Langevin random forces

Following the addition of water molecules by several consecutive procedures and after deletion of more remote parts of the protein, the protein was divided into a reaction region, a buffer region and a reservoir region, and separate equations of motion were applied to each. In the final model, about 40% of the enzyme was retained. Two trajectories have been investigated, which differ in the Langevin random force applied to water molecules that cross the boundary between the buffer and reservoir region. 

In this equation x, is the Cartesian coordinate x for atom i, /3, is the frictional drag on atom i and/, is the Langevin random force on atom i obtained from a Gaussian random distribution of zero mean and variance... 

Hereby, the first term is the conservative force on particle i due to interaction with other particles and the additional terms represent interactions with the continuum medium. R is the Langevin random force and is the friction coefficient. In DPD simulation, the polymer chain is represented by a coarse-grain model in which a coarse-grain particle (bead) represents a group of atoms. The total acting force on a DPD particle i is defined as ... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

We further note that the Langevin equation (which will not be discussed in detail here) is an intermediate between the Newton s equations and the Brownian dynamics. It includes in addition to an inertial part also a friction and a random force term ... 

Langevin dynamics simulates the effect of molecular collisions and the resulting dissipation of energy that occur in real solvents, without explicitly including solvent molecules. This is accomplished by adding a random force (to model the effect of collisions) and a frictional force (to model dissipative losses) to each atom at each time step. Mathematically, this is expressed by the Langevin equation of motion (compare to Equation (22) in the previous chapter) ... 

In Langevin dynamics, we simulate the effect of a solvent by making two modifications to equation 15.1. First of all, we take account of random collisions between the solute and the solvent by adding a random force R. It is usual to assume that there is no correlation between this random force and the particle velocities and positions, and it is often taken to obey a Gaussian distribution with zero mean. 

In the Langevin description, one assumes that the degrees of freedom within the system that are not explicitly considered in the simulation, exert, on average, a damping force that is linear in velocity y,-f, along with additional random forces Ti t). This leads to the following equation of motion for particle number i ... 

A disadvantage of Langevin thermostats is that they require a (local) reference system. Dissipative particle dynamics (DPD) overcomes this problem by assuming that damping and random forces act on the center-of-mass system of a pair of atoms. The DPD equations of motion read as... 

The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

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. 

We now instead calculate the drift velocity and diffusivity by directly integrating the traditional formulation of the Langevin equation in terms of random forces, and compare the results to those obtained above by rewriting the Langevin equation as a standard Stratonovich SDE. As in the analysis of the Stratonovich SDE, we calculate the first and second moments of an increment AX (f) = Z (f) — X (0) by integrating Eq. (2.262) from a known initial condition at f = 0. 

The results given above are essentially identical to those obtained by Hinch [10] by a similar method, except for the fact that Hinch did not retain any of the terms involving the force bias (tIv)o which he presumably assumed to vanish. An apparent contradiction in Hinch s results may be resolved by correcting his neglect of this bias. In a traditional interpretation of the Langevin equation as a limit of an underlying ODE, the bead velocities are rigorously independent of the hard components of the random forces, since the random forces in Eq. (2.291) appear contracted with K , which has nonzero components only in the soft subspace. Physically, the hard components of the random forces are instantaneously canceled by the constraint forces, and thus can have no effect... 

Equation (2.315) may be interpreted as a peculiar discretization of the Langevin equation, which is constructed so as to avoid the force bias that arises in the traditional interpretation. We identify random forces... 

The kinetic interpretation thus corresponds to a discretization of Langevin equation (2.262) in which the mobility is evaluated at a midstep position, as first suggested by Fixman [9], but in which the random force Pp is constructed at the... 

Stratonovich interpretation of the Langevin equation, it is the use of a midstep value of C (X) that causes the unwanted bias in the random forces. 

In both traditional and kinetic interpretations of the Cartesian Langevin equation for a constrained system, one retains some freedom to specify the hard and mixed components of the force variance tensor Several forms for Z v have been considered in previous work, corresponding to different types of random force, which generally require the use of different corrective pseudo forces ... 

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. 

We now show that the same drift velocity is obtained from a Stratonovich interpretation of the Langevin equation with unprojected and projected random forces that have the same soft components but different hard components. We consider an unprojected random force ri (t) and a corresponding projected random force Tj (f) that are related by a generalization of Eq. (2.300), in which... 

In Kramers classical one dimensional model, a particle (with mass m) is subjected to a potential force, a frictional force and a related random force. The classical equation of motion of the particle is the Generalized Langevin Equation (GLE) ... 

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