Drift, diffusion, and third order derivatives in FokkerPlanck equations: one specific case
Abstract
I present a case where there is an exact reinterpretation for the third order derivative term in a FokkerPlanck equation, purely in terms of ordinary drift and diffusion.
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I Introduction
There are many situations in optics where we would like to treat system dynamics using those methods developed for FokkerPlanck equations, or their stochastic analogues. Some notable cases are for the quantumoptical parametric oscillator Kinsler and Drummond (1991) and the coldatom GrossPitaevski equation for BoseEinstein (and other) condensates Sinatra et al. (2002); Gardiner et al. (2002); Cherroret and Wellens (2011) However, some (many) systems in fact give rise to partial differential equations containing extra terms, notably derivative terms of higher order than second. Fortunately, there are often grounds for considering such terms to negligible, so they are neglected (“truncated”) – thus giving rise to the socalled “truncated Wigner” phase space Kinsler and Drummond (1991); Sinatra et al. (2002) descriptions of quantum optical dynamics. Nevertheless, just because a term is small, that does not preclude it gradually accumulating and so providing a significant distortion. An ideal test bed for such comparisons is the secondorder nonlinear interection present in the parametric ocsillator ….. to the dynamicsDrummond and Kinsler (1989); Kinsler and Drummond (1995); Kinsler (1996), so attempts have been made to estimate these effects. In this note I show that an exact comparison can be made between the distribution function of a system follwing a purely third order differential equation and one following a purely second order (FokkerPlank) form with modified drift and diffusion terms.
Ii FokkerPlanck equations
FokkerPlanck equations (FPE’s) are usually of the form Risken (1989); Gardiner (2004); Arnold (1974)
(1)  
(2) 
This partial differential equation describes how the probability distribution function evolves in space () as a function of time (). The form gven here is one dimensional, but of course multidimensional generalizations also exist. Here is a drift term causing probability to flow deterministically, and is a diffusion term causing probability to spread away from some give point.
However, in some contexts we can generate FPElike equations that have additional third order derivative terms. An example is when using the Wigner representation to derive a FPE for the optical parametic oscillatorKinsler and Drummond (1991). Such a FPE could be of the simple form
(3) 
It is not uncommon in simple quantum optics treatments to simply truncate (i.e. neglect) these third order terms. Even if we chose to take this step, we would often like to know what perturbations these terms would have made to our approximate solution. So, what interpretation should (or can) we place on the term ? This is a hard problem in general, but I show here that for a very simple and specific case based on Hermite polynomials, an exact identification can be made. This then can motivate physical intuition as to the likely effects of such thirdorder terms in more general cases.
Iii Hermite polynomials
Hermite polynomials are a set of orthogonal polynomials, and, as such, can be used to describe any arbitrary function using the form
(4) 
For the purpose of this short note, they have some useful recurrence properties,
(5)  
(6)  
(7) 
When used appropriately, these recurrence properties enable us to convert a FPE containing thirdorder derivative terms into a form containing only second order ones MathWorld .
Iv Transforming the third derivatives
Using eqn. (4) we can decompose the full FPE as follows
(8) 
Note that since there is no crosscoupling between the components, and because the are orthogonal, we can write
(9) 
iv.1 A single component
I now assume is a constant^{1}^{1}1??as it is for the truncatedWigner dpo case?, and consider the RH derivative part of eqn. (9). Considering just one component of the Hermite decomposition of , we have a RHS from (9) of
(10)  
(11)  
(12)  
(13)  
(14)  
(15) 
where
(16)  
(17) 
So for a distribution function decomposed into Hermite polynomials , a thirdorder derivative term with a constant prefactor has two effects:

The first, and presumably most important feature is that the drift term gains a constant positive contribution. We would typically expect such a directional property, since is clearly antisymmetric in nature. The strength of this effective drift is dependent on , the polynomial order, so that fine structure in (requiring higher order contributions) drifts faster than coarse features;

The second, more minor effect is to add an antisymmetric adjustment to the diffusion.
The transformed counterpart to eqn. (9) is
(18) 
This therefore is a nice specific example where we can recast a third order derivatve term into the readily understood first order (drift) and second order (diffusion) terms. If implemented in some numerical scheme, it would require repetition of the following steps:
 (a)

the distribution to be decomposed into with weights ,
 (b)

each to evolve away from an exact Hermite polynomial under the influence of the drift and diffusion for some suitably small time interval ,
 (c)

an evolved distribution to be calculated.
Whether or not this is useful in practise is left as an exercise for those dealing with such situations. However, even if such an implementation is not done, and the thirdorder derivatives are simple truncated as usual, the behaviour of can be used to put constraints on the size of spatial features on distributions evolved using a truncated FPE. Thus it might at least be put to use as an intermittently applied test on the validity of a simulation of a truncated FPE model.
iv.2 Rearranged component
We might attempt an alternative strategy that tries to avoid the requirement of decomposing the distribution function . However, the one given below will not work, but for the sake of completeness I give it here.
Firstly, assume is a constant and consider the RH derivative part of eqn. (9),
(19)  
(20)  
(21) 
Note the appearance of the . Since we aim to reinstate the summation over all , we might reassign this to , so
(22)  
(23) 
So we see no drift modification, but a diffusion reduction instead. However, the equations are still crosscoupled, since has gained a dependence on . In the case of an even the diffusion adjustment will depend on which is a measure of the oddness of (as projected onto ); the converse is true for odd . This has turned out to be just a rerepresentation of the effectivedrift adjustment seen for in the first method.
iv.3 Rebuilding the FPE
Another (unsucessful) attempt to make a useful application of the special case in Sec. IV.1 is to try to reinstate the summation over , to convert eqn. (18) back into a true FPE. Thus
(24)  
(25) 
Unfortunately this fails because is dependent on , and we cannot reach the desire independent form for seen in the target eqn. (25). And even if I instead attempt to remove that dependence, using eqn. (5), i.e. , I end up crosscoupling the contributions instead, which is no better.
V Conclusion
I have shown that in one specific case, the Hermite polynomial recurrence relations can be used to develop an exact relationship between a constant third order derivative term in an FPE, and the commonly understood diffusion and drift terms. Although there seems no clear path to use this as a basis to solve FPE’s with third order terms in general, it can still be useful in applying check to an ongoing numerical solution to a truncated (standard) FPE equation.
Note
This is a previously unpublished fragment of my PhD research, which I have dusted off and put here on the arXiv, in case someone finds it useful.
References
 Kinsler and Drummond (1991) P. Kinsler and P. D. Drummond, Phys. Rev. A 43, 6194 (1991), doi:10.1103/PhysRevLett.64.236.
 Sinatra et al. (2002) A. Sinatra, C. Lobo, and Y. Castin, J. Phys. B 35, 3599 (2002), doi:10.1088/09534075/35/17/301.
 Gardiner et al. (2002) C. W. Gardiner, J. R. Anglin, and T. I. A. Fudge, J. Phys. B 35, 1555 (2002), c, doi:10.1088/09534075/35/6/310.
 Cherroret and Wellens (2011) N. Cherroret and T. Wellens, Phys. Rev. E 84, 021114 (2011), doi:10.1103/PhysRevE.84.021114.
 Drummond and Kinsler (1989) P. D. Drummond and P. Kinsler, Phys. Rev. A 40, 4813 (1989), doi:10.1103/PhysRevA.40.4813.
 Kinsler and Drummond (1995) P. Kinsler and P. D. Drummond, Phys. Rev. A 52, 783 (1995), doi:10.1103/PhysRevA.52.783.
 Kinsler (1996) P. Kinsler, Phys. Rev. A 53, 2000 (1996), doi:10.1103/PhysRevA.53.2000.
 Risken (1989) H. Risken, The FokkerPlanck Equation Methods of Solution and Applications, vol. 18 of Springer Series in Synergetics (Springer, Berlin Heidelberg, 1989), ISBN 9783540615309, URL http://link.springer.com/book/10.1007/9783642615443/page/1.
 Gardiner (2004) C. W. Gardiner, Handbook of Stochastic Methods, vol. 13 of Springer Series in Synergetics (Springer, 2004), 3rd ed., ISBN 9783540208822.
 Arnold (1974) L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974), ISBN 0471033596.
 (11) MathWorld, Hermite polynomial, URL http://mathworld.wolfram.com/HermitePolynomial.html.