Langevin equation tensors

In this section, we introduce generalized definitions of sets of reciprocal basis vectors, and of corresponding projection tensors, which include the dynamical reciprocal vectors and the dynamical projection tensor introduced in Section VI as special cases. These definitions play an essential role in the analysis of the constrained Langevin equation given in Section IX. 

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. 

Note that the term involving a derivative of In / in Eq. (2.331) is identical to the velocity arising from the second term on the RHS of Eq. (2.286) for the transformed force bias in the traditional interpretation of the Langevin equation. The traditional interpretation of the Langevin equation yields a simple tensor transformation rule for the drift coefficient A , but also yields a contribution to Eq. (2.282) for the drift velocity that is driven by the force bias. The kinetic interpretation yields an expression for the drift velocity from which the term involving the force bias is absent, but, correspondingly, yields a nontrivial transformation mle for the overall drift coefficient. 

In both traditional and kinetic interpretations of the Cartesian Langevin equation for a constrained system, one retains some freedom to specify the hard and mixed components of the force variance tensor Several forms for Z v have been considered in previous work, corresponding to different types of random force, which generally require the use of different corrective pseudo forces ... 

In the preaveraged approximation for the Oseen hydrodynamic tensor [19, 20], the linear Langevin equation may be written as... 

The magnitude of that appears in Eq. (17.11) may also be studied in a dynamic way as described in the following Eq. (17.11) represents the time-correlation function of the stress tensor component Jxy t) in the long-time region described by the Langevin equation ... 

A rather efficient method to calculate the root of the hydrodynamic interaction tensor is Cholesky decomposition. The random displacements are then obtained via multiplying the root matrix with a vector of random numbers. The root is usually not unique, i.e., there are several matrices whose square is the diffusion tensor, but since any of these matrices yields random displacements which satisfy the condition eq. (3.22), this nonuniqueness averages out in the course of the simulation. These matrix operations become numerically rather intensive if the number of monomers becomes large (the number of operations is proportional to the third power of the number of monomers). The numerical algorithms for Langevin equations are well established, however, some details are still discussed today. ... 

Tsekov and Ruckenstein considered the dynamics of a mechanical subsystem interacting with crystalline and amorphous solids [39, 40]. Newton s equations of motion were transformed into a set of generalized Langevin equations governing the stochastic evolution of the atomic co-ordinates of the subsystem. They found an explicit expression for the memory function accounting for both the static subsystem—solid interaction and the dynamics of the thermal vibrations of the solid atoms. In the particular case of a subsystem consisting of a single particle, an expression for the fiiction tensor was derived in terms of the static interaction potential and Debye cut-off fi equency of the solid. 

Several renormalization group calculations of the dynamic exponent now exist, starting from a Langevin equation identical to equation (22), with either a preaveraged or a non-preaveraged Oseen tensor. These calculations provide an expansion of the exponents as a function of the small parameter 8=4—d, where d is the space dimension. So far all calculations are consistent with a value z=d (z = 3 in three dimensions). We will thus consider here that, whatever the quality of the solvent (good or 0 solvent), the dynamic exponent is z = 3 and that the characteristic time of a polymer coil is given by a Zimm formula (equations 31 or 32). 

Since describes the velocity response of particle i to the force acting on particle j, it must be identical to the mobility tensor appearing in the Langevin equation. 

Equation 3 can now be employed to calculate the linear and nonlinear susceptibilities in (1) and (2), revealing very specific tensor properties exhibited by poled polymers. With the foregoing model for the potential energy, the thermodynamic averages are sometimes also expressed in terms of the Langevin functions, which are defined as... 

Momentum conservation requires that an equal and opposite force be applied to the fluid. Both discrete and continuous degrees of freedom are subject to Langevin noise in order to balance the frictional and viscous losses, and thereby keep the temperature constant. The algorithm can be applied to any Navier-Stokes solver, not just to LB models. For this reason, we will discuss the coupling within a (continuum) Navier-Stokes framework, with a general equation of state p p). We use the abbreviations for the viscosity tensor (46), and... 

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