boundary problem

continues

continues

injection at shock

The motion of individual particles in diffusion models has always been viewed as a random walk since the beginning of theoretical efforts (e.g. Parker, 1965). However, it is only a recent development that the CR transport equation can be reformulated with stochastic differential equations (Zhang, 1999b). In this approach the guiding center position of the particle and its momentum (energy) follow a set of Ito stochastic differential equations

a where wa(t) is a Wiener process (see below) and the sum of o runs over all required independent random noises. The probability density for the particle in the Markov process determined by Eq. 2.25.2 to appear in a unit volume at a particular location in q-space at time t, P((, q), follows the same Fokker- Planck diffusion equation as Eq. 2.25.1 (Zhang, 1999b) if we let apv = 'Lau,aav,a, fiju=fiju + (1/2)2da^v/dqv , (2.25.3)

a v and let the process be created at an exponential rate of c as a function of time, i.e. d (ln P )/dt = c. The probability density in q-space can be made proportional to the CR flux or distribution function. If the probability density starts with a S-function initially, i.e., the stochastic process starts from a single location point, the solution is the Green's function to the diffusion Eq. 2.25.1 (thus, the Green's function is often called the transition probability density or propagator). Therefore the stochastic differential Eq. 2.25.2 with an additional creation term can be used to describe diffusion.

Zhang (1999b) applied the Ito stochastic differential equation to studies of modulation, and the results from Monte Carlo simulation of the stochastic process completely agree with those by directly solving the diffusion equation. One obvious advantage of using the stochastic process approach is that it can reveal the physics of particle diffusion in more detail. For example, we can investigate the trajectory of simulated particles traveling through heliospheric or interstellar magnetic fields and when an ensemble of particles is simulated, we can find the distributions of source particles in terms of entry location at the boundary, initial momentum and propagation time (which is approximately proportional to path length). The path length distribution is particularly useful for studies of nuclear fragmentation during interstellar propagation.

2.25.3. Path integral representation for the transition probability of Markov processes

For simplicity Zhang (1999a) considers only a 1-dimensional stochastic diffusion process governed by an Ito stochastic differential equation dq = /3(t, q)dt + a(t, q)dw(t), (2.25.4)

with a creation rate c(t, q). The Wiener process has an associated probability for the process w(t) to transit from wo at time to to an interval w1 < w(t1 )< w1 + dw1 at t1

The Wiener process can be understood as the simplest diffusion with a coefficient of 1/2 and no convection. To calculate the transition probability density for the process described by Eq. 2.25.4 to get from qo at time to to q at t, we normally divide the time interval {to,t} into N small segments {to,ti,t2,...t^-i,tn}, where tN = t. This method is often called discretization. The probability for the process to go through a path

{qo,qi < q(i)< qi + dqi,q2 < q(h)< q2 + dq2, qN < q(tN)< qN + dqN},(2.25.6)

during which the driving Wiener process goes through

{wo,wj < )< wi + dwi,W2 < w(t2)< W2 + dw2, wN < w(tN)< wN + dwN},(2.25.7)

^exp

2A ti

where Atj = tj _ tjand Aw;- = wt _ Wj. When Atj ^ 0, Aw;- must be 0\y]Atl in order to have non-vanishing probability. The transition probability density from the initial point (to, qo) and the end point ((, q) can be obtained by integrating all the intermediate points, wj,w2, w^_i. However, the probability density, as obtained directly from Eq. 2.25.8, is for the w-space. To calculate the probability density in the q-space, we need to find the Jacobean for the transformation to q-coordinates, which can be obtained by finite expansion of the Ito stochastic differential Eq. 2.25.4 to the 6th order (Langouche et al., 1982). Replacing also the argument wy in the exponential of Eq. 2.25.8, we obtain a path integral representation for the transition probability density:

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