Stochastic forcing

FIGURE 3 Simulated Northern Hemisphere temperature varations for 300 years with the ECHAM4/OPYC3 coupled climate model. Annually averaged and 50-year low-pass filter. 

We may therefore conclude that stochastic forcing is the most likely explanation for the natural variability in the period prior to 1900. It is also concluded that stochastic forcing for this model cannot explain the large, sustained warming during the 20th century. Similar results have been obtained in other model studies (Manabe and Stouffer, 1997). 

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In the presence of a potential U(r) the system will feel a force F(rj,) = — ViT/(r) rj,. There will also be a stochastic or random force acting on the system. The magnitude of that stochastic force is related to the temperature, the mass of the system, and the diffusion constant D. For a short time, it is possible to write the probability that the system has moved to a new position rj,+i as being proportional to the Gaussian probability [43]... 

Fig. 59. Time dependence of phases /1a and for a realization of stochastic force at T = 7 c- Also shown are the straight lines of the zero-temperature behavior of /I (solid line) and A (dashed line). Time is measured in units 2I/h.

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,... 

One aspect of MD simulations is that all molecules, including the solvent, are specified in full detail. As detailed above, much of the CPU time in such a simulation is used up by following all the solvent (water) molecules. An alternative to the MD simulations is Brownian dynamics (BD) simulation. In this method, the solvent molecules are removed from the simulations. The effects of the solvent molecules are then reintroduced into the problem in an approximate way. Firstly, of course, the interaction parameters are adjusted, because the interactions should now include the effect of the solvent molecules. Furthermore, it is necessary to include a fluctuating force acting on the beads (atoms). These fluctuations represent the stochastic forces that result from the collisions of solvent molecules with the atoms. We know of no results using this technique on lipid bilayers. 

R. Van Zon and E. G. D. Cohen, Extended heat-fluctuation theorems for a system with deterministic and stochastic forces. Phys. Rev. E 69, 056121 (2004). 

These constructions are evidently phenomenological in that they rely on consistency between the stochastic forces and their correlations. A more rigorous construction of these terms is therefore desirable. This has led to the use of the Mori projections of large-dimensional Hamiltonian systems. In particular, the projection of the Hamiltonian, ... 

Equation (5.40) reveals the essence of the diffusion coefficient. Since d( 2)/d/ = 2 , then = 2 >/, The same conclusion was reached in Section4.3 where we used an ensemble averaging procedure instead of introducing F°, the time average of the stochastic force, F(t), acting upon a single particle. 

Let us further analyze the stochastic force F(t) with respect to the dynamics of the system in the pre-Brownian regime (t< /0). Before we introduce models, let us clarify the role of F(t) by averaging the particle velocities v. To this end, we rewrite Eqn. (5.37) in the following form... 

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 second direction is connected with numerical modeling of climate change on a scale of decades and longer (in particular, with an explanation of the secular change in global annual mean SAT). Apparently, in the case of long-term climatic variability the influence of stochastic forcing (ISF) can be considered a zero hypothesis. 

Institute of Radio Engineering and Electronics International Study of Arctic Change International Satellite Cloud Climatology Project Index of Sustainable Economic Welfare Influence of Stochastic Forcing International Organization for Standardization Independent Summary for Policy-Makers International Sea Year... 

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... 

These equations describe the reptation normal relaxation modes, which can be compared with the Rouse modes of the chain in a viscous liquid, described by equation (2.29). In contrast to equation (2.29) the stochastic forces (3.47) depend on the co-ordinates of particles, equation (3.48) describes anisotropic motion of beads along the contour of a macromolecule. 

Now, we have to discuss in some details the properties of the stochastic force (f), defined so that ( f(t)) = 0. The second-order moment... 

We reproduce the procedure used in Section 3.4 and consider the stochastic forces Cf in the above system of equations as the sum of two independent processes... 

Here F°(f) is the Langevin stochastic force, the average and the autocorrelation of which are given by the equations ... 

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