^{1}

^{*}

^{1}

Stochastic processes such as diffusion can be analyzed by means of a partial differential equation of the Fokker-Planck type (FPE), which yields a transition probability density, or by a stochastic differential equation of the Langevin type (LE), which yields the time evolution of a statistical process variable. Provided the stochastic process is continuous and certain boundary conditions are met, the two approaches yield equivalent information. However, Brownian motion of radioactively decaying particles is not a continuous process because the Brownian trajectories abruptly terminate when the particle decays. Recent analysis of the Brownian motion of decaying particles by both approaches has led to different mean-square displacements. In this paper, we demonstrate the complete equivalence of the two approaches by 1) showing quantitatively and operationally how the probability densities and statistical moments predicted by the FPE and LE relate to one another, 2) verifying that both approaches lead to identical statistical moments at all orders, and 3) confirming that the analytical solution to the FPE accurately describes the Brownian trajectories obtained by Monte Carlo simulations based on the LE. The analysis in this paper addresses both the spatial distribution of the particles (
*i.e.* the question of displacement as a function of diffusion time) and the temporal distribution (
*i.e.* the question of first-passage time to fixed absorbing boundaries).

The diffusion of radioactive particles, particularly in the gaseous and aerosol state, plays an important role in the dispersal, detection, and monitoring of a variety of radioactive isotopes affecting public health and safety. Examples include nuclides that arise naturally in the environment such as radon ^{222}Rn and thoron ^{220}Rn, or are associated with power reactor releases such as iodine ^{129}I and ^{131}I, cesium ^{131}Cs and strontium ^{90}Sr, or are produced in nuclear weapons manufacture and testing such as tritium ^{3}H, among many other possibilities [

Although diffusion as a physical process has been investigated theoretically since Einstein’s seminal work on Brownian motion in 1905 [

In a recent publication [

∂ p ( x , t | x 0 , t 0 ) ∂ t = D ∇ 2 p ( x , t | x 0 , t 0 ) − λ p ( x , t | x 0 , t 0 ) , (1)

leads to the transition probability density p ( x , t | x 0 , t 0 ) for finding a particle at location x at time t, given that it was at x 0 at time t 0 . The constant parameters of the equation are the diffusion coefficient (or diffusivity) D and the intrinsic nuclear decay rate λ. The solution to Equation (1) in the case of one-dimensional (1D) Brownian motion of a particle located at x 0 = 0 at t 0 = 0 was shown to be

p ( x , t | x 0 , t 0 ) = 1 4 π D ( t − t 0 ) exp ( − ( x − x 0 ) 2 / 4 D ( t − t 0 ) ) e − λ ( t − t 0 ) (2)

and is readily generalized for independent Brownian motion in three dimensions (as will be discussed in Section 2.3).

In contrast to the FPE, the Langevin equation (LE) expresses the time-variation of a process random variable such as displacement or velocity, rather than the transition probability density. In the case of 1D Brownian motion of a decaying particle, the LE derived in [

x ( t + d t ) = [ x ( t ) + 2 D d t n t ] ε t (3)

whose graphical solution shows the sequential displacements x ( t ) of the particle as a function of time and whose analytical solution yields the statistical distribution of the displacement variable X.^{1} The symbol n t , which governs particle displacement, represents a random sample from the unit normal distribution^{2} N t t + d t ( 0 , 1 ) where the subscript t and superscript t + d t explicitly denote the temporal range with respect to which n t is associated. In other words, two samples n t 1 and n t 2 corresponding to distributions N t 1 t 1 + d t ( 0 , 1 ) and N t 2 t 2 + d t ( 0 , 1 ) are independent for | t 2 − t 1 | > d t . The symbol ε t , which governs particle decay, represents a Bernoulli random variable^{3} defined by

ε t ≡ B t ( 1 , p s ) = { 1 withprobability p s = 1 − λ d t 0 withprobability 1 − p s = λ d t (4)

in which the probability of radioactive decay within the differentially small time interval dt is given by λ d t , the transformation law first recognized by Rutherford and Soddy and used by von Schweidler to derive the law of exponential decay [

X ( k d t ) ≡ X ( t ) = N ( 0 , 2 D d t Σ k 2 ) . (5)

Note that the expression in (5) is not a function, but a statistical distribution (Gaussian) whose variance 2 D d t Σ k 2 after k particle displacements is expressed in terms of another random variable Σ k 2 that was derived and explained in [

For a system of non-decaying particles, the statistical content of the FPE is ordinarily equivalent to that of the LE. It is a matter of choice which method to apply, and there is a standard formulaic procedure for generating the LE from the FPE and vice versa [

Δ x FPE 2 ( t ) = 2 D t e − λ t (6)

Δ x LE 2 ( t ) = 2 ( D / λ ) ( 1 − e − λ t ) (7)

where the mean-square displacement about the mean is defined by Δ x 2 = 〈 X 2 〉 − 〈 X 〉 2 .

Now two theoretical approaches to the same physical problem of Brownian motion of radioactive particles may yield complementary information, but they cannot yield inconsistent information and both be correct. In this paper we establish the complete equivalence of the FPE and LE approaches to the Brownian motion of radioactive particles by

1) deriving probability density functions from the FPE that account for the distribution of particle displacements obtained from the LE;

2) clarifying operationally the difference in predicted variances (6) and (7);

2) demonstrating by means of the random variable Σ k 2 that the two approaches lead to the same statistical moments at all orders; and

4) testing the distributions derived from the FPE by a Monte Carlo method based on the LE.

Solutions to the FPE and LE address the general question of how far a diffusing particle has gone in a given time. The converse question of how much time a particle takes to diffuse a given distance is part of stochastic theory that analyzes first-passage processes [

In [

S ( t ) = ∑ n = − ∞ ∞ 1 2 [ erf ( ( 4 n + 1 ) l 4 D t ) + erf ( ( 4 n − 3 ) l 4 D t ) − 2 erf ( ( 4 n − 1 ) l 4 D t ) ] e − λ t , (8)

is the probability that the particle, initially located at x 0 = 0 , has not reached either boundary at time t. The summation over an infinite number of terms in (8) arises from the requirement that the probability density p X ( x b , t ) of finding the particle at x b is 0 for all values of time t. Equation (8) was solved by a method of images [

In the analysis of first-passage processes of non-decaying particles, the probability density that a particle has not reached a boundary is obtained by differentiation of the survival function [

p T ( t ) = − d S ( t ) / d t . (9)

Correspondingly, the random variable T, for which expression (9) is the probability density (as indicated explicitly by the subscript on p), is referred to as the first-passage time (FPT). However, it is important to note that in the case of decaying particles, T takes on a broader meaning because the process of decay, as well as the process of diffusion, may be the cause of a particle failing to reach a boundary. Nevertheless, unless otherwise noted, for convenience of reference we will maintain the convention of referring to T as the first-passage time and to the density (9) as the FPT probability density.

In Ref. [

In the survival function (8), the decay parameter λ is present only in the exponential factor. One might therefore surmise erroneously that the effect of particle decay would simply be to shorten the mean FPT relative to what it would be for a stable particle. The effects, however, are more profound than that. In the present paper, Equation (9) is evaluated explicitly and shown to take a form substantially different from the corresponding FPT density for non-decaying particles. To clarify these matters we

1) calculate the FPT probability density from the FPE and interpret its mathematical structure, and

2) test the expressions derived from the FPE (relations (8) and (9)) by a Monte Carlo simulation based on the LE applied to Brownian motion of radioactive particles confined between absorbing boundaries.

This paper is organized as follows. In Section 2 spatial distributions (1D and 3D) obtained from the solution to the FPE are derived, the mean-square displacements of the particles are calculated, an explanation is given for the two statistical expressions (6) and (7), and the equivalence of the FPE and LE analyses of Brownian motion of radioactive particles is demonstrated by showing that both equations give rise to the same statistical moments at all orders. In Section 3 the distribution of first-passage times is obtained from the solution to the FPE, and the probability density is shown to comprise two components, one relating to removal of a particle from the system by absorption at the boundaries, and the other relating to removal of a particle by radioactive decay. In Section 4 a Monte Carlo simulation based on the stochastic LE is used to generate distributions of Brownian trajectories under specified conditions to test the density functions derived from the FPE in Sections 2 and 3. Conclusions regarding the equivalence of the two approaches are summarized in Section 5.

As discussed in [

There is, however, a mathematical connection between the different perspectives of Brownian motion afforded by the FPE and LE, which relates the mean-square displacements (6) and (7) consistently. Consider first the interpretation of solution (2) to the 1D FPE, which for emphasis is re-expressed in the following way (with initial conditions taken to be x 0 = 0 , t 0 = 0 for simplicity)

w S ( x , t ) = [ 1 4 π D t exp ( − x 2 / 4 D t ) ] e − λ t . (10)

The factor in square brackets is the probability density for the particle to reach location x at time t. The subsequent exponential decay factor is the independent probability that the particle has survived to time t. The product of the two factors in relation (10) is therefore the probability density w S for the Brownian particle to diffuse to x and survive―hence the subscript “S”.

Multiplying FPE solution (2) by λ d t leads to a differential expression with the same initial conditions as in (10)

d w D ( x , t ) = [ 1 4 π D t exp ( − x 2 / 4 D t ) e − λ t ] λ d t (11)

in which the bracketed factor is the survival probability density (10), and the differential factor that follows is the probability of subsequent radioactive decay within interval dt. The subscript D on the probability density w D in Equation (11) signifies that the particle has decayed at x at some time between t and t + d t . Thus, the probability density that a radioactive particle decayed at x irrespective of the moment of decay within the finite interval [ 0 , t ] is obtained by integration

w D ( x , t ) = ∫ 0 t d w D ( x , t ′ ) = ∫ 0 t ( d w D ( x , t ′ ) d t ′ ) d t ′ , (12)

leading to

w D ( x , t ) = 1 4 ζ [ e x / ζ erf ( x + 2 v d t 4 D t ) − e − x / ζ erf ( x − 2 v d t 4 D t ) − 2 sinh ( | x | ζ ) ] , (13)

where

ζ ≡ D / λ (14)

is the characteristic diffusion length and

v d ≡ D λ (15)

is the characteristic diffusion velocity (See Appendix 1 of Ref. [

Probability densities (10) and (13) are not individually normalized to unity because each density covers an incomplete set of possible outcomes. For example, the integral of density (10) over all available space

∫ − ∞ ∞ w S ( x , t ) d x = e − λ t (16)

yields the probability of survival to time t. The sum of the two densities, however,

w ( x , t ) = 1 4 ζ [ e x / ζ erf ( x + 2 v d t 4 D t ) − e − x / ζ erf ( x − 2 v d t 4 D t ) − 2 sinh ( | x | / ζ ) ] + 1 4 π D t exp ( − x 2 / 4 D t ) e − λ t (17)

is normalized to unity

∫ − ∞ ∞ w ( x , t ) d x = 1 (18)

because it covers a complete set of mutually exclusive, uncorrelated Brownian processes that originate at ( x 0 , t 0 ) = ( 0 , 0 ) : 1) survival at ( x , t ) , 2) decay at x at some instant ≤ t .

In the limit t → ∞ ,Equation (13) and therefore Equation (17) reduce to the asymptotic form

w ( x , ∞ ) ≡ w ( x ) = exp ( − | x | / ζ ) / 2 ζ , (19)

which is the probability density of a Laplace distribution [

The general forms of the densities w S ( x , t ) , w D ( x , t ) , w ( x , t ) are shown as functions of displacement in

graphical presentation. For example, given the designated value of λ, the probability of a single particle decay in time step d t = 1 is predicted from Equation (4) to be 1%. This is a convenient probability to use in the Monte Carlo simulations of Brownian motion presented in Section 4.

The first and second moments of the distributions (10), (13), and (17) are respectively calculated to be

〈 X 〉 S,D = ∫ − ∞ ∞ x w S,D ( x , t ) d x = 0 (20)

〈 X 2 〉 S = ∫ − ∞ ∞ x 2 w S ( x , t ) d x = 2 D t e − λ t (21)

〈 X 2 〉 D = ∫ − ∞ ∞ x 2 w D ( x , t ) d x = 2 ( D / λ ) ( 1 − e − λ t ) − 2 D t e − λ t (22)

〈 X 2 〉 = ∫ − ∞ ∞ x 2 w ( x , t ) d x = 2 ( D / λ ) ( 1 − e − λ t ) , (23)

from which it is seen from Equation (23) that the total variance Δ x FPE 2 = 〈 X 2 〉 of the Brownian displacements, derived from the solution to the 1D FPE (1), is exactly the same result (7) for Δ x LE 2 ( t ) obtained from the analytical solution of the 1D LE (3). The demonstration of this equality provides an operational interpretation of the difference between the two expressions for variance, Equations ((6) and (7)), which are displayed as functions of time in

An experimental procedure to test the foregoing predictions would be to follow the Brownian diffusion of a sample of radioactive particles initially concentrated

at the origin (in approximation to a delta function distribution) and record (a) the location of each particle as a function of time, as well as (b) the time at which any particle has decayed. An experiment of this kind is difficult to execute physically, but can be simulated on a fast computer by a Monte Carlo method. In general, the Monte Carlo method utilizes random numbers of a specified probability distribution to simulate the dynamics of a stochastic process by calculating and aggregating the outcomes of a large number of trials for a prescribed set of initial conditions. Commercially available symbolic mathematical software for science, engineering, and statistics usually includes random number generating algorithms for common probability distributions (e.g. binomial, Gaussian, Poisson, exponential) in their accompanying libraries of functions. The results of a Monte Carlo simulation of 1 million radioactively decaying particles undergoing 1D and 3D Brownian motion are discussed in Section 4.

It is also of interest to calculate the fourth moments of the displacement, since the variances of the mean-square distributions depend on them. Straightforward integration leads to

〈 X 4 〉 S = ∫ − ∞ ∞ x 4 w S ( x , t ) d x = 12 D 2 t 2 e − λ t (24)

〈 X 4 〉 D = ∫ − ∞ ∞ x 4 w D ( x , t ) d x = 24 ( D 2 / λ 2 ) ( 1 − ( 1 + λ t + 1 2 λ 2 t 2 ) e − λ t ) (25)

〈 X 4 〉 = ∫ − ∞ ∞ x 4 w ( x , t ) d x = 24 ( D 2 / λ 2 ) ( 1 − ( 1 + λ t ) e − λ t ) . (26)

It is again to be noted that the total fourth moment (26), derived above directly from solution of the FPE, is the result obtained previously in Ref. [^{4}.

Agreement of the FPE and LE predictions of the second and fourth moments demonstrates consistency of the two approaches at a practical level, since analysis of diffusion by Brownian motion rarely requires higher moments. However, for complete theoretical equivalence, which is essential if the two approaches to treating diffusion of radioactive matter are to be equally trusted, it is required to show that the FPE and LE yield equivalent statistical moments at all orders. This demonstration is given in the next section.

The general problem of whether a distribution function is uniquely determined by the totality of its moments is referred to in theoretical statistics as the “problem of moments” [^{th} century and has generated a copious mathematical literature whose content overall is beyond the scope of this paper. Of relevance to the present work, however, is the version of the problem associated with the name of Hamburger, who treated the case of random variables defined on the entire real axis. In brief, if the random variable has a finite moment-generating function (MGF), then the totality of the moments uniquely determines the distribution of that random variable [

The MGF g Z ( θ ) of a random variable Z is defined by the expectation value [

g Z ( θ ) ≡ 〈 e Z θ 〉 , (27)

provided it exists^{5}, where θ serves only as an expansion variable. A Taylor series expansion in θ of relation (27) leads to a superposition of the statistical moments

g Z ( θ ) ≡ ∑ n = 0 ∞ 〈 Z n 〉 θ n n ! (28)

from which follows the operational expression by which individual moments can be calculated

〈 Z k 〉 = ( d k g Z ( θ ) / d θ k ) | θ = 0 (29)

for non-negative integer k. The term k = 0 represents the completeness relation 〈 Z 0 〉 = g Z ( 0 ) = 1 .

From the form of Equation (5) for the displacement variable X derived from the LE, it follows that the even statistical moments of X are given by the product of factors

〈 X 2 k 〉 LE = 〈 ( Σ n 2 ) k 〉 〈 N ( 2 k ) 〉 (30)

( k = 0 , 1 , 2 , ⋯ ) , where the first factor is an expectation of the k^{th} power of the random variable Σ n 2 that characterizes the decay process, and the second factor is an expectation of the unit normal (Gaussian) distribution

〈 N ( k ) 〉 ≡ 〈 X k 〉 Gaus = 1 2 π ∫ − ∞ ∞ x k e − x 2 / 2 d x = ( 1 + ( − 1 ) k ) 2 k 2 − 1 π Γ ( k + 1 2 ) (31)

that characterizes the spatial displacement of the particle during its survival. The function Γ ( k + 1 2 ) is the standard gamma function defined by the integral

Γ ( k ) = ∫ 0 ∞ u k − 1 e − u d u . (32)

One sees from Equation (31) that all odd statistical moments of spatial displacement vanish. This is a consequence of the left-right symmetry of Brownian motion in the absence of advection.

In Appendix 1 of Ref. [

g Σ n 2 ( θ ) = p S n e n θ + ( 1 − p S ) ∑ k = 0 n − 1 ( p S k e k θ ) (33)

where p S = 1 − λ d t is the survival probability per step defined in Equation (4), and the subscript n signifies a Brownian trajectory of n discrete steps each of duration d t = t / n in which t is the total diffusion time. Substitution of MGF (33) into Equation (29), followed by taking the continuum limit ( n → ∞ ) , leads to a closed form analytical expression for the moments of spatial displacement as a function of time

〈 X ( t ) k 〉 LE = ( 1 + ( − 1 ) k ) 2 [ 2 k ( D t ) k / 2 π Γ ( k + 1 2 ) e − λ t + ( D / λ ) k / 2 ( Γ ( k + 1 ) − 2 k π Γ ( k + 1 2 ) Γ ( k 2 + 1 , λ t ) ) ] (34)

where the upper incomplete gamma function Γ ( k , z ) is defined by the integral [

Γ ( k , z ) = ∫ z ∞ u k − 1 e − u d u . (35)

An explicit listing of the five lowest non-vanishing moments shows an interesting series pattern arising primarily from the incomplete gamma function

〈 X ( t ) 2 〉 LE = 2 ( D / λ ) ( 1 − e − λ t ) (36)

〈 X ( t ) 4 〉 LE = 24 ( D / λ ) 2 ( 1 − ( 1 + λ t ) e − λ t ) (37)

〈 X ( t ) 6 〉 LE = 720 ( D / λ ) 3 ( 1 − ( 1 + λ t + 1 2 λ 2 t 2 ) e − λ t ) (38)

〈 X ( t ) 8 〉 LE = 40320 ( D / λ ) 4 ( 1 − ( 1 + λ t + 1 2 λ 2 t 2 + 1 6 λ 3 t 3 ) e − λ t ) (39)

〈 X ( t ) 10 〉 LE = 3628800 ( D / λ ) 5 ( 1 − ( 1 + λ t + 1 2 λ 2 t 2 + 1 6 λ 3 t 3 + 1 24 λ 4 t 4 ) e − λ t ) . (40)

In brief, apart from a numerical coefficient, the moments as a function of t take the form:

〈 X ( t ) 2 k 〉 LE ∝ ( D / λ ) k ( 1 − e − λ t ∑ j = 0 k − 1 ( λ t ) j j ! ) . (41)

In the limit t → ∞ , the first term in Equation (34) vanishes exponentially, and the incomplete gamma function (35) likewise vanishes since the upper and lower limits to the integral become identical. Equation (34) then reduces asymptotically to

〈 X ( ∞ ) k 〉 LE = ( 1 + ( − 1 ) k ) 2 ( D / λ ) k / 2 Γ ( k + 1 ) (42)

or

〈 X ( ∞ ) 2 k 〉 LE = ( D / λ ) k Γ ( 2 k + 1 ) . (43)

We consider next the moments deducible from the Fokker-Planck equation by direct integration using the density function w ( x , t ) , Equation (17), which is the sum of the survival density w S ( x , t ) , Equation (10), and the decay density w D ( x , t ) , Equation (13). Evaluating the integrals, one obtains

〈 X ( t ) k 〉 S = ∫ − ∞ ∞ x k w S ( x , t ) d x = 2 k − 1 ( 1 + ( − 1 ) k ) π ( D t ) k / 2 Γ ( k + 1 2 ) e − λ t (44)

〈 X ( t ) k 〉 D = ∫ − ∞ ∞ x k w D ( x , t ) d x = ( 1 + ( − 1 ) k ) 2 ( D / λ ) k / 2 ( Γ ( k + 1 ) − 2 k π Γ ( k + 1 2 ) Γ ( k 2 + 1 , λ t ) ) (45)

whose sum

〈 X ( t ) k 〉 FPE ≡ 〈 X ( t ) k 〉 S + 〈 X ( t ) k 〉 D = 〈 X ( t ) k 〉 LE (46)

is identical to Equation (34) derived from the LE moment-generating function.

Although the preceding demonstration is sufficient, it is useful to note that one can apply Equation (27) to obtain the moment-generating function g X ( θ ) directly from the probability densities derived from the FPE as follows

g X ( θ ) ≡ ∫ − ∞ ∞ w ( x , t ) e X θ d x = ∫ − ∞ ∞ ( w S ( x , t ) + w D ( x , t ) ) e X θ d x . (47)

The integrals on the right side reduce to

∫ − ∞ ∞ w S ( x , t ) e X θ d x = e D t θ 2 − λ t (48)

∫ − ∞ ∞ w D ( x , t ) e X θ d x = λ ( 1 − e D t θ 2 − λ t ) λ − D θ 2 (49)

which sum to the compact expression

g X ( θ ) = 1 − ζ 2 θ 2 e − λ t ( 1 − ζ 2 θ 2 ) 1 − ζ 2 θ 2 (50)

in terms of the characteristic diffusion length z. Applying Equation (29) to the MGF (50) generates the spatial moments given by Equations (34) and (46). One final remark to avoid any confusion: Although the LE and FPE lead to identical statistical moments 〈 X k 〉 for all non-negative integers k, the moment-generating functions (50) and (33) are not the same because they pertain to two different random variables, X and Σ k 2 , related by Equation (5).

We have therefore established that, despite the structural differences in form of the Fokker-Planck and Langevin equations derived in Ref. [

Since Brownian motions along the x, y, z axes are all uncorrelated, and since diffusion in the atmosphere without advection is isotropic, it is a straightforward matter to deduce the analogous 3D probability density w 3 ( r , t ) of the radial displacement r = x 2 + y 2 + z 2 , by extension of the analysis used to obtain the 1D density function (17). The procedure involves two parts: 1) calculation of the survival density w 3 S ( r , t ) for the particle to reach a radial distance r from the origin at time t, and 2) calculation of the density w 3 D ( r , t ) for the particle to decay upon reaching a radial distance r from the origin at some moment in the time interval between 0 and t. The two sets of events are mutually exclusive and complementary (i.e. complete the sample space), whereupon the total density

w 3 ( r , t ) = w 3 S ( r , t ) + w 3 D ( r , t ) (51)

is normalized to unity.

Consider first the density w 3 S ( r , t ) , which is defined by the differential relation

w 1S ( x , t ) w 1S ( y , t ) w 1S ( z , t ) d x d y d z ≡ w 3 S ( r , t ) d r (52)

where numerical subscripts explicitly signify the dimensionality of a Brownian process. In the left side of relation (52) it is understood that the survival probability e − λ t occurs as a factor only once (not three times) since the same time axis applies to motion along all three spatial dimensions. It then follows from Equation (10) that

w 3 S ( r , t ) = 4 π ( 4 π D t ) − 3 / 2 r 2 exp ( − r 2 / 4 D t ) e − λ t , (53)

which is equivalent to a chi-square distribution χ k 2 of k = 3 degrees of freedom. (The equivalence is established by the change of variable u → r 2 in the definition of the chi-square distribution d F ( u ) = f k ( u ) d u where f k ( u ) = u k 2 − 1 e − u / 2 .)

Next, in analogy to the 1D relation (11), the density w 3 D ( r , t ) is obtained by integration of the differential relation

d w 3 D ( r , t ) = w 3 S ( r , t ) λ d t (54)

to yield the density

w 3 D ( r , t ) = r 2 4π ζ 3 ∫ 0 λ t u − 3 / 2 e − u exp ( − r 2 4 ζ 2 1 u ) d u . (55)

The integral in Equation (55) is a generalized incomplete gamma function [

γ k ( x , b ) = ∫ 0 x u k − 1 e − u − b u − 1 d u . (56)

The total density for undergoing a radial displacement r by Brownian motion in time t is then

w 3 ( r , t ) = r 2 exp ( − r 2 / 4 D t ) 4 π ( D t ) 3 e − λ t + r 2 4π ζ 3 γ − 1 / 2 ( λ t , r 2 / 4 ζ 2 ) (57)

and satisfies the normalization requirement for completeness

∫ 0 ∞ w 3 ( r , t ) d r = 1 . (58)

In the limit of infinite time, all particles will have decayed, and w 3 reduces to the asymptotic form

w 3 D ( r , ∞ ) ≡ w 3 ( r ) = ( r / ζ 2 ) e − r / ζ . (59)

The function w 3 ( r ) is the probability density of a gamma distribution of index 2 [

asymptotic density w 3 ( r , ∞ ) (dashed black) given by Equation (59). Although the peak of w 3 ( r , ∞ ) is lower than that of w 3 ( r , t ) for finite t, its heavy tail falls off more slowly, and the area under both densities is 1.

The spatial moments of the foregoing 3D densities can be evaluated in closed form as given below (where k = 0 , 1 , 2 , ⋯ ):

〈 r k 〉 S = ∫ 0 ∞ r k w S ( r , t ) d r = 2 k + 1 π − 1 / 2 ( D t ) k Γ ( k + 3 2 ) e − λ t (60)

〈 r k 〉 D = ∫ 0 ∞ r k w D ( r , t ) d r = 2 k + 1 π − 1 / 2 ζ k Γ ( k + 3 2 ) γ ( k 2 + 1 , λ t ) , (61)

in which

γ ( k , z ) = 1 − Γ ( k , z ) = ∫ 0 z u k − 1 e − u d u (62)

is known as the lower incomplete gamma function [^{th} moment of r is then

〈 r k 〉 = 〈 r k 〉 S + 〈 r k 〉 D . (63)

The lowest five moments (including the completeness relation k = 0 ) calculated from Equation (63) are given in

〈 r t k 〉 S | 〈 r t k 〉 D | 〈 r t k 〉 | 〈 r ∞ k 〉 | |
---|---|---|---|---|

k = 0 | e − λ t | 1 − e − λ t | 1 | 1 |

k = 1 | 4 π ( D t ) 1 / 2 e − λ t | 2 ζ ( erf ( λ t ) − 2 λ t / π e − λ t ) | 2 ζ erf ( λ t ) | 2 ζ |

k = 2 | 6 D t e − λ t | 6 ζ 2 ( 1 − ( 1 + λ t ) e − λ t ) | 6 ζ 2 ( 1 − e − λ t ) | 6 ζ 2 |

k = 3 | 32 π ( D t ) 3 / 2 e − λ t | 24 ζ 3 ( erf ( λ t ) − 2 π ( ( λ t ) 1 / 2 + 2 3 ( λ t ) 3 / 2 ) e − λ t ) | 24 ζ 3 ( erf ( λ t ) − 2 π ( λ t ) 1 / 2 e − λ t ) | 24 ζ 3 |

k = 4 | 60 ( D t ) 2 e − λ t | 120 ζ 4 ( 1 − ( 1 + λ t + 1 2 ( λ t ) 2 ) e − λ t ) | 120 ζ 4 ( 1 − ( 1 + λ t ) e − λ t ) | 120 ζ 4 |

D = 0.5 , λ = 0.01 . The asymptotic values are given by the relations in column 5 of

An alternative approach, in analogy to Equation (50), is to determine the statistical moments by differentiation of the 3D moment-generating function (MGF) defined by the integral

g r ( θ ) ≡ ∫ 0 ∞ w 3 ( r , t ) e r θ d r , (64)

which reduces to the expression

g r ( θ ) = [ 1 + ζ 2 θ 2 + 2 ζ θ erf ( λ t ) + 2 π − 1 / 2 λ t ( ζ 5 θ 5 − ζ 3 θ 3 ) e − λ t + e ( ζ 2 θ 2 − 1 ) λ t ( 2 λ t ζ 6 θ 6 ( 1 + erf ( λ t ζ θ ) ) + ζ 4 θ 4 ( ( 1 − 2 λ t ) ( 1 + erf ( λ t ζ θ ) ) ) − 3 ζ 2 θ 2 ( 1 + erf ( λ t ζ θ ) ) ) ] ( ζ 2 θ 2 − 1 ) − 2 . (65)

Although Equation (65) is complicated in appearance compared to relations (60)-(62), there are potential advantages to having the MGF at one’s disposal. In regard to the analytical or numerical calculation of moments, symbolic mathematical software can generally perform derivative operations more quickly and efficiently than integrations. And secondly, the MGF can be useful, as shown in the preceding sections, for deriving properties and establishing equivalence of statistical distributions.

The probability density for a decaying particle to diffuse in time t from the origin 0 to a location x between two absorbing boundaries at x b = ± l is derivable from the FPE by the method of images [

p X ( x , t ) = lim N → ∞ ∑ n = − N N 1 4 π D t ( exp ( − ( x + 4 n l ) 2 4 D t ) − exp ( − ( x + ( 4 n − 2 ) l ) 2 4 D t ) ) e − λ t . (66)

Although Equation (66) is not a closed form, since the summation includes an infinite number of terms, in practice the number of terms required for an accurate calculation can be relatively few depending on the values of the parameters ( D , λ , l ) and the duration t of the Brownian motion. For example, for the parameters D = 0.5 , λ = 0.01 used in the preceding section, the density p X ( x b , t ) vanishes for t ≤ 100 at both absorbing boundaries x b = ± 10 for just the N = 1 set of images (i.e. summation indices n = − 1 , 0 , + 1 ), as shown in

The survival function S ( t ) given explicitly in Equation (8) was obtained by integration of the density (66) over the range ( − l , + l ) . As such, S (t)

represents the probability that the diffusing particle has not reached boundary points − l or + l by time t―hence the usual term “survival function”. The probability density for termination of a Brownian trajectory is given by Equation (9) and can be expressed as a sum of two terms

p T ( t ) = p T ( 0 ) ( t ) + p T ( 1 ) ( t ) (67)

in which

p T ( 0 ) ( t ) = e − λ t 4 π D t 3 × ∑ n = − ∞ ∞ ( ( 4 n + 1 2 ) l exp ( − ( 4 n + 1 ) 2 l 2 4 D t ) + ( 4 n − 3 2 ) l exp ( − ( 4 n − 3 ) 2 l 2 4 D t ) − ( 4 n − 1 ) l exp ( − ( 4 n − 1 ) 2 l 2 4 D t ) ) (68)

p T ( 1 ) ( t ) = λ e − λ t ∑ n = − ∞ ∞ ( 1 2 erf ( ( 4 n + 1 ) l 4 D t ) + 1 2 erf ( ( 4 n − 3 ) l 4 D t ) − erf ( ( 4 n − 1 ) l 4 D t ) ) . (69)

The first function, expressed by (68), is interpretable as the actual FPT probability density, i.e. the density function for a particle to reach one of the two absorbing boundaries for the first time at t. In other words, p T ( 0 ) ( t ) describes a particle that has survived radioactive decay to reach a designated boundary, at which point the particle is removed from the system because the boundary is absorbing. We designate density (68) by the label “Absorption” in

The second function expressed by (69) is the original survival function S ( t ) multiplied by the intrinsic nuclear decay constant λ. In other words, p T ( 1 ) ( t ) describes a particle that has decayed at time t before reaching a designated boundary. Bear in mind that it is probability (or, in practical terms, relative frequency) that is measurable, rather than probability density which is a mathematical function to be summed or integrated. Thus, to interpret the two expressions comprising density (67) one must actually examine the differential expression

p T ( t ) d t , whereupon the factor λ d t is recognized from Equation (4) as the probability of radioactive decay. The plot of density (69) (red curve) in

The sum of the two contributions in Equation (67), which together represent the loss of the particle from the system by either absorption or decay, is designated “Total” and shown (black curve) in

The mean time 〈 T 〉 for a decaying particle to be removed from the system by reaching either of two symmetrically disposed absorbing boundaries at ± l or by radioactive decay is the first moment of the total density (67)

〈 T 〉 = ∫ 0 ∞ t p T ( t ) d t = ( e l / ζ − 1 ) 2 λ ( e 2 l / ζ + 1 ) = 1 λ ( 1 − 1 cosh ( l / ζ ) ) (70)

where the characteristic diffusion length z is defined by relation (14). Two other ways of deriving relation (70) were shown in [

In the limit of vanishing decay rate, the mean time (70) reduces to

lim λ → 0 〈 T 〉 = l 2 / 2 D , (71)

which is the same mean time obtained for decaying particles in the limit ( l / ζ ) ≪ 1 . In the opposite limit ( l / ζ ) ≫ 1 , the mean first-passage time of a decaying particle asymptotically approaches the mean particle lifetime λ − 1 . This behavior is illustrated in

The mean time 〈 T 〉 0 for just the process of particle absorption at the boundaries (blue curve in

〈 T 〉 0 = ∫ 0 ∞ t p T ( 0 ) ( t ) d t ∫ 0 ∞ p T ( 0 ) ( t ) d t = l 2 v d tanh ( l ζ ) (72)

where the characteristic diffusion velocity v d is defined by relation (15) and the normalization constant is

∫ 0 ∞ p T ( 0 ) ( t ) d t = 1 cosh ( l / ζ ) . (73)

As in the case of the spatial distributions analyzed in the previous section, it is useful to have the second moment of the temporal distribution

〈 T 2 〉 = ∫ 0 ∞ t 2 p T ( t ) d t = 2 λ 2 ( 1 − 1 cosh ( l / ζ ) − l 2 ζ tanh ( l / ζ ) cosh ( l / ζ ) ) . (74)

Comparison of relations (74), (70), and (72) leads to the identity

〈 T 2 〉 = 2 λ ( 〈 T 〉 − 〈 T 〉 0 ) , (75)

which can be derived directly from the survival function (8), based on calculation of statistical moments by means of the cumulative distribution function rather than the probability density function; see Appendix 2 of Ref [

The first application of a statistical method to study the dynamics of a physical system subject to numerous random interactions is attributable to Enrico Fermi’s unpublished investigations of neutron diffusion in the 1930s. Designated the Monte Carlo method in the 1940s by Los Alamos scientists Metropolis and Ulam [^{6}, this approach now comprises the most widely used class of numerical methods to solve statistical problems in physics, chemistry, finance, and other fields [

To study the diffusion of a radioactive gas, we used a Monte Carlo method to create a large set of Brownian trajectories arising from the Langevin stochastic differential Equation (3) which incorporates two independent random number generators (RNG): 1) a Gaussian RNG to account for spatial displacement at each time step, and 2) a Bernoulli RNG to account for the possibility of decay at each step. The simulations comprised two parts: Part 1 was to determine the distribution of displacements as a function of time; Part 2 was to determine the distribution of first-passage times from the point of origin x 0 = 0 to one or the other of two symmetrically located absorbing boundaries at x b = ± l . Because of radioactive decay, it is possible that a particle will disintegrate before reaching any specified location. Under the conditions of the simulation, the Brownian trajectories in the first part were always finite since they terminated at a decay. The events in the second part fell into one of two categories: a) those for which first passage to a boundary preceded decay, and b) those for which decay eliminated the possibility of first passage to a boundary.

In implementation of Part 1, N p = 10 6 particles were sequentially created at the origin (a new particle being generated upon decay of a preceding one) and tracked in time as they underwent 3D Brownian motion in time steps Δ t = 1 (arbitrary units) for up to a maximum duration T max = K Δ t with K = 5000 . For the parameters used in these simulations, the probability of a particle surviving to T max was infinitesimal as shown in Appendix 2. At each time step, the displacement along each Cartesian axis was determined by three independent samples ( n x , n y , n z ) of a N ( 0 , 1 ) Gaussian RNG in accordance with Equation (3). The numbers of particles reaching locations ( x , y , z ) at some time step within a specified interval t k = k Δ t for K ≥ k ≥ 0 were then tallied and recorded in histograms (plots of frequency vs displacement) comprising N b bins (statistical categories) of width Δ x (same value for Δ y and Δ z ). N b and Δ x were selected to facilitate analysis and the graphical display of data, and could vary for different diffusion times t k .

Simulations were performed for different values of the diffusion time, diffusivity D, and radioactive decay rate λ. Histograms of 1D Brownian motion displayed in

The three panels of

Quantitative statistical tests, employing the ratio of observed to predicted frequencies as described in Appendix 3, showed for all histograms that deviations between computer-simulated outcomes and theoretical predictions could be attributable to pure chance.

The Fokker-Planck equation (FPE) and the Langevin equation (LE) provide different mathematical approaches to the analysis of Brownian motion of radioactive particles. Although the two equations differ structurally―the FPE is a multivariable partial differential equation, the LE takes the form of a stochastic differential equation based on Wiener and Bernoulli processes―we have demonstrated analytically that they are fully equivalent. The demonstration of complete equivalence was based on showing that the probability density functions derived from the FPE and the moment-generating function derived from the LE led to the identical complete sequence of statistical moments for 1D and 3D Brownian motion. Although such equivalence is expected for a continuous stochastic process that meets appropriate boundary conditions [

Mean (Monte Carlo Simulation) | Mean (Theory Equation (20)) | Std. Dev. (Monte Carlo Simulation) | Std. Dev. (Theory Equation (23)) | |
---|---|---|---|---|

t = 10 | ||||

x | −0.0123 | 3.2095 | ||

y | −0.0080 | 3.2205 | ||

z | 0.0389 | 3.2338 | ||

Average | 0.0062 | 0.0 | 3.2213 | 3.0848 |

r | 5.0802 | 4.8830 | 2.3070 | 2.1692 |

t = 50 | ||||

x | −0.0345 | 6.2989 | ||

y | −0.0209 | 6.3027 | ||

z | 0.0225 | 6.3196 | ||

Average | −0.0110 | 0.0 | 6.3071 | 6.2727 |

r | 9.6762 | 9.6547 | 5.0705 | 4.9828 |

t = 100 | ||||

x | 0.0215 | 7.9626 | ||

y | 0.0259 | 7.9500 | ||

z | −0.0037 | 7.9225 | ||

Average | 0.0146 | 0.0 | 7.9450 | 7.9506 |

r | 11.8653 | 11.9176 | 6.9704 | 6.8998 |

t = 1000 | ||||

x | 0.0122 | 9.9410 | ||

y | −0.0252 | 9.9394 | ||

z | −0.0035 | 9.9085 | ||

Average | −0.0055 | 0.0 | 9.9296 | 9.9998 |

r | 14.0043 | 14.1421 | 9.9835 | 9.9995 |

methods was not a priori evident or demonstrated at the time of publication of Ref. [

Besides the analytical demonstration, we have tested the two approaches by means of a Monte Carlo method that generated histograms of Brownian trajectories from the LE whose empirical profiles were matched nearly perfectly by the probability density functions derived from the FPE. Statistical tests applied to the sets of sufficiently occupied bins showed that any deviations between theory and observation could be attributable to pure chance.

In the course of these demonstrations, we have derived and reported in this paper explicit mathematical expressions for the probability density functions, spatial moments, and moment-generating functions of 1D and 3D Brownian trajectories of radioactive particles. We have also derived an explicit expression for the probability density function for the distribution of first-passage times of a radioactively decaying particle to diffuse to specified absorbing boundaries. These functions, which should prove useful in the measurement and analysis of radioactive particles dispersed in the environment or investigated in the laboratory, differ markedly from corresponding statistical functions for Brownian motion of non-decaying particles.

In statistical terminology, radioactive decay as described by the Rutherford-Soddy law, whereby the intrinsic decay rate is constant and independent of interactions external to the nucleus, is one of the simplest examples of a birth-and-death process of which there exists an extensive literature [

Silverman, M.P. and Mudvari, A. (2018) Brownian Motion of Radioactive Particles: Derivation and Monte Carlo Test of Spatial and Temporal Distributions. World Journal of Nuclear Science and Technology, 8, 86-119. https://doi.org/10.4236/wjnst.2018.82009

Relation between First and Second Moments of the First-Passage Time

In Appendix 2 of Reference [

〈 T 〉 = ∫ 0 ∞ ( 1 − F ( t ) ) d t (76)

〈 T 2 〉 = 2 ∫ 0 ∞ t ( 1 − F ( t ) ) d t (77)

in which the cumulative distribution function F ( t ) , defined in terms of the probability density function p T ( t ) , is

F ( t ) = ∫ 0 t p T ( u ) d u . (78)

By definition, the survival function S ( t ) is the probability that the diffusing particle has not reached a boundary―or, in other words, the probability that the random variable T, the first-passage time (FPT), exceeds t. Therefore

S ( t ) = Pr ( T > t ) = 1 − F ( t ) . (79)

Substitution of Equation (79) into Equations (67), (76), (77), with recognition that Equation (69) has the form

p T ( 1 ) ( t ) = λ S ( t ) (80)

leads to

〈 T 〉 = 〈 T 〉 0 + λ 2 〈 T 2 〉 (81)

which is equivalent to Equation (75).

Probability of Survival: The Geometric Distribution

Let p be the probability of surviving a single time step and q = 1 − p the corresponding probability of decay. If survival or decay at each time step is independent of the outcome at any other time step, then the probability of surviving k − 1 time steps and decaying on the k^{th} is given by the geometric probability law

P ( k ) = p k − 1 q . (82)

The first few moments of the distribution P ( k ) are

Completeness:

∑ k = 1 ∞ P ( k ) = 1 (83)

Mean number of steps:

〈 k 〉 = ∑ k = 1 ∞ k P ( k ) = 1 q (84)

Mean square number of steps:

〈 k 2 〉 = ∑ k = 1 ∞ k 2 P ( k ) = 1 + p q 2 (85)

Standard deviation:

σ k = 〈 k 2 〉 − 〈 k 〉 2 = p q . (86)

Given probabilities of survival and decay to be p = 0.99 and q = 0.01 respectively at each time step, the probability of surviving 5,000 time steps is

P ( 5000 ) = ( 0.99 ) 4999 ( 0.01 ) = 1.51 × 10 − 24 ,

and the number of surviving particles out of a sample of size of 10^{6} is then 1.51 × 10 − 18 or effectively 0. The mean number of time steps to a decay event is 〈 k 〉 = ( 0.01 ) − 1 = 100 . The standard deviation in the number of time steps to a decay event is 0.99 / 0.01 = 99.5 .

Statistical Tests of the Monte-Carlo Histograms

Operationally, a histogram is a plot of event frequency as a function of event class. Mathematically, a histogram is a multinomial distribution M ( { n k } , { p k } ) ≡ M ( n , p ) of outcomes (classes or bins) k = 1 , ⋯ , K with set of frequencies { n k } and probabilities { p k } governed by the discrete probability function (Ref. [

f M ( n ; p ) = n ! ∏ k = 1 K p k n k n k ! (87)

subject to the constraint

∑ k = 1 K n k = n . (88)

For a radioactive particle undergoing Brownian motion, the probability that the particle is located between coordinates x k − 1 2 Δ x and x k + 1 2 Δ x that defines the k^{th} bin of constant width Δ x is obtained from Equation (17)

p k = ∫ x k − 1 2 Δ x x k + 1 2 Δ x w ( x , t ) d x . (89)

The mean frequency of the k^{th} bin is then predicted to be

m k = n p k (90)

where n is the total sample size (i.e. number of particles). For a multinomial distribution, the variance of m k is given by

σ k 2 = n p k ( 1 − p k ) = m k ( 1 − m k n ) . (91)

For large sample size and numerous bins the transformation of observed frequencies

z k = n k − m k σ k ∼ N k ( 0 , 1 ) (92)

is by the Central Limit Theorem (Ref [

R k ≡ n k m k ∼ N k ( 1 , σ k 2 m k 2 ) , (93)

which is distributed as a Gaussian of mean 1 and variance σ k 2 / m k 2 . Equation (93) follows from the identity (Ref [

N ( μ , σ 2 ) = μ + σ N ( 0 , 1 ) . (94)

Summing Equation (93) over all K classes generates the ensemble random variable R

R ≡ ∑ k = 1 K R k ~ N ( K , S 2 ) (95)

with variance

S 2 = ∑ k = 1 K ( σ k 2 m k 2 ) . (96)

It then follows from relations (93) and (95) that the probability of obtaining under repeated trials a value R k greater or equal to the observed value R k obs for the k^{th} bin, or an ensemble value R greater or equal to the observed value R obs is given by

Pr ( R k ≥ R k obs ) = 1 2 π σ k 2 ∫ R k obs ∞ exp ( − ( u − 1 ) 2 2 σ k 2 ) d u (97)

Pr ( R ≥ R obs ) = 1 2 π S 2 ∫ R obs ∞ exp ( − ( u − K ) 2 2 S 2 ) d u . (98)

The numerical values associated with integrals (97) and (98) are generally referred to as P-values, in which P ( x ) ≡ 1 − F ( x ) for some specified cumulative distribution function F ( x ) .

Consider, as an example, histogram D of

Up to this stage, the preceding test is similar in procedure to that of a standard Pearson chi-square test, except that it involves the ratio of observed to predicted frequencies, whereas a chi-square test involves the square of frequency differences. The chi-square test, although widely used, is based on a chain of assumptions that do not necessarily apply to Brownian motion (e.g. that the probability of an outcome is approximately Poissonian; see Ref. [^{7}.

Furthermore, to discern whether an empirical histogram is a good representation of the overall profile of a proposed probability density function, it is of utility to go beyond the ensemble P-value and to calculate the P-values (97) individually for all sufficiently occupied bins. In the above example for histogram D of

Statistical tests for goodness-of-fit can never prove that a hypothesized theory is the “true” explanation of some set of empirically derived results. What the preceding statistical tests show, however, is that they provide no statistical grounds for rejecting the conclusion, reached by rigorous analysis, that the Monte-Carlo simulations of the Brownian motion of radioactive particles, based on the Langevin Equation (3), are accounted for by the theoretical probabilities derived from the Fokker-Planck Equation (1).