Brownian equation of motion

Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). 

As with Newtonian molecular dynamics, a number of different algorithms have been developed to calculate the diffusional trajectories. An efficient algorithm for solving the Brownian equation of motion was introduced by Ermak and McCammon [21]. A detailed survey of this and other algorithms as well as their application can be found in Ref. 2. 

For some problems, such as the motion of heavy particles in aqueous solvent (e.g., conformational transitions of exposed amino acid sidechains, the diffusional encounter of an enzyme-substrate pair), either inertial effects are unimportant or specific details of the dynamics are not of interest e.g., the solvent damping is so large that inertial memory is lost in a very short time. The relevant approximate equation of motion that is applicable to these cases is called the Brownian equation of motion,... 

It should be borne in mind that for ordinary Brownian equations of motion Fokker-Planck methods [3.47] are often attractive computational possibilities. 

In the general case, (A3.2.23) caimot hold because it leads to (A3.2.24) which requires GE = (GE ) which is m general not true. Indeed, the simple example of the Brownian motion of a hannonic oscillator suffices to make the point [7,14,18]. In this case the equations of motion are [3, 7]... 

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. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

The definition of the above conditional probability for the case of Brownian trajectories can be found in textbooks [12], However, the definition of the conditional probability for the Newton s equations of motion is subtler than that. 

The conceptual forerunner to mesoscale dynamics is Brownian dynamics. Brownian simulations used equations of motion modified by a random force... 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

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... 

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... 

For calculating the time-dependent properties of biopolymers, the equations of motion of the molecule in a viscous medium (i.e., water) under the influence of thermal motion must be solved. This can be done numerically by the method of Brownian dynamics (BD) [83]. Allison and co-workers [61,62,84] and later others [85-88] have employed BD calculations to simulate the dynamics of linear and superhelical DNA BD models for the chromatin chain will be discussed below. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

There has been very much effort devoted to the solution of the diffusion equation of motion for a reactant particle executing Brownian motion. The Euler equation of diffusion... 

After the work of Einstein and Smoluchowski an alternative treatment of Brownian motion was initiated by Langevin.Consider the velocity of the Brownian particle, as in VIII.4. When the mass is taken to be unity it obeys the equation of motion... 

To account for Brownian motion, the equation of motion for u given equation... 

The angle brackets denote averaging over the ensemble of the realisation of random forces in the equations of motion of the particles. The sum in (6.2) and (6.3) is evaluated over all the Brownian particles. The double index aa consists of the label of a chain a and the label of a particle a in the chain. 

In equation (6.33), the stresses in the moving viscoelastic liquid (6.31) are added to the stresses in the continuum of Brownian particles. When the equations of motion are formulated, we have to take into account the presence of the two interacting and interpenetrating continuous media formed by the viscoelastic liquid carrier and the interacting Brownian particles that model the macromolecules. However, the contribution of the carrier in the case of a concentrated solution is slight, and we shall ignore it henceforth. 

For molecules and colloids the forces due to molecular impacts are large, but they are short in duration and random in direction, as can be surmised from the observation of Brownian motion. These forces can be described by a rapidly fluctuating term A(t) in the equation of motion. With this Brownian term added to the other forces on the right, Eq. 3.8 assumes the form... 

The computationally intensity of the MD methods led to the development of Brownian Dynamics methods (BD), which use approximate expressions in place of the exact equations of motion. BD methods use randomness to simulate Brownian motion, and thus are stochastic in nature. Over sufficiently long time periods, Brownian motion appears random, and is amenable to this type of treatment. 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

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