Langevin term

Because of the formal equivalence of Fokker-Planck and Langevin methods, there is no intrinsic difficulty in translating Kirkwood s theory into Langevin terms. As far as 1 am aware, this has not yet been done. The main purpose of this article is to perform the translation. 

Equations (19)—(23) provide a complete translation of Kirkwood s Fokker-Planck theory into Langevin terms, when the entire 3JV-dimen-sional configuration space is used. 

More generally, suppose one has a physical system with a nonlinear equation of motion y = A(y) and, following the Langevin approach, one adds a Langevin term to describe the fluctuations in the system,... 

If one adds to this equation a Langevin term to describe the noise produced by the rectifier one obtains for the average of Q... 

However small the effect, it violates the second law of thermodynamics. An array of 1012 diodes would be able to run a flashlight. The answer to this paradox is that one cannot trust the phenomenological equation to the extent that one may use it for deducing a result that is itself of the order of the fluctuations. This example demonstrates the danger of adding a Langevin term to a nonlinear equation. ... 

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

Since gm is related to the orbital angular momentum vector L the second term is often written in terms of this operator. The first term is the Langevin term it is the expectation value of a one-electron operator 2 (xt2 + jV), and can therefore be obtained directly from the ground-state wavefunction. It has formed the basis for a quite successful additivity scheme, Pascal s rules, which work well except for the conjugated hydrocarbons where non-additivity is ascribed to ring currents. Nevertheless, as it depends on the square of the electron co-ordinates, xL will be sensitive to the basis set used in variational calculations. 

The first term is always negative and is referred to as the diamagnetic contribution or the Langevin term. The second, sum-over-states term is known as the paramagnetic contribution and is always positive or zero for ground states. 

The first term is seen to be positive, corresponding to a diamagnetic contribution to the susceptibility, y,, called the Langevin term. The second is seen to be negative, corresponding to a paramagnetic contribution Xp, called the Van Vleck term. We may obtain these terms directly by multiplying Eq. (5-23) by the electron... 

The matrix element in the Langevin term, Eq. (5-24), will clearly be of the order of the value it would have if the charge were concentrated at the bond... 

It is interesting to notice that if we substitute the expression V2 = 2A6h lmd into Eq. (5-30), we may combine the Van Vleck and Langevin terms to obtain... 

Since pm is related to the orbital angular momentum vector L the second term is often written in terms of this operator. The first term is the Langevin term it is the expectation value of a one-electron operator 2 + yp), and can therefore be... 

Normally the TIP contribution (the van Vleck term) is only a minor correction to the underlying diamagnetism (the Langevin term). In complex systems like metalloproteins the diamagnetic contribution may also dominate the temperature-dependent paramagnetism. 

The matrix element in the Langevin term, Eq. (5-24), will clearly be of the order of the value djl /sy it would have if the charge were concentrated at the bond extremities. Therefore it is convenient to introduce a magnetic ym > analogous to the y introduced for the electric susceptibility. Then (b (z — z,) b,) can be written d /12. (D. J. Chadi and R. M. White have been able to write an approximate relation for y , in terms of y by using a sum rule. However, their analysis is intricate, and their corrections to y are only of the order of 5 percent, so we will not reproduce it here.) The Langevin term becomes simply... 

Fig. 4. Demonstration of the validity of the Langevin term in the relationship M XhH rHLL iLH kBT ...

In O Eq. 11.185, a is the (frequency-dependent) electric polarizability tensor, and i, the magne-tizabUity tensor. The two contributions to the molar Cotton-Mouton constant in O Eq. 11.183 are, therefore, a temperature-independent term related to the hypermagnetizabUity and a temperature-dependent molecular orientational part - the Langevin term. ... 

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