_{1}

^{*}

Recent developments in the measurement of radioactive gases in passive diffusion motivate the analysis of Brownian motion of decaying particles, a subject that has received little previous attention. This paper reports the derivation and solution of equations comparable to the Fokker-Planck and Langevin equations for one-dimensional diffusion and decay of unstable particles. In marked contrast to the case of stable particles, the two equations are not equivalent, but provide different information regarding the same stochastic process. The differences arise because Brownian motion with particle decay is not a continuous process. The discontinuity is readily apparent in the computer-simulated trajectories of the Langevin equation that incorporate both a Wiener process for displacement fluctuations and a Bernoulli process for random decay. This paper also reports the derivation of the mean time of first passage of the decaying particle to absorbing boundaries. Here, too, particle decay can lead to an outcome markedly different from that for stable particles. In particular, the first-passage time of the decaying particle is always finite, whereas the time for a stable particle to reach a single absorbing boundary is theoretically infinite due to the heavy tail of the inverse Gaussian density. The methodology developed in this paper should prove useful in the investigation of radioactive gases, aerosols of radioactive atoms, dust particles to which adhere radioactive ions, as well as diffusing gases and liquids of unstable molecules.

Brownian motion, one of the simplest examples of a random walk, is a nonequilibrium statistical process the mathematics of which serves to model a wide variety of stochastic processes throughout the physical and social sciences. From the earliest applications of Einstein to the random motion of small particles in a fluid [

Despite their great diversity, nearly all such studies known to the author have in common the feature that the diffusing particles maintain their identity throughout the stochastic process. There are a few exceptions, such neutron diffusion with beta decay in reactor materials [

Under the assumption, usually justified by the Fermi Golden Rule of time- dependent perturbation theory in quantum mechanics [

p ( Δ t ) = λ Δ t ≪ 1 , (1)

where λ is the intrinsic decay rate, it is readily deducible [

d p ( t ) / d t = − λ p ( t ) (2)

with normalized solution

p ( t ) = λ e − λ t . (3)

From Equation (3) it follows that the mean lifetime τ is the reciprocal of λ

τ = ∫ 0 ∞ t p ( t ) d t = λ − 1 , (4)

and that the half-life, i.e. duration within which the survival probability is 50%, is

τ 1 / 2 = λ − 1 ln 2 = τ ln 2 . (5)

The premise that the probability (1), with λ independent of environmental interactions, accurately characterizes the transmutation of radioactive nuclei has been challenged on various grounds during the past 20 years. However, in each case the theoretical consequences of relation (1) were validated experimentally and shown to be in accord with currently known laws of physics [

Because unstable particles are removed from the system at random instances during the time they are moving randomly within a specified sample space, the familiar standard equations (e.g. Langevin and Fokker-Planck) for Brownian motion must be re-derived on the basis of a conservation law that takes particle loss into account. This more general relation, presented in Section 2, leads to significant differences in the analytical structure of the stochastic equations and statistical moments compared with corresponding expressions derived for the random walk of a stable particle.

In general, there are two mathematically different approaches to analyzing Brownian motion. One method, to be referred to as the Langevin approach, is to examine the process variables directly. For a particle undergoing a one-dimensional random walk, the process variable of interest in this paper is the displacement X ( t ) as a function of time t. Conventional symbolic notation, which is used in this paper unless stated otherwise, employs an upper-case letter (e.g. X) to designate a random variable and a corresponding lower-case letter (e.g. x) to represent a realization or measurement of the variable (referred to as a variate in statistical terminology). The Langevin equation of motion for the time-evolution of the coordinate variable usually takes the form of an Ito stochastic differential equation [

d X ( t ) ≡ X ( t + d t ) − X ( t ) = A ( X , t ) d t + B ( X , t ) d W ( t ) (6)

in which A ( X , t ) is the drift function, B ( X , t ) is the diffusivity, and d W ( t ) = d t N ( 0 , 1 ) is the differential Wiener process in which N ( 0 , 1 ) symbolizes a Gaussian distribution of mean μ = 0 and variance σ 2 = 1 . The drift A ( X , t ) is a deterministic function, whereas the Wiener term is the source of fluctuations.

It is clear from Equation (6) that the analytical solution, if one can be found, is not a deterministic function of a dynamical variable, but a probability distribution. A distribution is said to be stable if two independent random variables characterized by this distribution sum to form a random variable governed by a distribution of the same kind [

p X ( x | μ , σ ) = 1 2 π σ 2 exp ( − ( x − μ ) 2 / 2 σ 2 ) , (7)

one can express the corresponding random variable as

X = N ( μ , σ 2 ) = μ + σ N ( 0 , 1 ) (8)

and the sum of two independent Gaussian random variables as

X 1 + X 2 = N 1 ( μ 1 , σ 1 2 ) + N 2 ( μ 2 , σ 2 2 ) = N ( μ 1 + μ 2 , σ 1 2 + σ 2 2 ) = μ 1 + μ 2 + σ 1 2 + σ 2 2 N ( 0 , 1 ) . (9)

It follows from the stability relation (9) that the solution to (6) is itself a Gaussian distribution.

The second method, to be referred to as the Fokker-Planck approach, is to solve for the transition probability function (TPF) p ( x , t | x 0 , t 0 ) of finding the particle at location x at time t, given that the particle was initially at x 0 at time t 0 < t . The partial differential equation corresponding to the stochastic differential Equation (6) is the forward Fokker-Planck equation [Ref [

∂ p ( x , t | x 0 , t 0 ) ∂ t = − ∂ ( A ( x , t ) p ( x , t | x 0 , t 0 ) ) ∂ x + 1 2 ∂ 2 ( B ( x , t ) p ( x , t | x 0 , t 0 ) ) ∂ x 2 . (10)

For non-decaying particles, Equations ((6) and (10)) have the same information content, but provide different perspectives on the same stochastic process. For example, the stochastic differential Equation (6) is particularly suitable for numerical solution by an iterative up-dating algorithm, thereby permitting computer simulation of particle displacement and visualization of the process variables in real time. On the other hand, the forward Fokker-Planck Equation (10) yields a probability density function from which all statistical moments can be calculated. The solution p ( x , t | x 0 , t 0 ) also solves the backward Fokker- Planck equation [Ref [

∂ p ( x , t | x 0 , t 0 ) ∂ t 0 = − A ( x 0 , t 0 ) ∂ p ( x , t | x 0 , t 0 ) ∂ x 0 − 1 2 B ( x 0 , t 0 ) ∂ 2 p ( x , t | x 0 , t 0 ) ∂ x 0 2 , (11)

in which derivatives are taken with respect to the initial coordinates. Equation (11) provides the same information as the forward Fokker-Planck Equation (10), but is of particular utility in the analysis of Brownian motion within a region confined by boundaries. The structure of Equation (11) facilitates treatment of the problem of first-passage times to be analyzed in Section 3 by alternative, simpler methods.

Generally speaking, one solves a Brownian motion problem to answer two kinds of questions: 1) How far has a particle randomly walked in a given time? and 2) In how much time will a particle randomly walk to a given location? It is to be understood, of course, that answers to questions like these are probabilistic, not deterministic, statements. Questions of the first kind are perhaps more familiar, but there are numerous circumstances that call for answers to questions of the second kind. These are usually referred to as first-passage processes [

This paper is concerned primarily with one-dimensional Brownian motion of a particle of finite statistical lifetime in free (unconstrained) space and in a space with absorbing boundaries. Derivation and analysis of the equations corresponding to the Fokker-Planck equation and Langevin equation will show that these equations do not lead to equivalent descriptions of the Brownian motion of a decaying particle, in marked contrast to the case of stable particles. This inequivalence is traceable to the fact that particle decay is a discontinuous process. The Fokker-Planck equation, which yields a transition probability density, remains a continuous differential equation, but the Langevin equation, which describes a process variable, and not a probability density, must directly manifest the discontinuous nature of decay.

The distinction between the Brownian motion of stable and unstable particles becomes very apparent in the problem of first-passage time to absorbing boundaries. Upon reaching an absorbing boundary for the first (and therefore only) time, the particle is removed from the system and the process is terminated. Likewise, upon decay at some unpredictable moment, the particle is also removed and the process is terminated. Thus, the instability of the particle introduces into a first-passage problem an additional time element that can radically modify expectation values. From the perspective of physics, this modification extends the applicability of first-passage time theory to a broader class of physical systems. As a physical model, the solution to a first-passage time problem with particle decay has been applied to the diffusion of the radioactive gas radon-222 in the atmosphere, and should likewise prove useful in the study of diffusion of other radioactive gases, radioactive ions that form as daughter products in radioactive decay, as well as unstable molecules that change their identity by chemical transformation.

As a purely stochastic problem, the analysis undertaken in this paper

・ derives and provides an exact solution to the Fokker-Planck and Langevin equations of Brownian motion of an unstable particle,

・ extends to unstable particles the two principal methods of calculating first-passage times,

・ demonstrates how to simulate by computer the Brownian motion of an unstable particle, and

・ clarifies a number of confusing issues that arise in the case of unstable particles (but not stable particles) regarding Fokker-Planck and Langevin equations, expectation values, probability density functions, and transition probability functions.

This paper is organized as follows. Section 2 is concerned with spatial aspects of a decaying particle undergoing a one-dimensional random walk. Derivation and solution of the Fokker-Planck equation to obtain the transition probability density and associated statistical moments are given in Section 2.1. In Section 2.2 the mean-square displacement of the decaying particle is reconsidered from the perspective of a random walk on a discrete lattice and shown to coincide in the appropriate limit with the result obtained in Section 2.1. The derivation, numerical solution, and computer simulation of the Langevin equation to obtain Brownian motion trajectories of the decaying particle are given in Section 2.3. In Section 2.4 the Langevin equation is solved analytically to obtain the distribution function of the particle displacement. Section 3 is concerned with temporal aspects of a decaying particle undergoing a one-dimensional random walk. The mean first-passage time to absorbing boundaries is solved in Section 3.1 by the method of image functions and in Section 3.2 by solution of a screened Poisson equation. Section 3.3 illustrates the critical role of particle decay in leading to first-passage time results that differ markedly from corresponding results for a stable particle. Section 4 examines the validity of the stochastic model, based on a Wiener process or Fick’s law, to account for fluctuations in the spatial displacement of a decaying particle. Finally, conclusions drawn from these analyses are summarized in Section 5.

Consider a quantity Q ( t ) = ∭ V n ( x , t ) d V of particles with decay constant λ and number density n ( x , t ) within a volume V bound by surface S. Since loss of Q can occur either by intrinsic decay at the rate − λ Q or by diffusion of a current density j ( x , t ) outward across surface S, macroscopic mass balance requires that

∂ ∂ t ∭ V n d V = − ∯ S j ⋅ d S − λ ∭ V n d V . (12)

Application of the divergence theorem then leads from the integral relation (12) to the differential equation

∂ n ∂ t = − ∇ ⋅ j − λ n (13)

for the conservation law of a disintegrating quantity.

Upon dividing Equation (13) by the initial number of particles N 0 ≫ 1 and relating the current density to the gradient of the particle density by Fick’s law

j = − D ∇ n , (14)

with diffusion coefficient D (here taken to be a constant), one obtains an equation of the form

∂ w ∂ t = D ∇ 2 w − λ w (15)

in which w ( x , t ) = n ( x , t ) / N 0 is interpretable as the probability density for diffusion of a single decaying particle. However, it is demonstrable that the solution w ( x , t ) under the special initial condition at t 0

w ( x , t 0 ) = δ ( x − x 0 ) (16)

is identical to the transition probability density p ( x , t | x 0 , t 0 ) , which is the conditional probability for an unstable particle to have reached position x at time t given that it was at x 0 at t 0 < t . The equivalence is readily established by substitution of condition (16) into the Chapman-Kolmogorov equation [

w ( x , t ) = ∫ − ∞ ∞ p ( x , t | x 1 , t 0 ) w ( x 1 , t 0 ) d x 1 = ∫ − ∞ ∞ p ( x , t | x 1 , t 0 ) δ ( x 1 − x 0 ) d x 1 = p ( x , t | x 0 , t 0 ) . (17)

In the case of one-dimensional Brownian motion treated in this paper, Equation (15) then takes the form

∂ p ( x , t | x 0 , t 0 ) ∂ t = D ∂ 2 p ( x , t | x 0 , t 0 ) ∂ 2 x − λ p ( x , t | x 0 , t 0 ) , (18)

which, together with the initial delta-function condition (16)

p ( x , t 0 | x 0 , t 0 ) = δ ( x − x 0 ) (19)

comprises the equation for the Green’s function of a one-dimensional random walk with decay [

The solution to Equations ((18) and (19)) can be obtained in several ways. One method is to take the Fourier transform with respect to the spatial coordinate, which converts the partial differential Equation (18) of two variables (space, time) into an ordinary differential equation in one variable (time). The simplest method, however, which has been used to solve the Schroedinger equation for transitions between excited atomic states [

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 ) . (20)

From the form of relation (20), one sees that the transition probability is a function of the time interval t − t 0 , and not the separate time coordinates ( t , t 0 ) . From this point on, it will be assumed that t 0 = 0 and t will represent a time interval.

Although the functions w ( x , t ) and p ( x , t | x 0 , 0 ) are mathematically identical in form, they serve different purposes and are used to calculate different quantities. For example, the probability density function (PDF) is employed to calculate the statistical moments { m k } of a distribution, where the k th moment (expectation value) with respect to the origin is defined by

m k = ∫ x k w ( x , t ) d x / ∫ w ( x , t ) d x , (21)

and the range of integration covers the defined sample space. If the PDF is normalized, then the denominator in relation (21) is unity. However, the normalization of PDF (20) is not unity

∫ − ∞ ∞ w ( x , t ) d x = ∫ − ∞ ∞ p ( x , t | x 0 ) d x = e − λ t , (22)

but yields, instead, the survival probability to time t of the unstable particle. In contrast to (21), the TPF is employed to calculate transition probabilities and statistical moments of a different nature. For example, the expectation value of the k th power of displacement of a particle in time interval t starting from position x 0 is

〈 x k 〉 = ∫ x k p ( x , t 0 | x 0 , 0 ) d x = m k e − λ t . (23)

Because the TPF is a conditional probability, the expectation value (23) does not include a normalizing denominator as in Equation (21). For a stable particle, relations (21) and (23) yield the same result, but this is not the case for a decaying particle.

The distinction between PDF and TPF leads to different results for the mean- square displacement. Employing solution (20) as the Gaussian PDF w ( x , t ) in the expectation value (21), one obtains the mean location m 1 = x 0 and variance about the mean

m 2 − m 1 2 = 2 D t (24)

of a non-decaying diffusing particle. However, calculation of expectation values (23) with the TPF p ( x , t | x 0 , 0 ) leads to the initial location 〈 x 〉 = x 0 of the diffusing particle and its mean-square distance from the point of origin

σ x 2 ( t ) = 2 D t e − λ t . (25)

Thus, the same mathematical function (20) generates expectation values that differ in interpretation and mathematical form depending on what is sought by the analyst. In the context of understanding how decay affects the probability of displacement of a single particle in continuous Brownian motion, relation (25) is the relevant quantity, as shown in the following section.

Consider a one-dimensional Gaussian random walk on a lattice with time step δ t and displacement δ X j during the jth time step given by

δ X j = δ X N j ( 0 , 1 ) (26)

where the lattice spacing δ X sets the scale of spatial displacement. Each displacement is taken to be an independent Gaussian random variable. However, a displacement can be made only if the particle has survived during that time step. From Equation (3) it is seen that the probability of survival during a time step δ t is p s = 1 − λ δ t , and the probability that the process ends at a particular time step is λ δ t . Thus, although the distance traveled in the jth time step is determined by a unit normal distribution N j ( 0 , 1 ) , the probability that the displacement is made at all is determined by a Bernoulli random variable

ε j = B j ( 1 , p s ) = { 1 withprobability p s = 1 − λ δ t 0 withprobability 1 − p s = λ δ t . (27)

A Bernoulli distribution is the special case n = 1 of the binomial distribution B ( n , p ) of n trials with probability of success p ; the corresponding discrete probability function is

p B ( k | n , p ) = ( n k ) p k ( 1 − p ) n − k ( n ≥ k ≥ 0 ) . (28)

The subscript j in relations (26) and (27) denotes that each random sample, whether Gaussian or Bernoulli, is independent of all the others.

The displacement after n time steps is given by the random variable X n

X n = ∑ j = 1 n δ X j = δ X ∑ j = 1 n N j ( 0 , 1 ) = δ X N ( 0 , n ) . (29)

The expectation value of the mean square displacement at time t = n δ t is therefore

〈 X n 2 〉 = ( δ X ) 2 〈 ( N ( 0 , n ) ) 2 〉 p s n = n ( δ X ) 2 ( 1 − λ δ t ) n , (30)

which can be re-expressed in the form

〈 X n 2 〉 = t ( ( δ X ) 2 δ t ) ( 1 − λ t n ) n . (31)

Upon defining the diffusion constant D in the standard way

2 D = ( δ X ) 2 / δ t (32)

and taking the limit δ t → 0 , δ X → 0 , with requirement that D remain finite, Equation (31) becomes

〈 X t 2 〉 = 2 D t e − λ t (33)

in accord with the expectation value (25) obtained directly from the TPF (20).

As a point of clarification, the reason for the factor of 2 in the conventional definition (32) of the diffusion coefficient is that the quantity 2 D corresponds to the fluctuation function B ( x , t ) in the Wiener term of the forward Fokker- Planck Equation (10). Under the conditions that B ( x , t ) = 2 D is constant and there is no drift, A ( x , t ) = 0 , Equation (10) then reduces to the standard one- dimensional diffusion equation

∂ p ( x , t | x 0 , t 0 ) ∂ t = D ∂ 2 p ( x , t | x 0 , t 0 ) ∂ x 2 (34)

in which D is the physically measurable diffusion constant first introduced by Einstein in his theory of Brownian motion.

The Langevin Equation (6) for one-dimensional Brownian motion of a non-decaying particle in the absence of drift is expressible in a form

x ( t + d t ) = x ( t ) + 2 D d t n t (35)

that facilitates numerical solution by an update algorithm. The lower-case letters x in Equation (35) signify numerical realizations of the random variable X in Equation (6); dt is the numerical time step; and n t is a random sample from the unit normal distribution 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. Thus, 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 . Given an initial value x ( 0 ) , a sequence of points { x ( 0 ) , x ( d t ) , x ( 2 d t ) , ⋯ , x ( n d t ) } is generated by iterative use of Equation (35) up to time t = n d t . (Note: The symbol n without subscript is the number of time steps, not a sample from a normal distribution.) The sequence of points is an approximation to the true Brownian motion in the limit d t → 0 . In that theoretical limit, the trajectory of Brownian motion is a curve that is everywhere continuous, but nowhere differentiable; in other words, a particle trajectory for which the particle velocity is undefined.

Although Equation (35) is simple enough to be solved analytically, a numerical solution provides a graphical visualization of Brownian motion paths. Moreover, starting from the same initial condition and generating numerous Brownian trajectories for a specified time interval t provides a Gibbs ensemble [

The question addressed in this section is this: What is the Langevin equation for Brownian motion of a decaying particle? Since the Langevin and Fokker- Planck equations of a stable particle ordinarily provide equivalent information, one can in principle start with either one and obtain the functions A ( x , t ) and B ( x , t ) needed for the other. Thus, for a stable particle the Langevin Equation (35) leads to the corresponding Fokker-Planck Equation (34), and vice-versa. It is to be stressed, however, that only a continuous Markov process is completely and equivalently defined by either the Langevin equation or Fokker-Planck equation [Ref [

Theoretically, it is possible to transform Equation (18) into a Fokker-Planck equation. One merely needs to find a drift function A ( x , t ) that satisfies the differential equation

λ ∂ ∂ x ( A ( x , t ) p ( x , t | x 0 , t 0 ) ) = λ p ( x , t | x 0 , t 0 ) . (36)

Integration of Equation (36) yields

A ( x , t ) = π D t erf ( x 4 D t ) exp ( x 2 4 D t ) (37)

with boundary and initial conditions

A ( 0 , t ) = 0 , A ( x , 0 ) ∼ δ ( x ) . (38)

The error function is defined by the integral

erf ( x ) ≡ 2 π ∫ 0 x e − u 2 d u = − erf ( − x ) . (39)

However, the preceding solution (37)―or, indeed, any transformation that generates a Fokker-Planck equation from Equation (18) by finding a drift term―seriously misrepresents the physics of the problem. This is a stochastic process, as emphasized in the preceding sections, in which the physical particle (or the probability of particle existence), and not a process variable, is decaying. There is neither drift nor friction in this process.

The key point to recognize in constructing an appropriate stochastic differential equation―which, for the sake of conventional nomenclature, is referred to in this paper as a Langevin equation, even though rigorously it is not―is that the unstable particle, as long as it exists, undergoes Brownian motion as described by the standard diffusion Equation (35). The Brownian motion does not continue indefinitely, however, but is interrupted randomly by particle decay. The first instance of decay terminates the process; there is no further diffusion of that particle. (One could then introduce another particle and follow its Brownian motion if it is desired to acquire an ensemble of trajectories.) The stochastic process, therefore, entails sequences of two paired independent distributions: (a) the unit normal distribution N t t + d t ( 0 , 1 ) , which determines the direction and extent of displacement during the time period [ t , t + d t ] , and (b) the Bernoulli distribution B t t + d t ( 1 , p s ) , which determines whether or not the particle survives the interval dt with a survival probability p s given by Equation (27).

The appropriate Langevin equation would then take the update form

x ( t + d t ) = [ x ( t ) + 2 D d t n t ] ε t (40)

where the Bernoulli variate ε t is defined in relation (27). Note that an occurrence of ε t = 0 terminates the entire process by setting x ( t + d t ) to zero. Thus, just before the moment of decay, the particle will have reached location x ( t ) and no further. Although the Langevin Equation (40) and the TPF Equation (18) both describe Brownian motion of a decaying particle, the descriptions are not equivalent. The TPF is a continuous probability function for all displacements x of a given particle; there is no built-in mechanism to terminate the process at decay. The Langevin Equation (40) has such a mechanism.

To get a sense of the progression of the process, the first few iterations of (40) for a particle starting at the origin x 0 = 0 are shown explicitly below:

x ( d t ) = 2 D d t n 1 ε 1 x ( 2 d t ) = 2 D d t ( n 1 ε 1 ε 2 + n 2 ε 2 ) x ( 3 d t ) = 2 D d t ( n 1 ε 1 ε 2 ε 3 + n 2 ε 2 ε 3 + n 3 ε 3 ) ⋮ x ( n d t ) = 2 D d t ( n 1 ∏ j = 1 n ε j + n 2 ∏ j = 2 n ε j + n 3 ∏ j = 3 n ε j + ⋯ + n n ε n ) . (41)

The mean displacement 〈 x ( n d t ) 〉 = 0 follows immediately from the properties of the unit Gaussian 〈 n t 〉 = 〈 N t t + d t ( 0 , 1 ) 〉 = 0 . The mean-square displacement is less obvious and leads to two results depending on whether one retains the structure of discrete time steps or takes the limit for continuous displacement in time.

Consider first the continuous-time limit of the mean-square displacement, where one sets d t = t / n and eventually takes the limit n → ∞ and d t → 0 such that t is a fixed quantity. Equation (41) then yields

〈 x 2 ( n d t ) 〉 = 2 D t n ( 〈 ( n 1 ) 2 〉 ∏ j = 1 n 〈 ( ε j ) 2 〉 + 〈 ( n 2 ) 2 〉 ∏ j = 2 n 〈 ( ε j ) 2 〉 + ⋯ + 〈 ( n n ) 2 〉 〈 ( ε n ) 2 〉 ) (42)

where the Bernoulli variates ε t = 1 at each time step (otherwise there would not be n steps). There are no cross-terms in the expectation values because all the Gaussian and Bernoulli samples are realizations of independent uncorrelated random variables. Upon insertion in Equation (42) of the expectation values

〈 ( n t ) 2 〉 = 1 〈 ( ε t ) 2 〉 = p s = ( 1 − λ t n ) , (43)

summing the terms in Equation (42), and taking the forementioned limit, one arrives at

〈 x 2 ( t ) 〉 = 2 D t n ∑ j = 1 n ( 1 − λ t n ) j → n → ∞ 2 D λ ( 1 − e − λ t ) . (44)

It should not be surprising that expression (44) differs from the mean-square displacement (25) derived from the TPF, because Equation (18) provides different information than the stochastic differential Equation (40). As shown in Section 2.2, the expectation value (25) is equivalent to the mean-square displacement of a particle prior to decay multiplied by the survival probability to time t. Statistically, it may be thought of as a compound stochastic process N ( 0 , n ) × B ( n , p s ) . By comparison, one can think of the outcome (44) as resulting from a sum of n compound stochastic processes of the form N ( 0 , 1 ) × B ( 1 , p s ) .

It is of interest to examine the two limiting cases of Equation (44):

〈 x 2 ( t ) 〉 → { 2 D t ( λ t ≪ 1 ) 2 D / λ ( λ t ≫ 1 ) . (45)

The first limit in (45) shows that the mean-square displacement of a particle whose statistical lifetime is much longer than the diffusion time is the same as for Brownian motion of a stable particle. The second limit shows that the root- mean-square distance 〈 x 2 ( t ) 〉 reached by a particle very likely to decay during the diffusion time is equivalent, within a factor 2 , to the characteristic diffusion length [

ζ = D / λ (46)

that occurs in the solution of the diffusion equation for radioactive gases [

Consider next the mean-square displacement as obtained from Equation (44) with n discrete time steps of duration δ t

〈 x 2 ( n δ t ) 〉 = 2 D δ t ∑ j = 1 n p s j = 2 D λ ( 1 − λ δ t ) [ 1 − ( 1 − λ δ t ) n ] , (47)

where use was made of the summation formula for a geometric series

∑ j = 1 n p j = ∑ j = 0 n p j − 1 = 1 − p n + 1 1 − p − 1 . (48)

For λ δ t ≪ 1 , in accordance with the assumption underlying Equation (1), Equation (47) can be reduced to

〈 x 2 ( n δ t ) 〉 ≈ 2 D n δ t = 2 D t (49)

by application of the approximation ( 1 − λ δ t ) n ≈ 1 − n λ δ t and neglect of the term λ δ t ≪ 1 where it occurs alone, i.e. not multiplied by n.

The two methods of taking limits led to two different outcomes, (44) vs. (49), because each method held a different quantity constant. In the approach leading to (44), the total diffusion time t was fixed, and the number of time steps n was taken to an infinite limit as time interval δ t approached zero. In other words, n was merely an intermediary variable; finite values of n did not define the duration t of the process. However, in the approach leading to (49), the number of time steps n is the quantity of interest, and the duration t = n δ t is determined by n.

Because the decay of the diffusing particle can occur randomly at any time step, the number n in Equation (49) is itself a random quantity unknown at the outset of a Brownian random walk. One can therefore regard n as a realization of a discrete random variable N (not to be confounded with the symbol N ( 0 , 1 ) for a unit normal distribution) that is subject to a geometric distribution with probability function

p N ( n | p s ) = p s n q s = p s n ( 1 − p s ) (50)

in which p s = 1 − λ δ t is the probability of success (or survival) at each time step and q s = λ δ t is the probability of failure (or decay). Thus, Equation (50) expresses the probability of a process with n successes in sequence followed by a single failure that terminates the process. The distribution is normalized

∑ n = 0 ∞ p N ( n | p s ) = 1 (51)

with mean n ¯

n ¯ = ∑ n = 0 ∞ n p N ( n | p s ) = p s 1 − p s = 1 − λ δ t λ δ t . (52)

The outcome of a large number of Brownian random walks of a decaying particle, each starting from the same initial condition x 0 = 0 , leads to a distribution of time steps n given by Equation (50). One can ask, therefore, for the ensemble-average of the mean-square displacement (49)

〈 x 2 〉 ≡ 〈 〈 x 2 ( n δ t ) 〉 〉 n = 2 D n ¯ δ t = 2 D λ ( 1 − λ δ t ) ≈ 2 D / λ (53)

where, again, the term λ δ t ≪ 1 was dropped. The result (53) of the ensemble average is a root-mean square displacement 〈 x 2 〉 equal to the mathematical diffusion length ζ m .

It is instructive to recalculate the ensemble average of the mean-square displacement starting with the second equality in (49), in which the continuous variable t, rather than the discrete time step n, is the variable of interest. The duration t of a Brownian random walk is a realization of a continuous random variable T governed by the exponential distribution E ( λ ) with PDF (see Equation (3))

p T ( t | λ ) = λ e − λ t . (54)

The distribution is normalized

∫ 0 ∞ p T ( t | λ ) d t = 1 (55)

with mean t ¯

t ¯ = ∫ 0 ∞ t p T ( t | λ ) d t = λ − 1 . (56)

The ensemble average of the mean-square displacement (49) is then

〈 x 2 〉 ≡ 〈 〈 x 2 ( t ) 〉 〉 t = 2 D t ¯ = 2 D / λ (57)

in necessary agreement with ensemble average (53) obtained from the geometric distribution. The geometric and exponential distributions are, respectively, the discrete and continuous distributions for the class of statistical problems referred to as waiting-time problems. For continuous or discrete Markov processes that are characterized by a lack of memory, i.e. that have the same probability of success or failure at each trial, the distribution of the duration of the process must take the form of either a geometric or exponential distribution [

The red trace in

A point of particular interest in

Moreover, there is no formal “law of averages”; the closest rigorous principle would be the law of large numbers [Ref [

The black trajectories of

It is possible to solve stochastic Equation (40) analytically in closed form for the displacement X ( t ) and PDF p X ( x , t ) of a particle undergoing Brownian motion with decay. It is worthwhile to do so for at least two reasons. First, it is easy to misconstrue the form of Equation (40) and thereby end up with a solution that is structurally incorrect. And second, the correct solution leads to a distribution with features that physicists do not ordinarily encounter.

Consider first the misleading (or at least incomplete) way to proceed. If one regards the numerical update algorithm (41) as a sum of n independent unit normal variates multiplied by Bernoulli variates all taken to have value ε = 1 , then X ( t ) must also be expressible in closed form as a normal random variable of zero mean because of the stability of the normal distribution. Moreover, since a normal distribution is uniquely determined by the mean and variance, it follows from relation (44) that the solution (in the limit n d t → t ) must be

X ( t ) = N ( 0 , 2 D λ ( 1 − e − λ t ) ) , (58)

whereupon the corresponding PDF, generalized on the basis of space and time translational symmetry to include initial conditions ( x 0 , t 0 ) , takes the form

p X ( L ) ( x , t | x 0 , t 0 ) = 1 4 π D ( 1 − e − λ ( t − t 0 ) ) / λ exp ( − ( x − x 0 ) 2 / 4 D ( 1 − e − λ ( t − t 0 ) ) / λ ) .(59)

The superscript L distinguishes solution (59) to the Langevin stochastic equation from solution (20) to the Fokker-Planck equation repeated below for ease of comparison

p X ( FP ) ( 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 ) (60)

and denoted by a superscript FP.

The problem with solution (58), however, is that it no longer describes a process that can be interrupted at any time by the decay of the particle. The transformation from a discontinuous to a continuous stochastic process arose by confounding the Bernoulli random variables, which can be either 1 or 0, with the specific outcomes, or variates, all taken to be 1 in the previous calculation of expectation values. Re-examining the last line in Equation (41)―i.e. before expectation values are taken―shows, together with the stability of the normal distribution, that Langevin Equation (40), expressed in terms of random variables, can be written as

X ( n d t ) ≡ X ( t ) = N ( 0 , 2 D d t Σ n 2 ) (61)

in which

Σ n 2 = ∏ j = 1 n ε j 2 + ∏ j = 2 n ε j 2 + ∏ j = 3 n ε j 2 + ⋯ + ε n 2 (62)

is a random variable, not an expectation value. So as not to complicate the notation unnecessarily, the symbol for a Bernoulli random variable will remain ε , in departure from the conventional notation to use an upper-case letter.

The question then becomes: What kind of random variable is Σ n 2 and what are its properties? Details of the analysis are left to Appendix 1, but the salient points are summarized as follows. Any random variable Z is uniquely characterized by its moment-generating function (MGF)

g Z ( θ ) ≡ 〈 exp ( Z θ ) 〉 (63)

(if it exists), or its characteristic function (CF)

h Z ( θ ) ≡ 〈 exp ( i Z θ ) 〉 (64)

(which always exists), or its probability function (for discrete outcomes) or probability density (for continuous outcomes) [

ε j 2 = ε j = B ( 1 , p s ) ( j = 1 , 2 , ⋯ , n ) (65)

ω j ≡ ∏ k = j n ε k 2 = ∏ k = j n ε k = B ( 1 , p s n − j + 1 ) ( j = 1 , 2 , ⋯ , n ) (66)

and

Σ n 2 ≡ ∑ j = 1 n ω j . (67)

If the set of random variables { ω j } were mutually independent, the MGF of the sum in (67) could be easily calculated. However, by virtue of the defining relation (66), any pair of the ω variables, e.g. ω j = ε j ε j + 1 ⋯ ε k ε k + 1 ⋯ ε n and ω k = ε k ε k + 1 ⋯ ε n , could contain some identical factors and be highly correlated. Nevertheless, although less easily done, the MGF of Σ n 2 can be shown to be

g Σ n 2 ( θ ) = 1 − p s + p s n + 1 e n θ ( 1 − e θ ) ( 1 − p s e θ ) . (68)

Although MGF (68) does not correspond to any of the tabulated random variables known to the author, all the moments of Σ n 2 can be determined by differentiation

〈 ( Σ n 2 ) k 〉 = d k g Σ 2 ( θ ) d θ k | θ = 0 ≡ g Σ 2 ( k ) ( 0 ) ( k = 0 , 1 , 2 , ⋯ ) , (69)

thereby characterizing Σ n 2 uniquely.

For example, moments k = 0 , 1 , 2 of Σ n 2 calculated from (68) and (69) are

〈 ( Σ n 2 ) 0 〉 = g Σ 2 ( 0 ) ( 0 ) = 1 〈 ( Σ n 2 ) 1 〉 = g Σ 2 ( 1 ) ( 0 ) = p s ( 1 − p s n ) 1 − p s 〈 ( Σ n 2 ) 2 〉 = g Σ 2 ( 2 ) ( 0 ) = p s ( 1 + p s ) ( 1 − p s n ) ( 1 − p s ) 2 − 2 n p s n + 1 1 − p s (70)

The first line of Equation (70) expresses the completeness relation for probabilities, required of the MGF. The second line reproduces expression (48). Thus, calculation of the mean-square displacement from Equation (61)

〈 ( X ( t ) ) 2 〉 = 〈 ( N ( 0 , 2 D d t Σ n 2 ) ) 2 〉 = 2 D d t 〈 Σ n 2 〉 = 2 D d t ( p s ( 1 − p s n ) 1 − p s ) , (71)

leads to

〈 ( X ( t ) ) 2 〉 = lim n → ∞ d t → 0 [ 2 D d t ( ( 1 − λ d t ) ( 1 − ( 1 − λ d t ) n ) λ d t ) ] = 2 D λ ( 1 − e − λ t ) (72)

upon substitution of relation (27) for p s and transforming from discrete time steps to continuous time. The expectation value (72) is precisely the result obtained by a different procedure in (44) and justifies the form of PDF (59). The third line of (70) enables one to calculate the variance of Σ n 2 and the 4^{th} moment of the displacement

〈 ( X ( t ) ) 4 〉 = lim n → ∞ d t → 0 [ 4 D 2 ( d t ) 2 〈 ( Σ n 2 ) 2 〉 ] = 8 D 2 λ 2 ( 1 − ( 1 + λ t ) e − λ t ) (73)

by means of the same substitution and limiting process employed in Equation (72).

In short, therefore, the solution (61) takes the form of a normal distribution whose variance is itself a random variable of un-named variety (as far as the author is aware), but which is completely and uniquely specified by its MGF (68). In principle, the PDF of the distribution of Σ n 2 can be calculated by taking the inverse Fourier transform of the CF; the CF itself is obtained simply by substituting i θ for θ in the MGF (68). For the purposes of this paper, however, the PDF of Σ n 2 is not required.

Solution (61), in contrast to solution (58), incorporates the Bernoulli processes that generate particle decay. If, in a simulation of Brownian motion to be implemented for n time steps from Equation (61), the Bernoulli variate ε k = 0 at time step k ≤ n , it follows from Equation (66) that all the variates ω j = 0 for j = 1 , 2 , ⋯ , k , and therefore Σ k 2 = 0 , and X ( t = k d t ) = N ( 0 , 0 ) = 0 . (Note: N ( 0 , 0 ) = 0 signifies that lim σ → 0 ( σ − 1 exp ( − x 2 / 2 σ 2 ) ) = 0 of the Gaussian PDF.) Thus, the random walk of the decayed particle has been realistically terminated at the randomly selected k th time step. With regard to notation, the subscript on Σ n 2 can be taken to represent the full length (i.e. number of time steps) of a simulated random walk, and not (as before) a pre-determined number of sequential steps survived by the particle. In the limit of an infinitely long random walk, it follows from Equation (72) that the (infinitely) many particle trajectories arising from the introduction of a new particle after decay of each previous one, yield a root-mean-square (RMS) displacement equal to the mathematical diffusion length 2 D / λ , in accord with the ensemble averages (53) and (57).

An empirical test of the ensemble-averaged root-mean-square (RMS) displacement (53) can be made using the computer-generated Brownian trajectories of

Decay Event | Time of Decay | Net Displacement Between Times of Birth and Decay | Square of Displacement |
---|---|---|---|

1 | 3 | 0.5071 | 0.26 |

2 | 22 | −0.2076 | 0.04 |

3 | 63 | −1.0791 | 1.16 |

4 | 99 | 7.8968 | 62.36 |

5 | 163 | 14.9510 | 223.53 |

6 | 244 | −16.6969 | 278.79 |

7 | 254 | −1.2919 | 1.67 |

8 | 328 | −0.8794 | 0.77 |

9 | 496 | −11.4934 | 132.10 |

10 | 696 | −14.1404 | 199.95 |

11 | 755 | −3.7749 | 14.25 |

Sample RMS Displacement 9.49 Theoretical RMS Displacement 10.00 |

chronologically the decay events and corresponding times of decay. Columns 3 and 4 show the net displacement (to 4 significant figures) and square of displacement (truncated to 2 significant figures) between the time of creation of the particle at the origin x 0 = 0 up to the time step just prior to its decay. As shown, the mean of the 11 RMS values is very close to the theoretical prediction calculated from Equation (53) with the parameters used in the simulations of

Now that the stochastically correct solution (61) to the Langevin equation has been derived and shown to justify PDF (59), it is informative to compare the latter with PDF (60) derived from the Fokker-Planck Equation (18).

・ The Fokker-Planck Equation (18), although it includes the particle decay rate, describes one long continuous process, i.e. the evolution of the probability of a particular diffusing particle to be found at an arbitrary location x at time t, provided the particle survived to time t. As t increases, the survival probability of that particular particle decreases exponentially as e − λ t , and the mean- square displacement of its Brownian motion spreads as a power law 2 D t . However, only after an infinitely long time is the probability for the particle to reach any location precisely zero.

・ In contrast to the preceding, the Langevin Equation (40) describes a potentially infinite number of independent Brownian trajectories, disconnected one from the other by events of particle decay. As t increases, the ensemble of independent trajectories come from particles that have survived for different lengths of time. In the limit of an infinitely long time, the probability density (59) does not vanish, but describes, instead, the ensemble-averaged statistics characterized by a Gaussian distribution with mean-square displacement 2 D / λ .

Further perspective on the differences between the two approaches (Langevin vs Fokker-Planck) is given by

It is to be emphasized that the Fokker-Planck and Langevin descriptions are both valid. It is not that one is right and the other wrong; rather, the two analytical methods provide different perspectives on the process of Brownian motion with decay. The Fokker-Planck equation describes a continuous process; it gives a statistical description of the displacement of a decaying particle for as long as the particle might survive in the course of an infinite time span. The Langevin equation describes a sequence of discontinuous processes; it gives a statistical description of the displacement of an ensemble of particles up to the instant each actually fails to survive. It is understandable, therefore, why the theoretical mean-square displacements obtained by the two approaches are different. A significant point of this paper, however, is that for stable particles the two approaches lead to identical results because both would then be describing a single continuous process of infinite duration.

The horizontal dashed blue lines in

In

FPT problems have been studied in great detail for non-decaying diffusers; see, for example [

T FPT ≡ 〈 T 〉 = ∫ 0 ∞ t p T ( t ) d t = ∫ 0 ∞ ( 1 − F T ( t ) ) d t (74)

in which p T ( t ) is the probability density function (PDF) of T and F T ( t ) is the cumulative probability function (CPF)

F T ( t ) = ∫ 0 t p T ( t ′ ) d t ′ . (75)

A proof of the second equality in relation (74) for random variables such as T defined on the non-negative real numbers is given in Appendix 2 of [

In general, two mathematically different approaches have been used to find T FPT for a stable particle. One approach involves calculation of F T ( t ) , starting with the solution to the Fokker-Planck equation. The other approach yields T FPT directly as the solution to a Poisson equation. These two methods are generalized in the following two sub-sections so as to apply to Brownian motion with decay. In the example analyzed, the particle is absorbed upon reaching either of the symmetrically located boundaries x b = ± l before decaying. With the methods developed below the boundary conditions can be easily modified to apply to other systems.

The problem is to find a PDF p x ( x , t ) of the diffusing particle with initial condition (19) for x 0 = 0

p x ( x , 0 ) = δ ( x ) (76)

that satisfies boundary conditions p x ( l , t ) = p x ( − l , t ) = 0 . Since the Fokker- Planck Equation (18) is linear, the desired PDF can be constructed from solution (20) by the method of images [

p X ( x , t ) = lim N → ∞ ∑ n = − N N p X ( x , t , n ) = 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 (77)

in which the second line of (77) defines the function p X ( x , t , n ) .

A sense of the structure of solution (77) is provided by Figures 6-10. The figures depict the probability density of an unstable particle with diffusivity D = 1.0 cm 2 ⋅ s − 1 and decay rate λ = 2.0 × 10 − 5 s − 1 undergoing Brownian motion between absorbing barriers at l = 1 m to the left and right of the origin. In each figure the dashed black plot is the Gaussian PDF p X ( FP ) ( x , t ) , Equation (60), of the unconstrained particle at stated time t; the solid red plot is the specified linear superposition of functions p X ( x , t , n ) contributing to solution (77); the vertical dashed blue lines mark the boundaries, and the horizontal solid blue line marks the physical space along the axis within which the particle is required to be confined.

As time increases, the particle can diffuse to greater distances from the origin, and the probability density functions contributing to PDF (77) spread. The N = 1 solution that satisfied the boundary conditions at t = 4000 s , no longer satisfies these conditions at the later time t = 20000 s , as shown in

left boundary condition. In

as required. However, since time extends over an infinite range ( ∞ ≥ t ≥ 0 ) , the necessity for an infinite number of image functions in (77) becomes apparent.

The integral of PDF (77) over the physically allowed region defines the survival function

S ( t ) = ∫ − l l p X ( x , t ) d x = ∑ 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 (78)

which is the probability that the particle has not reached either boundary at time t. The probability S ( t ) that the particle remains in the interval [ − l , l ] after time t means that T > t , or, in terms of the definition (75) of cumulative probability, one has

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

It then follows from relation (74) that the expectation value T FPT is given by the integral of the survival function over time

T FPT = ∫ 0 ∞ S ( t ) d t = ( e l / ζ − 1 ) 2 λ ( e 2 l / ζ + 1 ) (80)

where ζ , the characteristic diffusion length, is defined by relation (46).

Decay Event | Time of Decay | Time Step to Reach Boundary ±15 | First-Passage Time To Boundary |
---|---|---|---|

1 | 3 | ― | ― |

2 | 22 | ― | ― |

3 | 63 | 158 | 59 |

4 | 99 | 203 | 40 |

5 | 163 | ― | ― |

6 | 244 | ― | ― |

7 | 254 | 450 | 122 |

8 | 328 | 633 | 137 |

9 | 496 | ― | ― |

10 | 696 | 788 | 33 |

11 | 755 | ― | ― |

Sample Mean FPT 78.2 Theoretical FPT 76.4 |

the instances of particle decay and the time step at which decay occurred. The third column shows the time step at which a particle first reached or exceeded the right ( l = + 15 ) or left ( l = − 15 ) absorbing boundary. A dash signifies that the particle decayed before reaching either boundary. The fourth column is the difference between columns 3 and 2, which gives the time interval measured from the point at which the particle was placed at the origin x = 0 . As shown, the mean of the five FPT values is very close to the theoretical prediction 76.4 calculated from Eq. (80) with the parameters used in the simulations shown in

Although the PDF p T ( t ) of the random variable T was not required to derive the mean first-passage time (80), it can be obtained, if needed, by differentiating the survival function

p T ( t ) = d F T ( t ) / d t = − d S ( t ) / d t . (81)

The resulting expression, however, is complicated and not needed in this paper.

The method employed in Section 3.1 yielded the CPF F T ( t ) (or, equivalently, the survival function S ( t ) ) from which all statistical properties of the FPT can be calculated. However, if all one wanted was T FPT , it can be derived directly from a master equation by a method that has been employed for stable particles, such as the diffusion of molecules of biological significance [

Consider an unstable particle in one-dimensional Brownian motion with time steps of δ t , on a lattice with coordinate spacing δ x and absorbing boundaries at x b = ± l . The probability to step either left or right is equal to 1/2, and the particle does not jump over any lattice points. If T ( x ) is the time to reach either boundary from point x, it must satisfy the following discrete equation

T ( x ) = δ t + 1 2 ( T ( x + δ x ) + T ( x − δ x ) ) ( 1 − λ δ t ) (82)

because: a) point x can be reached only from points x + δ x and x − δ x , b) the probability of a transition from these two points is the same (1/2), and c) the transition to x during interval δ t can be made only if the particle has survived the transition. Recall from Equation (27) that the survival probably per step is p s = 1 − λ δ t .

Rearrangement of Equation (82) to the form

1 2 [ ( T ( x + δ x ) − T ( x ) ) − ( T ( x ) − T ( x − δ x ) ) ] = − δ t + λ δ t 1 2 [ T ( x + δ x ) + T ( x − δ x ) ] (83)

which is expressible as a second derivative

d 2 T ( x ) d x 2 ( δ x ) 2 = − 2 δ t ( 1 − λ T ( x ) ) , (84)

leads in the limit δ x → 0 and δ t → 0 to a screened Poisson equation, encountered in the physics of ionized gases [

D d 2 T ( x ) d x 2 − λ T ( x ) = − 1 (85)

in which the ratio ( δ x ) 2 / 2 δ t is again taken to be the diffusion coefficient D as defined in Equation (32). In the absence of the term proportional to λ , Equation (85) takes the standard form of Poisson’s equation [

Although a derivation will not be given here, one can also arrive in several steps at Equation (85) by integrating the backward Fokker-Planck equation

∂ p ( x , t | x 0 , t 0 ) ∂ t 0 = − D ∂ 2 p ( x , t | x 0 , t 0 ) ∂ x 0 2 + λ p ( x , t | x 0 , t 0 ) (86)

for which Equation (18) is the associated forward Fokker-Planck equation. It is important to note, however, that the coordinate x appearing in Equation (85) and in all expressions involving T ( x ) refers to the initial point from which the particle diffuses to a boundary, and therefore actually corresponds to coordinate x 0 in Equation (86). Where there is no confusion, it is standard notation to use the variable x rather than x 0 as the argument of the FPT Equation (85) and solution.

The solution to Equation (85) with implementation of boundary conditions T ( ± l ) = 0 is

T ( x ) = 1 λ ( 1 − cosh ( x / ζ ) cosh ( l / ζ ) ) . (87)

Comparison with solution (80) of Section 3.1 can be made by setting the initial location x = 0 , whereupon Equation (87) reduces to

T ( 0 ) = 1 λ ( 1 − 1 cosh ( l / ζ ) ) = ( e l / ζ − 1 ) 2 λ ( e 2 l / ζ + 1 ) = T FPT . (88)

Methods 1 and 2, although very different, lead to the same final result, as they must for consistency.

The Fokker-Planck Equation (18) incorporates particle instability by means of a decay term − λ p ( x , t | x 0 , t 0 ) that affects the resulting transition probability density (20) only through a global exponential factor e − λ t . Thus, letting λ approach 0 smoothly generates the probability density of the unconstrained stable particle. Likewise, the vanishing of λ in the survival function (78) smoothly generates the survival function of a stable particle confined between absorbing boundaries. This ostensible continuity between decay and stability gives a misleading impression of the effect of particle decay on the FPT problem.

To illustrate the radical change in outcome that can be engendered by the decay process, consider again by method 2 the FPT problem of a decaying particle in Brownian motion between two non-symmetrically placed absorbing boundaries b > a . The solution to Equation (85) with boundary conditions T ( a ) = T ( b ) = 0 is

T a b ( x ) = 1 λ [ ( e − a / ζ − e − b / ζ e ( a − b ) / ζ − e ( b − a ) / ζ ) e x / ζ + ( e b / ζ − e a / ζ e ( a − b ) / ζ − e ( b − a ) / ζ ) e − x / ζ + 1 ] . (89)

In the limit a → − ∞ , there is only a single boundary b on the right, and the FPT (89) reduces in this limit to

T b ( x ) = 1 λ ( 1 − e − λ / D ( b − x ) ) (90)

in which the λ dependence is explicitly shown. Note that T b ( x ) (90) is a finite quantity although the decaying particle can be located at any point in the infinite range to the left of b. In the limit that b → ∞ as well, the particle is free to walk the entire x-axis, and the FPT reduces to the statistical lifetime (4) λ − 1 as expected.

For a stable particle ( λ = 0 ) , however, the limit of the FPT (90) is infinite no matter how close to b the particle is located initially. This is evident from the leading term in the series expansion of T b ( x )

T b ( x ) ∼ b − x D λ − ( b − x ) 2 2 D + ( b − x ) 3 6 λ D 3 + ⋯ . (91)

The first term of (91), in which the denominator is the characteristic diffusion velocity [Ref [

lim λ → 0 T a b ( x ) → ( b − x ) ( x − a ) 2 D . (92)

For a particle initially located at x = 0 , the left boundary − a > 0 , and therefore lim λ → 0 T a b ( 0 ) → | a b | / 2 D .

Further insight is gained by examining the probability density of the FPT in the case of a single absorbing boundary at b > 0 and a particle initially located at 0. p T ( t ) is obtained by method 1 of the preceding section as applied to PDF (77) truncated to include only the component p X ( x , t , 0 ) , since a Gaussian centered on 0 and a negative Gaussian image centered on 2 b suffice to maintain p X ( b , t ) = 0 for all time t. The calculation yields

p T ( t ) = [ b exp ( − b 2 / 4 D t ) 4 π D t 3 + λ erf ( b 4 D t ) ] e − λ t . (93)

The first term in brackets, known as the Smirnov density [

〈 T 〉 | λ = 0 = ∫ 0 ∞ b t exp ( − b 2 / 4 D t ) 4 π D t 3 d t = ∞ . (94)

However, the Smirnov density multiplied by an exponential e − λ t yields a finite first moment

∫ 0 ∞ b t exp ( − b 2 / 4 D t ) e − λ t 4 π D t 3 d t = b e − b / ζ 4 D λ (95)

because the exponential function decreases faster than a power law over the infinite extent of the tail, as shown in

∫ 0 ∞ λ t erf ( b 4 D t ) e − λ t d t = 1 λ [ 1 + ( b + 2 ζ 2 ζ ) e − b / ζ ] ,

which, together with expression (95), sum to λ − 1 ( 1 − e − b / ζ ) , in agreement with Equation (90) for T b ( 0 ) obtained directly from the screened Poisson Equation (85).

This paper is concerned principally with effects of particle instability (decay) on Brownian motion. Therefore, to have included a frictional force in the Langevin Equation (LE) or a drift term in the Fokker-Planck Equation (FPE) would have unnecessarily complicated the problem, increased the length of the analysis, and detracted from the primary objective.

Nevertheless, the physical cause of fluctuations in Brownian motion is in fact ascribable to random impacts on the observed particle by collisions with other particles of the medium. In the diffusion of radon gas in air, for example, a particular radon atom (or a particular dust particle to which is attached a radioactive polonium ion from a radon decay) is buffeted by impacts from ambient oxygen and nitrogen molecules. In keeping with the fluctuation-dissipation theorem [

^{1}Mobility μ p is the proportionality factor in V = μ p F relating velocity V and dissipative force F.

Since neglect of friction in the analysis of Brownian motion is the essence of the approach taken by Einstein [Ref. [^{1}, is large enough that the displacement x ( t + τ ) is independent of the displacement x ( t ) , but small compared to the time interval Δ t between observations [

Δ t > m D / k B T e (96)

in which D is the particle diffusion coefficient, k B is Boltzmann’s constant, and T e is the equilibrium temperature. For a radon-222 atom diffusing at room temperature T e ≈ 300 K , Equation (96) becomes Δ t > 1 ns , which is readily satisfied in most experiments or measurement protocols.

Another issue that also has arisen in the past is the validity of using Fick’s law (14) to model diffusion. Serber [

λ − 1 ≫ 4 m D / 3 k B T e . (97)

Using radon-222 at room temperature as an example, one has λ Rn-222 − 1 ≈ 4.8 × 10 5 s ≫ 1.3 ns , which is very well satisfied. In fact, relation (97) is well satisfied even for the short-lived progeny of radon-222, such as polonium- 214 for which λ Po-214 − 1 ≈ 3.4 × 10 − 4 s ≫ 1.3 ns One can conclude, therefore, that the stochastic model in this paper is valid for the Brownian motion of radioactive atoms in air probed at time intervals in excess of a few nanoseconds. Moreover, Fick’s law as applied to radon diffusion has been confirmed experimentally [

Motivated by new experimental methods to measure radioactive atoms and ions diffusing in gas, on dust, or in liquids, this paper analyzed the Brownian motion of decaying particles, a process that has received relatively little prior attention. In particular, equations to be interpreted as the Fokker-Planck and Langevin equations of an unstable particle were derived and solved. Also, the equations of time of first passage of unstable particles to absorbing boundaries were derived and solved.

The phenomenon of particle decay introduces into Brownian motion an additional time parameter (the decay rate λ or statistical lifetime λ − 1 ) that leads to marked differences in the analysis of unstable, compared to stable, diffusing particles. Whereas Brownian motion of a stable particle is a continuous Markov process that can in principle be followed for an infinite length of time, the process terminates abruptly at the decay of the unstable particle. A mathematical consequence of this discontinuity is reflected in the different analytical content of the Fokker-Planck Equation (FPE), which yields a transition probability density p ( x , t | x 0 , t 0 ) , and the Langevin Equation (LE), which yields the distribution function and trajectories of the corresponding process variable X ( t ) . For stable particles, the FPE and LE both describe a continuous Wiener process and provide entirely equivalent information. For decaying particles, however, the former (FPE) describes the probability of displacement of a single particle throughout the course of its existence, which terminates with 100% probability only after an infinite time interval. In contrast, the latter (LE) gives an ensemble statistical description of the discontinuous trajectories of numerous sequential particles that have undergone Brownian motion up to the instant of their actual, randomly occurring decays. The two approaches are both valid, but provide different, and differently interpreted, statistical results relating to means, variances, and higher moments.

In regard to the statistical description of the stochastic process, there is a critical difference in mathematical structure of the LE and its solution for an unstable particle compared to the LE and solution of a stable particle. Assuming that Brownian motion of a stable particle is modeled by a frictionless Wiener process with constant diffusivity D, the resulting solution is a normal distribution of zero mean and variance 2 D t . However, the LE for the decaying particle entails a mixture of Wiener processes for displacement and Bernoulli processes for decay and leads to a solution in the form of a normal distribution of zero mean but with a variance that is itself a random variable. This random variable, Σ n 2 , although not among any known to the author, depends on the number of time steps n and survival probability p s per step and is completely characterized by its moment-generating function. It is shown in Appendix 1 that in the limit of an infinite number of time steps, Σ ∞ 2 behaves increasingly like an exponential distribution as p s approaches unity.

Another fundamental distinction in the Brownian motion of unstable, compared to stable, particles concerns the first-passage time (FPT) to absorbing boundaries. The FPT of a stable particle is infinite irrespective of how close the initial position is to a single absorbing boundary. Mathematically, this is a consequence of the heavy tail―or asymptotic power law behavior―of the associated probability density (Smirnov density). In contrast, the FPT of an unstable particle is finite because the exponential decrease in time of the probability density is faster than that of a power law. Indeed, even if the unstable particle is free to walk the entire real axis, the FPT to reach − ∞ or + ∞ is finite and equal to the statistical lifetime λ − 1 of the particle.

Silverman, M.P. (2017) Brownian Motion of Decaying Particles: Transition Probability, Computer Simulation, and First-Passage Times. Journal of Modern Physics, 8, 1809-1849. https://doi.org/10.4236/jmp.2017.811108

Random variable Σ n 2 is defined in Equation (62) as a sum of products of Bernoulli random variables ε j ( j = 1 , ⋯ , n ) for a discrete Brownian motion of n time steps. The moment generating function (MGF) is defined by the expectation value

g Σ n 2 ( θ ) ≡ 〈 exp ( Σ n 2 θ ) 〉 = ∑ { ε } ( ∏ j = 1 n e θ ( ε j ⋯ ε n ) ∏ j = 1 n π j ) = ∑ { ε } ( e θ ( ε 1 ⋯ ε n ) e θ ( ε 2 ⋯ ε n ) e θ ( ε 3 ⋯ ε n ) ⋯ e θ ε n ) π 1 π 2 ⋯ π n (98)

in which π j is the probability of the j th Bernoulli outcome

π j = { p if ε j = 1 q = 1 − p if ε j = 0 (99)

and the sum represented by { ε } is over all possible outcomes of the Bernoulli variables.

An illustration of the simple case n = 3 can give a sense of the content of Equation (98). All possible outcomes with corresponding expectations are given in

Summing the results of

g Σ 3 2 ( θ ) = p 3 e 3 θ + p 2 ( 1 − p ) e 2 θ + p ( 1 − p ) e θ + ( 1 − p ) . (100)

Proceeding in the same way for n = 4 leads to g Σ 4 2 ( θ )

g Σ 4 2 ( θ ) = p 4 e 4 θ + p 3 ( 1 − p ) e 3 θ + p 2 ( 1 − p ) e 2 θ + p ( 1 − p ) e θ + ( 1 − p ) . (101)

The above pattern holds for arbitrary n, whereupon Equation (98) reduces to the form

g Σ n 2 ( θ ) = p n e n θ + ( 1 − p ) ∑ k = 1 n − 1 ( p k e k θ ) + ( 1 − p ) , (102)

which, upon summing the geometric series in Equation (102) yields the compact

ε 1 | ε 2 | ε 3 | Expectation |
---|---|---|---|

1 | 1 | 1 | p 3 e 3 θ |

0 | 1 | 1 | p 2 q e 2 θ |

1 | 0 | 1 | p 2 q e θ |

1 | 1 | 0 | p 2 q |

0 | 0 | 1 | p q 2 e θ |

0 | 1 | 0 | p q 2 |

1 | 0 | 0 | p q 2 |

0 | 0 | 0 | q 3 |

expression

g Σ n 2 ( θ ) = ( 1 − p ) + p n + 1 e n θ ( 1 − e θ ) 1 − p e θ . (103)

In the analyses of Sections 2.3 and 2.4 the survival probability p was eventually replaced by the expression (27), leading to the limit p s n = ( 1 − λ t / n ) n → n → ∞ e − λ t . However, if in Equation (103) p is taken to be a fixed finite parameter satisfying the requirement of a probability 1 ≥ p ≥ 0 , then p n → 0 for n → ∞ . In that limit, MGF (103) reduces to the approximate form

lim n → ∞ g Σ n 2 ( θ ) = ( 1 − p ) 1 − p e θ (104)

which can be approximated further

lim n → ∞ θ ≪ 1 g Σ n 2 ( θ ) = ( 1 − p ) 1 − p ( 1 + θ ) = 1 1 − ( p / q ) θ (105)

by expansion to first order of the exponential about θ = 0 . (Note: θ is always set to 0 after differentiation of the MGF.) Expression (105) is the MGF of the exponential distribution E ( p / q ) [Ref. [

A test of this relation is shown in

and σ ( p ) (dashed blue), given by the exact expressions (70) derivable from the MGF (103), over the full range of p for n = 1000 . The equality is very close at the scale of the figure, but the higher-resolution insert shows that σ ( p ) exceeds μ ( p ) by a small amount that decreases as p approaches 1. The theoretical ratio μ ( p ) / σ ( p ) (dotted black), equals p (dashed green) in the limit n ≫ 1 . As a sum of correlated products of Bernoulli random variables, the compound random variable Σ n 2 does not lend itself to an obvious physical interpretation. Nevertheless, its approximate exponential distribution in the asymptotic limit p → 1 (or q = ( 1 − p ) → 0 ) might be understood in the following way. A negative exponential distribution characterizes the time intervals between events in a Poisson process. The Poisson distribution itself, however, evolves from a binomial distribution under the circumstances that the number of samples n ≫ 1 and the probability of an event (e.g. the event of particle decay in the present context) q ≪ 1 such that n q approaches a constant mean number of events. Given that a Bernoulli random variable is a special case of binomial random variable, the asymptotic limit at which Σ n 2 is well-represented by a negative exponential distribution is precisely the condition for a Poisson process.

The point at issue is whether diffusion theory based on Fick’s law (14) is valid for a particle with a finite statistical lifetime. Serber [Ref. [

In terms of notation used in this paper, the criterion derived by Serber for the validity of Fick’s law is the following

1 3 l | d j ( x ) d x | ≪ 1 2 j ( x ) (106)

^{2}Jeans, J. (1954) The Dynamical Theory of Gases. 4^{th} Ed., Dover Publications, New York, 307-310.

^{3}Reif, F. (1965) Fundamentals of Statistical and Thermal Physics. McGraw-Hill, Boston, 483-486.

in which j ( x ) is the particle flux to the right or to the left along the x-axis, and l is the mean free path between collisions with particles of the medium. The diffusion coefficient D, derivable from the elementary kinetic theory of gases to an approximation adequate for the purpose of this appendix, is^{2}^{,}^{3}

D = 1 3 l v rms (107)

in which the root-mean-square speed of particles of mass m in equilibrium at temperature T e is

v rms = 3 k B T e / m (108)

where k B is Boltzmann’s constant. The self-diffusion coefficient (107) follows in the simple case of a dilute gas by calculation of the net flux j ( x ) = v rms [ n ( x − l ) − n ( x + l ) ] / 6 of particles originating from a distance l right and left of a plane perpendicular to the flow. n ( x ) is the number density of particles; the factor 1/6 arises because on average 1/3 of the particles move along the x-axis, and, of these, 1/2 move in the positive (negative) direction. Expansion in a Taylor series to first order in l then leads to

j x ≈ − 1 3 v rms l ( d n / d x ) ≡ − D ( d n / d x ) (109)

which defines the diffusion coefficient D.

From the macroscopic theory of diffusion of radioactive particles [Ref. [

| 1 j ( x ) d j ( x ) d x | = D λ ≡ ζ (110)

where ζ is the characteristic diffusion length. Combining relations (106) through (110) leads to the succinct expression for the validity of Fick’s law applied to radioactive particles

2 v d ≪ v rms (111)

in which

v d ≡ D λ (112)

is the characteristic diffusion velocity. Substitution of relations (112) and (108) into (111) yields λ − 1 ≫ 4 m D / 3 k B T , which is the inequality (97).