Stochastic forces definition

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

In order to have a unique identification, a constraint on the stochastic force is imposed so the symmetric matrix in Eqn (15.102) to be semipositive definite ... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Dhar modelled the stretching of a polymer using the stochastic Rouse model, for which distributions of various definitions of the work can be obtained. Two mechanisms for the stretching were considered one where the force on the end of the polymer was constrained and the other where its end was constrained. Dhar commented that the variable selected for the work was only clearly identified as the entropy production in the latter case. In the former case they argue that the average work is non-zero for an adiabatic process, and therefore should not be considered as an entropy production, however we note that the expression is consistent with a product of flux and field as used in linear irreversible thermodynamics. 

For K 2kT and /Km, the collinear result of Eq. (5.13) differs from the three-dimensional model of this work by a factor of 3. This difference is interpreted in terms of the projection of forces along the internuclear axis. The slightly different kinematic factors arise, in part, from the definition of the collision frequency that is used to derive, Eq. (5.11). The hard-sphere model gave excellent agreement with simulations for a very steep exponential repulsive potential with exponent 2a = 256h, where b is that of the Morse oscillator. It is to be remembered that Eq. (5.12) was derived from a stochastic model with three major assumptions ... 

Because the rupture event is a stochastic process, the rupture forces are distributed in a certain range. Therefore, as in the experimental studies, we performed a large number of simulations and analyzed the rupture force distributions and the rejoin force distributions. The mean values as a function of the pulling velocity represent the so-called force spectmm. Experimentally, often a logarithmic dependence of the mean rupture force on v is observed, but the data collected by Schlesier et al. [98] allow no definite conclusion regarding this dependence. 

An alternative approach is to employ stochastic boundary conditions where the finite molecular system employed in the simulation is not duplicated, but rather, a boundary force is applied to interact with atoms of the sys-tem. 2-67 Yhe boundary force is chosen to mimic the bulk solvent regions that have been neglected. However, a difficulty with the use of this method is due to the ambiguity associated with the definition and parameterization of the boundary forces. Thus, there is sometimes the impression that the method is not rigorous. Recently, Beglov and Roux provided a formal definition and suggested a useful implementation of the boundary forces. Their results are very promising. 

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