Tensor friction

These results show that hydrodynamic interactions and the spatial dependence of the friction tensor can be investigated in regimes where continuum descriptions are questionable. One of the main advantages of MPC dynamics studies of hydrodynamic interactions is that the spatial dependence of the friction tensor need not be specified a priori as in Langevin dynamics. Instead, these interactions automatically enter the dynamics from the mesoscopic particle-based description of the bath molecules. 

The interpretation of the Langevin equation presents conceptual difficulties that are not present in the Ito and Stratonovich interpretation. These difficulties are the result of the fact that the probability distribution for the random force rip(f) cannot be fully specihed a priori when the diffusivity and friction tensors are functions of the system coordinates. The resulting dependence of the statistical properties of the random forces on the system s trajectories is not present in the Ito and Stratonovich interpretations, in which the randomness is generated by standard Wiener processes Wm(f) whose complete probability distribution is known a priori. 

Geometrically projected random forces, which were introduced by Hinch [10], have a variance given by the geometrically projected friction tensor... 

Cartesian kinetic SDEs with unprojected, geometrically projected, and inertially projected random forces require the same correction forces in certain special cases. Inertial and geometric projections are completely equivalent for models with an equal bead mass m for all beads, for which the mass tensor m v = is proportional to the identity. Unprojected and geometrically projected random forces require identical correction forces in the case of local, isotropic friction with an equal friction coefficient for all beads, as in the Rouse or Kramers model, for which the friction tensor ... 

The projection ab of the friction tensor onto the / dimensional soft subspace may be divided into corresponding blocks... 

BCAH refer to as the modified effective friction tensor, which they denote by the symbol i p. (See p. 188 of Ref. 4 for a table of these and related definitions.)... 

The expression given by BCAH for elements of the constrained mobility within the internal subspace is based on inversion of the projection of the modified mobility within the internal subspace, rather than inversion of the projection (at of the mobility within the entire soft subspace. BCAH first define a tensor given by the projection of the modified friction tensor onto the internal subspace, which they denote by the symbol gat and refer to as a modified covariant metric tensor, which is equivalent to our CaT - They then define an inverse of this quantity within the subspace of internal coordinates, which they denote by g and refer to as a modified contravariant metric tensor, which is equivalent to our for afi = 1,..., / — 3. It is this last quantity that appears in their diffusion equation, given in Eq. (16.2-6) of Ref. 4, in place of our constrained mobility Within the space of internal coordinates, the two quantities are completely equivalent. 

Equation (A.99) may, however, also be obtained by explicitly using projected random forces, as Hinch did in order to reproduce Frxman s result. The use of the projected friction tensor for Z v in Eq. (2.306), rather than the unprojected tensor has the same effect in that equation as did Fixman s neglect of the hard components of because the RHS of Eq. (2.306) depends only on the dot... 

Suppose for example that there is no unperturbed velocity field, V°(R) =0. Then the frictional force on the, /th bead is determined by not only its own velocity ft , but also by the velocities ft, of all the other beads. The only exception can be when the friction tensor , is strictly diagonal and this may be expected to be true only in the complete absence of hydrodynamic interaction. 

The error concerned an explicit formula for the translational diffusion coefficient. Kirkwood calculated the diffusion tensor as the projection onto chain space of the inverse of the complete friction tensor he should have projected the friction tensor first, and then taken the inverse. This was pointed out by Y. Ikeda, Kobayashi Rigaku Kenkyushu Hokoku, 6, 44 (1956) and also by J. J. Erpenbeck and J. G. Kirkwood, J. Chem. Phys., 38, 1023 (1963). An example of the effects of the error was given by R. Zwanzig, J. Chem. Phys., 45, 1858 (1966). In the present article this question does not come up because we use the complete configuration space. 

Here, m and rai are the mass and position vector of beads, respectively. is the friction tensor, which is assumed to be isotropic for simplicity in our simulation, that is, = Fl, where I is the unit dyad and r = 0.5t 1 (t = cr(m/ )° 5j (Grest, 1996). Further, f aj is the Brownian random force, which obeys the Gaussian white noise, and is generated according to the fluctuation—dissipation theorem ... 

The term is the frictional tensor. We assume it to be isotropic ( IT (I is unit tensor), is assumed to be Gaussian white noise that is generated according to the fluctuation-dissipation theorem [173,174]... 

The components of the frictional tensor do not generally disappear between the two spaces, but in the complimentary space — 0. 

Let us suppose that the liquid system is described by a MFPKE in N + 1 rigid bodies (the solute, or body 1 and N rotational solvent modes or bodies ), each characterized by inertia and friction tensors I and a set of Euler angles ft , and an angular momentum vector L (n = 1,..., N -I-1) plus K fields, each defined by a generalized mass tensor and friction tensor and a position vector and the conjugate linear momentum k = 1,..., K). The time evolution of the joint conditional probability x", L , P° 11, X, L, P, t) (where ft, X, etc. stand for the collection of Euler angles, field coordinates etc.) for the system is governed by the multivariate Fokker-Planck-Kramers equation... 

But the major contribution of the projected fast field to the resulting operator is given by a new frictional tensor (or collisional frequency tensor), which includes coupling terms between body 1 and 2 that are of a purely dynamic nature that is, they do not affect the final equilibrium distribution. The collisional matrices, modified by the averaged action of the fast field, may be expressed in the following way ... 

Since one-body models fail to reproduce such behavior, even if large mean field potentials are included, one must turn to a many-body description. One would expect that the solute body should be described as coupled to a collective solvent body in such a way that the potential energy of the system is not affected, in order to maintain the normal diffusive behavior of Tj (i.e., proportionality to tj). We may then introduce a friction tensor affecting the motion of the molecule and the first... 

Under equilibrium conditions, ° = 0 and this term can be dropped under the assumptions which have been introduced concerning the environment of the first molecule. The friction tensor is given by... 

The introduction of Eq. 34 with time-smoothing in the Kirkwood method eliminates detailed consideration of the higher-order distributions such as /<8> in the continuity equation for /(2). In a sense, however, they implicitly reappear in the evaluation of the frictional coefficients. Kirkwood avoids the reintroduction of the triplet density /(3) by the assumption that the pair frictional tensors and may be equated to the singlet friction tensor U which is evaluated in terms of the pair densities /<2). 

The pair kinetic theory equation given in Section VII.D can be used to extend the Smoluchowski results outlined earlier. In this section, we present the microscopic derivation of the Smoluchlowski equation from the kinetic theory and also obtain expressions for the space and time nonlocal diffusion and friction tensors, which appear in this theory. 

In the terms discussed above, the soft force between the solute molecule enters only in an indirect fashion. It does not enter at all in (9.33), and only through the propagator in (9.34). The second contribution to the friction tensor contains this force in a direct and explicit manner. Hence, when a strong direct chemical force operates between the molecules, one might expect this term to play an important role. If we evaluate the action of the... 

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