The Langevin equation

An alternative description of Brownian motion is to study the equation of motion of the Brownian particle writing the random force f(t) explicitly in the Langevin form  

Physically, the random force f t) represents the sum of the forces due to the incessant collision of the fluid molecules with the Brownian particle. As we cannot know the precise time-dependence of such a force, we regard it as a stochastic variable, and assume a plausible distribution P /(f)] for it. In this scheme, the average of a physical quantity A(x(f)) is calculated, in principle, by the following procedure. First eqn (3.27) is solved for any given/(f), and A x(t)) is expressed by/(r). The average is then taken with respect to/(f) for Ae given distribution hmction (/(f)]. 

Though various distributions can be conceived for /(f) depending on physical modelling, here we shall consider only the process which is equivalent to that described by the Smoluchowski equation. It is shown in Appendix 3.II that if the distribution of /(f) is assumed to be Gaussian characterized by the moment 

Consider the Brownian motion of a free particle ( / = 0) for which the Langevin equation reads 

Equation (3.31) indicates that x t)—x is a linear combination of Gaussian random variables f t). Therefore, according to the theorem in Appendix 2.1, the distribution of x(t) must be Gaussian. Hence the probability distribution of x(t) is written as 

An alternative way for describing Brownian motion is to study the equation of motion for the Brownian particle with the random force f t) written explicitly. This equation — the Langevin equation with the inertial term neglected (i.e. zero acceleration) — expresses the balance of the external, friction, and fluctuation forces  

The Langevin equation corresponding to the Smoluchowski equation of multiple stochastic variables — Eq. (3.20) — with the condition dLnmjdxm = 0, wMch is often true, is given by sa 

The Langevin equation and the Smoluchowski equation represent the same motion. Each of the equations has its advantages and disadvantages in solving the dynamic problems of interest. One would choose one over the other for its convenience in treating the problem of interest. 

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If we now average the Langevin equation, (A3.1.56). we obtam a very simple equation for (v(0), whose solution is clearly... 

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

Fig. 7. Writhe distributions for closed circular DNA as obtained by LI (see Section 4.1) versus explicit integration of the Langevin equations. Data are from [36].

Discretizing the Langevin equation (2,3) by IE produces the following system which implicitly, rather than explicitly, defines in terms of quantities... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

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 ... 

To improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

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) ... 

To integrate the Langevin equation, HyperChem uses the method of M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987 Ch.9, page 261 ... 

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

The basic equation of motion for stochastic dynamics is the Langevin equation. 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

The classical motion of a particle interacting with its environment can be phenomenologically described by the Langevin equation... 

Huller and Baetz [1988] have undertaken a numerical study of the role played by shaking vibrations. The vibration was supposed to change the phase of the rotational potential V (p — a(t)). The phase a(t) was a stochastic classical variable subject to the Langevin equation... 

This relation is as general as the Langevin equation itself, i.e., it holds for collisions of any strength. When deriving Eq. (1.23) from Eq. (1.26),... 

Altenberger and Tirrell [11] utilized the Langevin equation for particle motion coupled with hydrodynamics described by the Navier-S takes equation to determine particle diffusion coefficients in porous media given by... 

In the lattice-gas model, as treated in Section IV.D above, ion transfer is viewed as an activated process. In an alternative view it is considered as a transport governed by the Nernst-Planck or the Langevin equation. These two models are not necessarily contra-dictive for high ionic concentrations the space-charge regions and the interface have similar widths, and then the barrier for ion transfer may vanish. So the activated mechanism may operate at low and the transport mechanism at high ionic concentrations. 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Brownian motion theory may be generalized to treat systems with many interacting B particles. Such many-particle Langevin equations have been investigated at a molecular level by Deutch and Oppenheim [58], A simple system in which to study hydrodynamic interactions is two particles fixed in solution at a distance Rn- The Langevin equations for the momenta P, (i = 1,2)... 

The concept of the TS trajectory was first introduced [37] in the context of stochastically driven dynamics described by the Langevin equation of motion... 

These equations are formally identical to the equations (19) for the diagonal coordinates of the Langevin equation. The diagonal coordinates are in both cases determined only by the deterministic part of the dynamics. In the Hamiltonian setting they can naturally be identified as coordinates or momenta, which is impossible for the dissipative dynamics. 

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