Langevin statistics

Above this stretch, between a fractional stretch of 0.15 and the fully stretched limit of Ac/v = 1.0, the statistics of chain stretch is described by a different formalism referred to as Langevin statistics, which we present in Section 6.5 below. 

The transition of bulk network from Gaussian to Langevin statistics accompanied with transfer of the part of a load to inner hard polymer layer at the filler surface (in the range of Mooney-Rivlin curve minimum). 

The stress of an elastomer obeying inverse Langevin statistics can be written (42,63)... 

The use of the inverse Langevin function, as for example to derive eqn (3.20) overcomes the objection that the Gaussian analysis does not take into account the finite extensibility of the network chains. However in order to make the Langevin statistics reasonably mathematically tractable assumptions have to be introduced which strictly make the statistics invalid under non-Gaussian conditions. Their justification must lie in the fact that the resultant expressions give far better fits to the experimental data than the Gaussian statistics. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

Thus, unlike molecular dynamics or Langevin dynamics, which calculate ensemble averages by calculating averages over time, Monte Carlo calculations evaluate ensemble averages directly by sampling configurations from the statistical ensemble. 

In order to proceed now to a statistical mechanical description of the corresponding relaxation process, it is convenient to solve the equation of motion for the creation and destruction operators and cast them in a form ressembling a Generalized Langevin equation. We will only sketch the procedure. 

The set of equations (50) can be formally considered as generalized Langevin equations if the operator Fj(t) can be interpreted as a stochastic quantity in the statistical mechanical sense. If the memory function does not correlate different solute modes, namely, if Kjj =Sjj Kj, then a Langevin-type equation follows for each mode ... 

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. 

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

This is the Langevin equation [271, 490]. The fluctuating force, f, varies so rapidly that it cannot be described as a function of time directly. Instead, only certain statistical properties can be defined. 

The plan of the article is as follows. First, we discuss the phenomenon of hydrodynamic interaction in general terms, and at the same time, we present some convenient notation. Then, we give the usual argument leading to the Fokker-Planck equation. After that we derive the Langevin equation that is formally equivalent to the Fokker-Planck equation, together with a statistical description of the fluctuating force. 

Equations (28) and (29) are derived from the statistical theory based on the Gaussian statistics which describes the network behaviour if the network is not deformed beyond the limit of the applicability of the Gaussian approximation33). For long chains, this limit is close to 30 % of the maximum chain extension. For values of r, which are comparable with rmax, the force-strain dependence is usually expressed using the inverse Langevin function 33,34)... 

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

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