complex eigenvalues of the relaxation time. Hence, they must come from a non-Hermitian master equation. A model is constructed which includes sound waves. In this case, the proof of the finiteness of transport coefficients fails. Aside from formal questions, the main physical point of this paper is the speculation that infinities in transport coefficients might be tied to the existence of oscillatory transport modes (like sound waves) coupled into the dynamics of the phase transition.
The long-time behavior of autocorrelation functions of conserved variables and of their corresponding dissipative fluxes is studied using the Mori formalism. A set of variables including linear and bilinear forms of the conserved variables is treated and consistent expansions to quadratic order in the wave-vector dependence of the appropriate correlation functions are carried out. The treatment is self-consistent and does not involve factorization of time-dependent correlation functions. Explicit calculations for the wave vector k⃗ and the frequency-dependent shear-viscosity coefficient are carried out in two dimensions. The asymptotic time dependence of the shear-viscosity correlation function is proportional to t-1 when k→0. The effect of higher-order nonlinear forms of the conserved variables has not been studied; such a study is essential to ensure the validity of the asymptotic time dependence found here.
The autocorrelation function of the density of a tagged particle is studied using the Mori formalism. The variables used are the collective conserved variables, the tagged-particle density, and bilinear products thereof. The case of point particles is considered in two dimensions, and, in three dimensions, self-diffusion by a particle of arbitrary size is treated. It is found that the bilinear-hydrodynamic approach automatically separates the self-diffusion coefficient of the tagged particle into a nonhydrodynamic part, and a hydrodynamic part which resembles the Stokes-Einstein law. In two dimensions, it is found that the mean-square displacement of a particle increases as tlnt, and that certain natural redefinitions of the diffusion and friction coefficients leave Einstein's law invariant. In three dimensions, for a large particle, the Stokes-Einstein law is reproduced. The relation between the well-known t(-3/2) "tails" on correlation functions, and the Stokes-Einstein law, is discussed.
Friction coefficients for the uniform rotation of prolate and oblate spheroids in a viscous fluid, with the slipping boundary condition, are computed numerically and reported in tabular form.
By using the Mori generalized Langevin equation an expression for the velocity autocorrelation function of a Brownian particle is obtained. By an appropriate choice of the potential of interaction between the particle and the fluid molecules the frequency dependent friction coefficient reduces to the value found when the slip boundary condition is applied in a hydrodynamic calculation; a different choice gives almost exactly the stick value. The reasons why the previous ’’mode‐coupling’’ calculations give an incorrect form for the friction coefficient are discussed.
The coupling of rotational and translation Brownian motions is examined from several points of view. The first is a phenomenological theory based upon generalized Langevin equations of motion and a Markoff integral equation. Next, a more detailed statistical‐mechanical theory is fashioned after the pattern of Kirkwood's theory for nonequilibrium processes in monatomic liquids. Both schemes lead to a generalized Fokker—Planck—Chandrasekhar equation for the singlet‐distribution function. This equation includes terms that account for separate rotational and translational motions as well as two mutually symmetric contributions which are descriptive of their coupling. The friction tensors associated with the uncoupled components of these motions are found to be proportional to the autocorrelations of the environmental force and torque which act upon a given molecule. The frictional coupling is related to the cross correlation of force and torque. From the principle of microreversibility it is possible to establish a reciprocal relationship between the two coupling tensors. A third approach is to derive the generalized Fokker—Planck equation from the Boltzmann equation for a dilute solution of rotating molecules. This has been done for the model of perfectly rough spheres and also for ``loaded spherocylinders.'' In both cases explicit formulas are obtained for the various friction tensors. The last section of the paper is devoted to the application of these theories to problems of diffusion.
A theory is given of the sharp viscosity rise in mixtures in the critical mixing region. Good agreement with the experimental dependence of viscosity on composition and temperature is found. The method involves a calculation of the entropy production through diffusion which results when a mixture in a state of composition fluctuation is caused to have a velocity gradient. The long wavelength part of the spectrum of composition fluctuations is intense and very easily distorted by a velocity gradient in the critical region. The return to uniform composition through diffusion dissipates energy, and the loss is interpreted as an excess viscosity. It is shown that this method, which requires no knowledge of the intermolecular potential but only of the long‐range part of the radial distribution function, correctly gives the viscosity of a dilute 1–1 electrolyte.
Near the critical point the characteristic time of motion associated with certain degrees of freedom experiences an enormous slowing-down. This circumstance gives rise to a possibility of constructing a kinetic equation to describe such slowed-down motions of the system, which corresponds to the Boltzmann equation of dilute gases which describes slow variations of the single-particle distribution function. We accomplish the derivation of kinetic equations with the aid of a generalized Langevin equation due to Mori. The theory is illustrated by deriving kinetic equations obeyed by critical fluctuations in isotropic and planar Heisenberg ferromagnets, an isotropic Heisenberg antiferromagnet, the liquid helium near the λ point, and a binary critical mixture. The kinetic equations conform to the dynamical scaling whenever it holds, and are valid in the hydrodynamic as well as in the critical regimes. The kinetic equations are then used to obtain selfconsistent closed equations to determine time correlation functions of critical fluctuations. In particular, in the case of a binary critical mixture, the selfconsistent equation can be used to obtain the diffusion constant and the decay rate of concentration fluctuation in the critical regime, which are expressed in terms of shear viscosity, and the results are in good numerical agreement with those of the recent light scattering experiments. The theory also leads to the modification of the Fixman correction to when q ⪡ k where q and k are the wavenumber and inverse correlation range of the concentration fluctuation, respectively.