Point Transformations and the Relationships Among Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

We present new connections among anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem. This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Diffusion equations of O'Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we are able to tell) be expressed as examples of the O'Shaughnessy and Procaccia equations. These equations also possess asymptotics that are related to a CLT but with bi-modal distributions as limits. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. Finally, we show that there is a new, anomalous diffusion possible for bi-modal probability distributions that also display attractor behavior which is the consequence of an underlying CLT.

, the process is termed "super diffusion" and if 1 < β, the process is termed "sub-diffusion". If 25 . 0 0    , the diffusion is termed ballistic. O'Shaughnessy and Procaccia were the first to show that many AD systems have exact, attractor solutions, called "stretched Gaussians": SHAUGHNESSY 1985). Here, D is a "constant effective diffusion coefficient". We postulate that the fact that these are computationally observed to be attractor solutions of the relevant differential equations implies that the standard CLT is "hidden" and applies also to AD associated with the diffusion equations we derive. In this paper, we show that this is indeed true.
The paper is organized as follows. In Section II, we summarize the solution of the ND equation, emphasizing its dependence on the Fourier transform (FT) of the probability function (the characteristic function) and noting that the characteristic function plays a key role in the proof of the CLT. In Section III, we discuss important connections among super symmetric quantum mechanics (SUSY), Heisenberg's uncertainty principle (HUP), point transformations (PT) (in order to deduce generalizations of the FT and the related Laplacians), and the CLT for diffusion in the new canonically conjugate "position". In Section IV, we explore the relations among AD, ND, scaling laws and PT's. In Section V, we provide a computational example for the point transformation, 3 ) , that illustrates the relation of the time dependence of the MSD to the functional form of the PT, W(x). Section VI discusses the detailed relationship between our diffusion equations and those in (O'SHAUGHNESSY 1985). Finally, in Section VII, we summarize our results.

The Normal Diffusion Equation, Laplacian Semigroup and the Central Limit Theorem
In the case of ND as a random process, the proof of the CLT is most readily based on the characteristic function, i.e., the FT of the probability distribution (KLEUKE 2014). At its heart is the fact that the Gaussian is invariant under the FT. This is interesting because the ND equation is also exactly solved using the FT of the diffusion equation. This makes use of the fact that the Laplacian satisfies The inverse FT of a product of functions of k yields the exact convolution solution the diffusing particle is localized   initially at   0   x x  , with time dependence given by ). For any initial probability distribution whose FT is differentiable, the long-time behavior will be dominated by the overall Gaussian envelope (simply expand the square in the exponent in Eq. (5) and factor out ) ). The 2 / 1  t in the normalization arises from requiring that the probability distribution be normalized to 1 under the measure dx. Then the MSD varies as t D 2 (note, the slope of the MSD is not necessarily equal to one). We thus observe the role of the CLT as implying that the time dependence of any initial distribution, evolving according to Eq. (2), will tend to a Gaussian distribution envelope after a sufficiently long time. We stress that the characteristic function proof of the CLT also rests on the facts that the Gaussian is invariant under the FT and the FT satisfies the convolution theorem. These properties are related to the semigroup structure of the standard Laplacian.
The uncertainty product for such operators as The 1-D SUSY formalism is built on a program to make all systems (on the domain -∞ < x < ∞) look as much as possible like the harmonic oscillator, whose ground state also satisfies Eq. (6). In the ladder operator notation, this equation is In the coordinate representation, In 1-D SUSY, the ground state is restricted to the form for W's such that the ground state is 2 L . Choosing the zero of energy to be that of the ground state, one finds that the potential of the system is given by which is constant only for W = x + constant, the desired transform kernel cannot be the simple FT kernel. As a consequence, we do not actually use the SUSY expressions directly to derive the desired transform. We remark that in the SUSY community, W is interpreted as a "super potential", based on Eq. (9). In our approach, we shall interpret W to be a generalized "position" variable, based the facts that: (1) 0  minimizes k W   and (2) the quintessential choice of W is that for the harmonic oscillator, where W is precisely the particle's "generalized position" (CHOU 2012, KLAUDER 2015, WILLIAMS 2017a, 2017b, 2018).
Recently, in KOURI (2017), we explored choosing W to represent a point transformation of the usual position, such that (1) the domain of W is -∞ < W < ∞, (extension to the half line can be done) (2) the domain of the canonically conjugate momentum is also -∞ < W P < ∞, (3) the transformation is invertible and (4) both W and W P can be interpreted as Cartesianlike coordinates. One expression that satisfies the above is the polynomial KOURI (2017) in which all coefficients are positive semi-definite and each even-power coefficient is always less than the next higher odd power coefficient. This results in a monotonic increasing behavior that is invertible almost everywhere (if one requires that the coefficient of the first power of x is positive definite, then the expression is invertible everywhere).
Assuming that W(x) satisfies the above conditions KOURI (2017), the classical canonically conjugate momentum, W P , is required to satisfy Invoking Occam's razor, we set the integration constant along a constant-x integration path, g(x), equal to zero. Since W P above satisfies Eq. (15) for all α, we then apply Dirac's canonical quantization to obtain (in the W-representation). This quantization consists of replacing the observables by operators, the Poisson bracket by the commutator of the operators and the scalar 1 by  i times the identity operator DIRAC (1958): We stress that these operators are manifestly self adjoint in the W-representation, with the measure dW. In addition, the minimizer of As in the case of the original position and momentum operators, This is analogous to the FT in terms of x and k. This transform satisfies the convolution theorem under the measure dW or dK. We note that in the W-representation, since W P is self adjoint, we have the four equivalent, exact expressions: which are all obviously self adjoint under the measure dW. The above relations are at the heart of our approach. The transformation that preserves the W-Gaussian ( Again, the W-Laplacian possesses a negative semi-definite spectrum so it is evident that this transform kernel possesses all the properties of the FT, including the fact that it supports a semigroup property.
We also note that there exists a generalized W-harmonic oscillator, with ground state satisfying (WILLIAMS (2017a), KOURI (2017)) ). 26 ( The diffusion equation describing independent random motion in W is obviously where here, D is a constant diffusion coefficient. It follows that the exact solution of Eq. (27) is obtained using the W-FT kernel Eq. (21), yielding The "characteristic function" underlying the above expression is . The analysis of the CLT for probability distributions in W is identical to that for the normal probability distribution (KLEUKE 2014).
We next consider the situation where we quantize Eqs. (23) - (24) in the x-representation (we do not invoke the chain rule). Because of the ambiguity of operator ordering, we consider the following explicit expressions for a generalized momentum operator and its adjoint, involving the parameter α: But these operators are obviously not self adjoint under the measure dx (except in the case where 2 / 1   )! Normally, they are discarded and replaced by their arithmetic average. However, we know that any self adjoint operator remains such, regardless of the representation. Of course, the point is that the above are self adjoint in the x-representation but the measure now includes the effect of the point transformation. Thus, Eq. (30) is self adjoint under the measure and Eq. (31) is self adjoint under the measure . When alpha equals ½, the measure for self adjointness reduces to simply dx.
We remark that, as shown above in Eq. (23), in the W-domain, all four Laplacians possess the W-Gaussian as the ground state solution of the W-HO. Also, all four Laplacians satisfy Eq. (25) in the W-domain. Thus, all four Laplacians are not only self adjoint in the xrepresentation but they also are all negative semi-definite operators which have an underlying semigroup structure regardless of their explicit form in the x-domain. Each of the four Laplacians will be self adjoint in the x-domain but only with appropriate measures. We now explicitly explore the forms of these Laplacians in the x-domain. It is convenient to group them in two pairs of two. This is because the forms are already manifestly self adjoint under the measure dx. The other pair is We note that the first two Hamiltonians are self adjoint under the measure d x. The next two are self adjoint under the measures discussed above for Laplacians 3 and 4. It is easily seen that Hamiltonians 1 and 3 will have identical ground states, satisfying Hamiltonians 2 and 4 will also have identical ground states, satisfying These follow from the fact that the ground states are annihilated by the right hand operator in the respective factored Hamiltonians. We stress that, in the general case, none are simple Gaussians. Rather, the influence of the point transformation is clearly evident. Of course, if one sets 0   or 1, then we see that the "stretched Gaussian" is one of the ground states.
However, there is a second (bi-orthogonal partner) ground state which, in general, will be a multi-modal distribution. Because of the monotonic increasing character of the PT, x d W d is never negative and the ground state is always positive semi-definite. This is new and it reflects the fact that our formalism is closely related to the "Coupled SUSY" formalism introduced elsewhere (WILLIAMS 2017b, 2018).
The next question of importance is: What are the eigenstates of the four x-domain Laplacians? These will provide the transformations under which the various generalized HO ground states are invariant (in analogy to the FT and the Gaussian distribution). Of course, in the "W-world", there is only the W-FT, resulting from the W-Laplacian, having only the simple W-Gaussian as the solution of the diffusion and HO equations. We begin by considering the first two Laplacians. In this case, the explicit, analytical transforms have been derived in WILLIAMS (2017a) for monomial W's and 0   . In that case, the first equation (the Laplacian is negative semi-definite and self adjoint under the measure dx). The unitary transformation was derived restricting W to the monomial form In this case, it is the, in general, multi-modal distribution   The transformations for the second two Laplacians are easily obtained for general values of The result for the Laplacian 4  is given by It is easily seen that these are bi-orthogonally related, since and that (in Dirac notation), Thus, the bi-orthogonal transforms serve to carry out Fourier-like transformations of the ground states of the GHOs. The ground state solution,   , when transformed using K  under the measure d x, will produce the stretched Gaussian. The same is true when   is transformed by K  under the measure d x. This explicitly shows how the underlying CLT for the W-domain Laplacian and W-Gaussian will be manifested in the x-representation.

Anomalous Diffusion and Normal Diffusion
We now discuss the implications of the above for diffusion. We consider the four linear diffusion equations In the cases of j = 3 and 4, we take care to recall the biorthogonal character of the transforms but it again results in equations of the same exact form as above, but now for all α. Once the time dependent equation is in the K-domain, the solution can be obtained using the original W-domain transform, Eq. (17). This transformation possesses a convolution theorem and in the W-domain, all behave in time like normal diffusion! We conclude that all of the linear diffusion equations derived herein are subject to the CLT associated with Gaussian distributions. The above analysis holds in general for any proper choice of PT but the above analysis for the j = 1,2 cases has only been explicitly realized for the monomial case with alpha equal to zero. In the case of j = 3,4, everything applies for general point PT's, including polynomials (KOURI 2017).
For simplicity, we shall illustrate these ideas using the specific example of the j = 3 case for alpha equal to zero. The transform equation is then The corresponding x-domain GFT kernel from Eq. (21) is given by When applied in the form Eq. (56), the measure is . We note that requiring K to have the same functional dependence on k as W(x) on x ensures the argument of the exponential is dimensionless and that the measure for integration over K in the k-domain is After applying the GFT and using Eq. (25), the exact solution (in the k- The inverse GFT (which satisfies the convolution theorem in the variable W) then leads to the probability distribution at time t: We stress that as derived, this is a probability distribution in the variable W(x), not the variable x. The measure for normalization will be )) ( ( x W d , the normalization constant will be and the MSD will be proportional to 1 t . The diffusion is normal in the variable W. However, Eq. (58) can also be interpreted as a probability distribution in the variable x. It is a positive definite function of x and can be normalized to 1 under the measure dx: , the integration will be transformed to the variable and the normalization will be proportional to The average value of x will be zero (for symmetric initial distributions) and the MSD will scale as This is, of course, the scaling for AD: the probability distribution is interpreted as applying to the physical position variable, x, rather than to the canonical variable, 1   corresponds to sub-diffusion. However, because the Laplacian satisfies Eq. (25), the time evolution will still be such that the CLT is satisfied. The normalization has nothing to do with the fact that the time evolution is governed by the attractor, whether we express it in the W-or the x-domain. Similar scaling results apply in the general cases.

A Computational Example for the Polynomial
We illustrate our results by the numerical example of a polynomial choice of PT: We assume the initial distribution to be   , W is, to a very good approximation, equal to x but as the particle diffuses to larger magnitude values of x, it quickly transitions to being dominated by the cubic x-dependence. The W-domain results are shown in Figs. (1) - (2), where it is obvious that the diffusion is normal, since Eq. (63) becomes Eq. (27).    It is clear that at the earliest time, the linear x-dependence dominates the Laplacian and the diffusion is essentially normal, with the normal MSD t-scaling becoming more and more accurate as t approaches zero. At longer times, the cubic x-dependence dominates the Laplacian and we see that the MSD behavior tends to that of 3 / 1 t , just as one expects. While for more complicated polynomial PTs, such analysis will be more difficult, these results suggest that experimental measurements of the scaling as a function of time can provide insight into the appropriate PT for which the diffusion will be normal. More important, this will give information as to the best "effective" displacement variable with which to characterize and interpret the diffusion dynamics.

Relationship between Our Diffusion Equations and the O'Shaughnessy-Procaccia Equations
As formulated, our analysis and diffusion equations are very general and applicable both to fractal and non-fractal AD processes. Furthermore, for specific choices of our parameters, our equations exactly incorporate the radial diffusion equations of (O'SHAUGHNESSY 1985). This might appear somewhat strange since our equations are defined on the entire real line while those of O'Shaughnessy and Procaccia are for radial variables on the half line. In fact, it is only a subset of our equations that incorporate the O'Shaughnessy and Procaccia equations (WILLIAMS 2017a). The Laplacians are, explicitly, Eq. (32) for α = zero, were first derived on the half line and then extended to the full real axis and these are the particular equations that capture those of reference (O'SHAUGHNESSY 1985). We explicitly prove this below. The equations of interest are given by Here, c is the dimension (non-integer c's correspond to fractal cases  First, we note that our analysis shows that for diffusion according to any of the four diffusion equations, AD will be observed in the variable x but ND will be observed in the variable W. For ND to be manifested, one must analyze the results using the relevant generalized coordinate. AD as a function of x is simply "disguised ND" in W. Second, we have shown that when linear anomalous diffusion is analyzed in terms of the appropriate PT displacement variable, the usual CLT applies. Of course, this in no way alters the fact that the situation is much more complex in the case of nonlinear diffusion (TSALLIS 2005(TSALLIS , 2009, PLASTINO 2011).
Third, our results show that, for such AD systems, experimental determination of the anomalous scaling leads directly to the identification of the relevant generalized coordinate, W(x) (in the case of the monomial choice of W(x)). Certainly, for the example case of , it is also quite easy to extract the relevant PT from the numerical data. For more general polynomial PTs, it will, naturally, be more complicated but the experimental scaling should still give information as to the relevant PT and effective displacement variable.
Fourth, we recall that the CLT is related to approximating the semigroup generated by the Laplacian operator. Because of this, we expect that any diffusion process for which a "proper" Laplacian operator exists will have an attractor solution that is invariant under the generalized transform and is an eigenfunction of the corresponding HO. It will be of interest to explore treatment of Levy processes using fractional powers of the generalized Laplacians. . It will be of interest to explore whether any experimental data corresponds to these new linear diffusion equations.
Sixth, in our analysis, we considered diffusion in a 1D "Cartesian system" with a constant diffusion coefficient, D. It readily generalizes to any number of Cartesian random variables by simply summing the generalized Laplacians for each degree of freedom. We also point out that the scaling need not be the same in each degree of freedom. This will be important for anisotropic diffusion processes. It is also possible to introduce non-Cartesian coordinates (e.g., in a 3-D Cartesian , along with angular variables to obtain a "spherically symmetric diffusion operator"). As in quantum mechanics, these should be derived by a coordinate transformation of the Cartesian-like Laplacians.
Finally, we are currently exploring the more general and difficult case of non-linear diffusion equations using our generalized Fourier transform methods.