Stochastic forcing function

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,... 

The phenomenological Langevin Eqs. (227) and (228) are only applicable to a very restricted class of physical processes. In particular, they are only valid when the stochastic forces and torques have infinitely short correlation times, i.e., their autocorrelation functions are proportional to Dirac delta functions. As was shown in the previous section, these restrictions can be removed by a suitable generalization of these Langevin equations. As we saw in the particular case of the velocity, the modified Langevin equation is... 

The fourth term on the right hand side of (3.4) represents the elastic forces on each Brownian particle due to its neighbours along the chain the forces ensure the integrity of the macromolecule. Note that this term in equation (3.4) can be taken to be identical to the similar term in equation for dynamic of a single macromolecule due to a remarkable phenomenon - screening of intramolecular interactions, which was already discussed in Section 1.6.2. The last term on the right hand side of (3.4) represents a stochastic thermal force. The correlation function of the stochastic forces is connected... 

The properties of the stochastic forces in the system of equations (3.31)-(3.35) are determined by the corresponding correlation functions which, usually (Chandrasekhar 1943), are found from the requirement that, at equilibrium, the set of equations must lead to well-known results. This condition leads to connection of the coefficients of friction with random-force correlation functions - the dissipation-fluctuation theorem. In the case under consideration, when matrixes f7 -7 and G 7 depend on the co-ordinates but not on the velocities of particles, the correlation functions of the stochastic forces in the system of equations (3.31) can be easily determined, according to the general rule (Diinweg 2003), as... 

F(f) is usually named multiplicative stochastic noise, since it multiplies the function g x), which depends on the variable of interest x. Due to the multiplicative nature of the stochastic force F t), it is not immediately... 

The behavior of (x y as a function of Q while keeping D fixed at several values of /i is shown in Fig. 6, whereas Fig. 7 shows the behavior of significantly influenced by the presence of an additive stochastic force. We are also in a position to state that in the special case where 0[Pg.462]

Note that the stochastic force has, for any value of Q = 0, a maximum at X = 1/2 and vanishes at x = 1 and x = 0. The potential, on the contrary, shows a more complex behavior as a function of x and Q. Its surface is shown in Fig. 3a. Let us discuss its properties together with the corre onding stationary distribution of gene frequency, which from Eq. (24) is found to be... 

The function y (Z), which is mathematically identical to the variable A of Section 6.5.1, represents a stochastic force that acts on the system coordinate X. Its stochastic nature stems from the lack of information about qjo and qjQ. All we know about these quantities is that, since the thermal bath is assumed to remain in equilibrium throughout the process, they should be sampled from... 

Nonstationarity can be introduced into the GLE through an auxiliary function g(t) so as to modulate the amplitude of the stochastic force [49,50]. The resulfing equation of motion has been called the iGLE where the "i" refers fo the irreversibly changing environment, and it takes the form. 

The difference between the GLE and iGLE is that the iGLE has the function g t) included in the friction kernel and random force. This g t) characterizes the amplitude of the stochastic force, and its changes allow to describe the irreversible processes in the nonequilibrium environment. 

Stochastic dynamic systems can be classified according to the very nature of/. Arnold Kliemann (1981) summarised the qualitative behaviour of x both for linear and nonlinear systems (for a condensed survey see Arnold (1981). The term linear is not specific here, since / can be linear either in state or in noise, even in both. In applications it is assumed very often that the forcing function has a systematic or deterministic part, and a term due to the rapidly varying, highly irregular random effects ... 

The friction coefficient C of the Brownian particle was determined by means of the stochastic force autocorrelation function (FACT) [4] ... 

With reference to earthquake engineering, we consider random support excitation in the above formulation, where the forcing function is modelled by a stationary stochastic process multiplied by a deterministic time-envelope-function, which accounts for the switch-on and die-off segments of typical earthquake records. As an example, we study the influence of the skewness angle of a trapezoidal simply supported plate on the random response characteristics. 

The key innovation in DPD is to apply the thermostat to particle pairs. A frictional damping is applied to the relative velocities between each neighboring pair, and a corresponding random force is added in a pairwise fashion also, such that Newton s third law holds exactly. The implementation is as follows. We define two functions (r) > 0, the relative friction coefficient for particle pairs with interparticle distance r, and a r) > 0, characterizing the strength of the stochastic force applied to the same particle pair. The fluctuation-dissipation theorem requires that... 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

The stochastic differential equation and the second moment of the random force are insufficient to determine which calculus is to be preferred. The two calculus correspond to different physical models [11,12]. It is beyond the scope of the present article to describe the difference in details. We only note that the Ito calculus consider r t) to be a function of the edge of the interval while the Stratonovich calculus takes an average value. Hence, in the Ito calculus using a discrete representation rf t) becomes r] tn) i]n — y n — A i) -I- j At. Developing the determinant of the Jacobian -... 

The second step is the molecular dynamics (MD) calculation that is based on the solution of the Newtonian equations of motion. An arbitrary starting conformation is chosen and the atoms in the molecule can move under the restriction of a certain force field using the thermal energy, distributed via Boltzmann functions to the atoms in the molecule in a stochastic manner. The aim is to find the conformation with minimal energy when the experimental distances and sometimes simultaneously the bond angles as derived from vicinal or direct coupling constants are used as constraints. 

Although this treatment does not explicitly involve interactions at a solid-liquid interface, the application of Green s function to find the stochastic friction force may be an excellent opportunity for modeling interfacial friction and coupling, in the presence of liquid. An interesting note made by the authors is that the stochastic friction mechanism is proportional to the square of the frequency. This will likely be the case for interfacial friction as well. 

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