Fluctuating force

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

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 this equation, m. is the effective mass of the reaction coordinate, q(t -1 q ) is the friction kernel calculated with the reaction coordinate clamped at the barrier top, and 5 F(t) is the fluctuating force from all other degrees of freedom with the reaction coordinate so configured. The friction kernel and force fluctuations are related by the fluctuation-dissipation relation... 

In the limit of a very rapidly fluctuating force, the above equation can sometimes be approximated by the simpler Langevin equation... 

VER occurs as a result of fluctuating forces exerted by the bath on the system at the system s oscillation frequency O [5]. Fluctuating dynamical forces are characterized by a force-force correlation function. The Fourier transfonn of this force correlation function at Q, denoted n(n), characterizes the quantum mechanical frequency-dependent friction exerted on the system by the bath [5, 8]. 

Consider an excited condensed-phase quantum oscillator Q, witli reduced mass p and nonnal coordinate q j. The batli exerts fluctuating forces on the oscillator. These fluctuating forces induce VER. The quantum mechanical Hamiltonian is [M, M]... 

The fluctuating forces F(t) on tire rigid oscillator D are characterized by a time-dependent force-force correlation Emotion [M, 55],... 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

This regime involves forces which are so strong that the ligand undergoes a drift motion governed by (3) in the limit that the fluctuating force aN t) is negligible compared to the applied force. In this case a force of about 800 pN would lead to rupture within 500 ps. 

There are two main reasons why a pump should not operate below its MCSF (/) the radial force (radial thmst) is increased as a pump operates at reduced flow (44,45). Depending on the specific speed of a pump, this radial force can be as much as 10 times greater near the shut off, as compared to that near the BEP and (2) the low flow operation results in increased turbulence and internal flow separation from impeller blades. As a result, highly unstable axial and radical fluctuating forces take place. 

Furthermore, we introduce additionally external forces f xj) and fluctuating forces and assume -correlations for the fluctuations ... 

Since the fate of a trajectory is determined not only by its initial condition but also by the fluctuating force acting on the system, the critical velocity v jjn depends on the instance a of the noise. For a fixed instance a, the criterion (51) allows one to calculate the probability for a trajectory randomly sampled from the ensemble (49) to be reactive ... 

To establish the relationship between dissipation and fluctuation, consider a system left in thermal equilibrium, without any applied force. It is expected that spontaneous fluctuation Q is associated with a spontaneously fluctuating force. 

Let (V2) be the mean square value of the fluctuating force, and (Q2) the mean square value of Q. Suppose that the system is known to be in the nth eigenstate. Since Hq is hermitian the expectation value of Q, (En Q En) vanishes and the mean square fluctuation of Q is given by the expectation value of Q2, i.e. 

To reiterate, a system with generalized resistance R lo) exhibits, in equilibrium, a fluctuating force given by (58), or at high temperature, by (59). 

The time dependent friction coefficient, per solute mass p, is related to the fluctuating forces exerted by the solvent on the solute coordinate x through their time correlation function ... 

So far, the solvent coordinate has not been defined. As noted at the beginning of this Section, the time dependent friction is to be found for the reacting solute fixed at the transition state value x of x. By (3.14), its dynamics were related to those of an (unspecified) solvent coordinate. v. One strategy to identify the solvent coordinate, its frequency, friction, etc., would be to derive an equation of motion for the relevant fluctuating force SF there. To this end, one can use a double-membered projection technique in terms of 8F and 8F. In particular, we define the projection operator... 

Let us make some connections to the results which came from the previous model development. First, if we compare (3.19)-(3.22) with (3.11)-(3.15), a natural identification of the solvent coordinate s in Sec. 3 is in fact just the fluctuating force SF on x at the transition state. (Note especially that this choice associates the solvent coordinate with a direct measure of the relevant solute-solvent interaction.) The solvent mass, force constant and frequencies in Sec. 3 would then be given molecular expressions via (3.19)-(3.21), while the solvent friction i (t) of Sec. 3 would be the friction per mass for Sf (3.22),... 

In fact, this sort of identification of the solvent coordinate s with the fluctuating force SF was first noticed in [5b], but its generality had not been pursued until Ref 45. It is important to remark that this force SF is a very highly nonlinear function of the coordinates of the solvent molecules, referenced to the solute location [5b] one should not at all think (as some evidently do) that the intermolecular forces can be untenably linearized. 

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. 

However, we are interested in situations in which the B-particle is not quite large enough to allow a continuous description of the fluid in other words, we want to express the fact that, supplementary to the systematic friction force introduced in Eq. (179), we have to consider a fluctuating force (with zero... 

Here, Pp = mpR is a Cartesian bead momentum, U is the internal potential energy of the system of interest, Xi,.. .,Xk are a set of AT Lagrange multiplier constraint fields, which must be chosen so as to satisfy the K constraints, and is the rapidly fluctuating force exerted on bead p by interactions with surrounding solvent molecules. The corresponding Hamiltonian equation of motion is... 

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