Mobility matrix

T is the free energy fiinctional, for which one can use equation (A3.3.52). The summation above corresponds to both the sum over the semi-macroscopic variables and an integration over the spatial variableThe mobility matrix consists of a synnnetric dissipative part and an antisyimnetric non-dissipative part. The syimnetric part corresponds to a set of generalized Onsager coefficients. 

Thus, the calculation of 2(Q) requires the knowledge of the partial static structure factors SaP(Q, t) and the elements PaP(Q) of the mobility matrix p(Q), which itself depend on SaP(Q,0). 

The evaluation of the generalized mobility matrix jj.(Q) follows the lines outlined in Section 5.1.1. The elements n1p(Q) can be approximated to high accuracy [100] by sums of a preaveraged term... 

Equations (125) and (126) explicitly show that in the initial slope approximation the elements of the generalized mobility matrix can be expressed only in terms of integrals over the corresponding partial static structure factor. Both equations are valid as long as one assumes a Gaussian distance distribution of the distances r between the monomers i on arm a and monomers j on arm p. 

Flory-Huggins parameters Xy do not influence the mobility. For the case of Rouse dynamics, which only depends on the local friction, the bare mobility matrix ... 

Mobility matrix Tube survival probability Neutron wavelength First cumulant matrix Rouse variable Osmometic pressure Conductivity... 

In the case of an invertible mobility matrix, expanding the required derivative of the matrix product on the RHS of Eq. (271) yields a drift velocity... 

The quantity Pp is a projection tensor, which reduces to the identity 8p in the case of an invertible mobility matrix, and which is always idempotent, since... 

X, ..., X. Such a transformation induces a trivial tensor transformation for the instantaneous force Tip(t). We show in the Appendix, Section H, by evaluating the time and ensemble average of the instantaneous force over a short time interval, that, in the case of a nonsingular mobility matrix, such a transformation creates a transformed force bias... 

The preceding definition of a kinetic SDE reduces to that given by Hiitter and Ottinger [34] in the case of an invertible mobility matrix X P, for which Eq. (2.268) reduces to the requirement that Zap = K. In the case of a singular mobility, the present definition requires that the projection of Z p onto the nonnull subspace of K (corresponding to the soft subspace of a constrained system) equal the inverse of within this subspace, while leaving the components of Z p outside this subspace unspecified. 

Change of Variables. It is shown in the Appendix, Section I that, for a set of kinetic SDEs with an nonsingular mobility matrix, a transformation from the L variables Z ,...,to another set of variables Z (X),..., X X) yields a transformed set of kinetic SDEs of the form... 

We have recently investigated another type of mobile matrix - a liquid metal [100, 102]. Here, we discovered that ion bombardment of the liquid metal surface, upon which sample particles were deposited, resulted in movement of the sample species towards the primary ion beam where they were desorbed and finally detected by the mass analyzer. 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

The coefficients L m are called the mobility matrix, and may be obtained using hydrodynamics. It can be proved that L , is a symmetric positive definite matrix ... 

Given the mobility matrix, the Smoluchowski equation is obtained from the continuity equation... 

To obtain the Smoluchowski equation for such a system, we first calculate the mobility matrix. Let Rt,Ri, . , i Ar = i be the positions of the spheres and Fi, J, ..., Fv be the forces acting on them. We assume that there eire no external torques acting on the particles. Then the velocities of the particles are written ast... 

In a very dilute suspension, the velocity of a particle is determined only by the force acting on it, and the mobility matrix becomes... 

To describe the dynamics of polymers in dilute solution, we have to take into account the hydrodynamic interaction, which is expressed by the mobility matrix calculated in Chapter 3,... 

A cautionary remark has to be made. The variational principle in this section is based on the positive definiteness of the mobility matrix, i.e., for any vector F ... 

This condition is guaranteed for the correct mobility matrix. However, the mobility matrix given by eqn (4.40) is an approximate one, and does not satisfy the inequality (4.1 ) in a certain configuration in which the beads are too close to each other. An improved formula which guarantees the inequality is proposed by Rotne and Prager. However, this correction is irrelevant for the asymptotic behaviour of N 1, which is determined by the hydrodynamic interaction between beads far apart from each other. Thus we shdl use eqn (4.40) for H, . 

Similarly, in the absence of any phase mixing, the hard phase is a hydrogen-bonded glassy or semicrystalline polymer. Experience of other polymers suggests that Gh i 000 MPa at room temperature [135], But this is a typical value for a macroscopic sample of polymer. In the thermoplastic PUs it must represent an upper bound on Gh, since in this case the hard segments are confined to such small domains that a large fraction of them reside at the particle surfaces, adjacent to the more mobile matrix, again as shown previously in Section 2.3.1. So a lower value is expected for Gh-... 

Multibody hydrodynamic interactions have generally been ignored in simulations (for reasons of computational cost) with the notable exception of [243,264] for a monolayer system involving a small number of particles. Satoh et al. [266,267] approximate the multibody hydrodynamic forces by assuming additivity of the velocities. This does, however, not guarantee positive definiteness of the mobility matrix (inverse of the resistance matrix), imless a short cutoff radius of the hydrodynamic interactions is used [266,267]. 

MIEZE technique 20 Mobility matrix 166 Mode coupling theory (MCT) 69,88,112, 141,142,146... 

It drives other monomers irtto motion. The net result is that the mobility matrix fi , introduced in eq. (VI.4), now becomes a very slowly decreasing function of the chemical intmval r — m this effect profoundly modifies the mode structure. ... 

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