Gaussian random force

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

Since Eq. (49) takes into account only the term of order Dt, the term of order in Eq. (51) is meaningless and the term linear in t in vanishes exactly. For T = 0, our result equals the well-known Smoluchowski rate. The main conclusion we can draw is that the activation rates for non-Markovian processes like Eq. (44) decrease as t increases the exact result of ref. 44 can thus be extended to the case of Gaussian random forces of finite correlation time as well. However, if we take Eq. (50) seriously, we obtain an Arrhenius factor, exp(A /Z)), of T(x) which does not exhibit a dependence on T. This is in contrast to the result found for telegr hic noises, where the Arrhenius factor increases with increasing autocorrelation time r (see ref. 44). The result of a numerical simulation for J(x) based on the bi-... 

The force arising from the potential is F, while R is a gaussian random force. The net effect of the collisions , i.e. dynamical interactions between the particle and solvent molecules, is thus approximately accounted for by the frictional, or damping force, Fj. = —fiC,x, where is a friction constant related to the time correlation of the random force ... 

Here, m is the mass of the particles, V(r) the potential energy of the system, 7 the friction constant, and. F is a Gaussian random force uncorrelated in time that satisfies the fluctuation dissipation theorem [20]... 

Using this result in Eq. (8.14) we find that the correlation function of the Gaussian random force R has the form... 

The Langevin equation (8.31), with R t) taken to be a Gaussian random force that satisfies (7 ) = 0 and (7 (0)7 (Z)) = ImykRTh t), is a model for the effect of a thermal environment on the motion of a classical harmonic oscillator, for example, the nuclear motion of the internal coordinate of a diatomic molecule in solution. 

The Gaussian random force (/) has zero mean but is correlated such that... 

The notation q denotes the time derivative, and yq(t) and yz(t) are the time-dependent friction forces acting upon q and z respectively. The Gaussian random forces are uncorrelated, with zero mean and obey the respective fluctuation dissipation relations ... 

As in Eq (1), f(t) is a Gaussian random force, but now, the angular friction coefficient Cr is proportional to the solvent-averaged time correlation function of f(t) ... 

The relation between the Langevin equations (3.49) [assuming -correlated gaussian random forces obeying (3.59, 61)] and the stochastic equation (3.29) has now to be established. The constitutive quantities of the latter equation are the moments, see (3.23, 28),... 

Another popular thermostat used in molecular dynamics simulations is the Langevin thermostat. It covers the heat-bath coupling part of the Langevin equation by friction and Gaussian random forces f. The Langevin equation basically describes the dynamics of a Brownian particle in solvent under the influence of external forces F. Its simplest form therefore reads ... 

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