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Mean square displacement and instantaneous
diffusion coefficient of charged particles in
stochastic motion
Gabriela Raluca Mocanu∗
keywords radiation: dynamics– methods: numerical
Abstract The mean square displacement and instantaneous diffusion coef- ficient for different configurations of charged particles in stochastic motion are calculated by numerically solving the associated equations of motion. The method is suitable for obtaining accurate descrip- tions of diffusion in both intermediate and long time regimes. It is also appropriate for studying a variety of astrophysical configurations since it may incorporate microscopic physics that analytical methods cannot cope with. The results show that, in the intermediary time regime, the diffusion coefficient has an irregular behavior, which can be described in terms of the complex interplay appearing between the physical parameters describing the configuration. The main conclu- sion is that such an approach may serve at differential diagnosis of different astrophysical configurations.
1 Introduction
In an astrophysical context, there is overwhelming observational evidence that the plasma emitting the recorded radiation is influenced by some stochas- tic component in its medium (for discussion of observational data and meth- ods that lead to this conclusion for, e.g., accretion disks around supermassive ∗Astronomical Observatory Cluj, Romanian Academy, Cluj-Napoca Branch, 19 Cire- silor Street, 400487, Cluj-Napoca, Romania, Email: [email protected]
arXiv:1906.10402v1 [cond-mat.stat-mech] 25 Jun 2019
black holes see Azarnia et al. (2005); Carini et al. (2011); Leung et al. (2011); Harko et al. (2014)). As this plasma moves, it emits radiation, which is our only source of information for diagnosis of the emitting system; we find it appropriate to describe this complicated framework by instantaneous quan- tities, such as the instantaneous diffusion coefficient. This approach takes into account all the regimes of the system evolution; the transitory regime is worth studying as it may be used to model and diagnose explosive astro- physical events. It is thus the purpose of this paper to use the advancements in numer- ical computation to study the instantaneous diffusion coefficient by solving the stochastic differential equation of motion, or sometimes called the A- Langevin equations, associated to the motion of the particles in the plasma. The trajectory x(t) of a particle undergoing a random motion is a solution of the stochastic differential equation (SDE)
d^2 x(t) dt^2
= adet + astoch,
where adet is the deterministic contribution to the equation of motion and astoch is the stochastic contribution to the equation of motion, such as ran- dom collisions causing momentum variations. The mean square displacement (msd) 〈x^2 (t)〉 exhibited by an ensemble of such particles can be numerically calculated by solving the associated equation of motion. The retrieval of the msd is necessary because the (macroscopic) instantaneous diffusion coeffi- cient is defined as (B˘alescu et al. , 1994, Eq. 6)
D(t) =
∂t〈x^2 (t)〉, (1)
with the equilibrium diffusion coefficient defined as
D = lim t→∞ D(t).
In the numerical treatment the motion will be split into a mean and a fluctuating part x(t) = ¯x(t) + δx(t), with δx(0) = δv(0) = 0. From hereon, the diffusion coefficient will thus study the diffusion with respect to a mean trajectory ¯x(t), where this mean trajectory is the solution to the deterministic equation of motion. The diffusion coefficient will obviously inherit the properties of both types of accelerations appearing in the equation of motion; these properties are in
2 Possible settings for random motion
2.1 Stochastic motion in the presence of an electric
field
The simplest possible stochastic motion of a charged particle with Z = 1 and mass m is the one-dimensional Brownian motion in the presence of an external electric field, E~ 6 = 0. In the following we restrict or analysis to the case of the constant electric field, E~ = constant. The random motion of the particle is described by the Langevin equation
d^2 x dt^2
= eE − ν
dx dt
where ξA is a random acceleration with properties defined through the en- semble averages (B˘alescu, 1997),
〈ξA(t)〉 = 0, 〈ξA (t 1 ) ξA (t 2 )〉 = Aδ (t 1 − t 2 ). (3) The physical quantities are split into their mean and fluctuating parts, and thus we obtain the SDE for δx as d^2 δx dt^2
= −ν
dδx dt
The differential equation is brought to a dimensionless form by the fol- lowing transformations
- dimensionless time: θ = νt; ν = 1/τ , where ν is the collision frequency in Brownian Motion;
- dimensionless fluctuation in displacement: q = δx (Aτ 3 )−^1 /^2. The dimensionless equation describing the motion of a Brownian particle in a constant electric field is
d^2 q(θ) dθ^2
dq dθ
(θ) + ξ¯A(θ), (5)
where (^) 〈 ξ¯A(θ)〉^ = 0, 〈^ ξ¯A (θ 1 ) ξ¯A (θ 2 )〉^ = Aδ¯ (θ 1 − θ 2 ). (6) Note that although the constant electric field is important to the overall energy emission (Harko & Mocanu, 2016), it does not change the characteris- tics of the fluctuating component of the trajectory and thus it will not affect the diffusion. The free parameter set in this case is given by { A¯}.
2.2 Stochastic motion in a harmonic external potential
The Langevin equation for the one dimensional motion of a charged particle with mass m and charge e in a harmonic potential with natural frequency ω 0 is given by d^2 x dt^2
dx dt
where the stochastic acceleration ξB has the properties
〈ξB (t)〉 = 0, 〈ξB (t 1 ) ξB (t 2 )〉 = Bδ (t 1 − t 2 ). (8)
Thus the fluctuating part of the trajectory will obey the SDE
d^2 δx dt^2
dδx dt
The same dimensionless variables are again used, together with
- dimensionless frequency: W = ω 0 τ.
The dimensionless form of the Langevin differential equation (9) becomes
d^2 q(θ) dθ^2
dq(θ) dθ
where (^) 〈 ξ¯B (θ)
ξ¯B (θ 1 ) ξ¯B (θ 2 )
= Bδ¯ (θ 1 − θ 2 ). (11) In this case, the characteristics of the outside medium (through W 2 ) do influence the msd and thus the diffusion coefficient. The free parameter set in this case is given by {W 2 , B¯}.
2.3 Stochastic motion described by the generalized Langevin
equation
In the presence of a non-trivially correlated noise and of a frictional force showing retarded effects, the motion of a charged particle in a harmonic potential is described by the generalized Langevin equation, which in the one-dimensional case can be written as (see, e.g., Harko & Mocanu (2016))
d^2 x dt^2
∫ (^) t
0
γ(t − t′)
dx (t′) dt′^
dt′^ + ω^20 x = ξC (t), (12)
Table 1: Parameter space tested by the simulations. In the first column the case is specified, in the second the parameter space tested in the simulations and the third column gives the total number of different settings covered by the simulations. Case Parameter space No. (A) A¯ ∈ { 0. 01 , 1 , 100 } 3 (B) B¯ ∈ { 0. 01 , 1 , 100 }, W 2 ∈ { 0. 01 , 0. 05 , 0. 1 , 0. 5 , 1 , 5 , 10 , 20 }
3 × 8
(C) C¯ ∈ { 0. 01 , 1 , 100 }, W 2 ∈
α¯ ∈ { 0. 01 , 0. 05 , 0. 1 , 0. 5 , 1 , 5 , 10 , 20 }
3 × 8 ×
(D) D¯ ∈ { 0. 01 , 1 , 100 }, Ω¯ ∈
3 × 8
- dimensionless magnetic frequency: Ω = Ω¯ τ = B (^0) mcZe τ.
Equation (17) is split into components and afterwards made dimensionless as d^2 X dθ^2
dY dθ
dX dθ
d^2 Y dθ^2
dX dθ
dY dθ
d^2 Z dθ^2
dZ dθ
where (^) 〈 ξ¯i (θ) ξ¯j (θ′)
= Dδ¯ ij (θ − θ′). (23) The free parameter set in this case is given by { Ω¯, D¯}.
3 Results
The stochastic differential equations (5)-(6), (10)-(11), (15)-(16), (20)-(23) were solved for q(θ), subsequently used to produce the time evolution of the mean square displacement and the instantaneous diffusion coefficient. Simu- lations were run for 10^5 realizations, 10^3 timesteps each, within an extended parameter set (Table 1).
timestep
MSD
A = 100
A = 1
A =0.
Figure 1: Mean square displacement of the charged particle undergoing Brow- nian Motion in a constant electric field (A), for various values of the noise amplitude A¯; for presentation purposes the curves were multiplied by 10^7 for the A¯ = 0.01 curve and by a factor of 5000 for the A¯ = 1 curve.
timestep
D
A = 100
A = 1
A =0.
Figure 2: Diffusion coefficient for case (A) (msd shown in Fig. 1); for pre- sentation purposes the curves were multiplied by 10^7 for the A¯ = 0.01 curve and by a factor of 5000 for the A¯ = 1 curve.
Due to the large number of resulting curves, we have grouped the most important results in Figures 1 and 2 for the electron in a constant electric
timestep
D
W^2 = 20
W^2 = 10
W^2 = 5
W^2 = 1
W^2 =0.
W^2 =0.
W^2 =0.
W^2 =0.
Figure 4: Diffusion coefficient for case (B) (msd shown in Fig. 3), for B¯ = 100.
For case (B) with B¯ = 100, the msd and diffusion coefficient are shown in Figures 3 and 4 respectively. An extra insight on the behavior of the system as the parameters are varied may be obtained by representing the maximum msd reached during the time evolution, as a function of W 2 ; this is done for three different noise amplitudes (Figure 5). For the dynamics of the diffusion and its potential applications, it is of great interest to see what is the maximum transitory value of the diffusion coefficient and at which timestep this appears; this is shown in Figure 6.
W^2
Maximum MSD
B = 100
B = 1
B =0.
Figure 5: Maximum MSD as a function of W 2 for (B), for various values of the noise amplitude B¯; for presentation purposes the curves were multiplied by 1. 5 × 107 for the B¯ = 0.01 case and by a factor of 1. 5 × 5000 for the B¯ = 1 case.
timestep
MSD
W^2 = 5
Α= 20
Α= 10
Α= 5
Α= 1
Α=0.
Α=0.
Α=0.
Α=0.
Figure 8: Mean square displacement for case (C), with W 2 = 5, C¯ = 100 and variable ¯α.
Due to the extended parameter set, it is not feasible to present all the curves obtained for case (C). The general principles resulting from the anal- ysis of the full set of curves will be discussed in the following section. Here we provide plots of some representative results. The msd for variable ¯α, with C¯ = 100 is shown in Figure 7 for W 2 = 0.05 and in Figure 8 for W 2 = 5. The diffusion coefficient for variable W 2 , with C¯ = 100 is shown in Figure 9 for ¯α = 1 and Figure 10 for ¯α = 10. The effect of varying ¯α and W 2 may be seen by analysis of changes in the maximum diffusion coefficient and the timestep at which this maximum occurs (Figures 11 and 12).
timestep
D
Α = 1
W^2 = 20
W^2 = 10
W^2 = 5
W^2 = 1
W^2 =0.
W^2 =0.
W^2 =0.
W^2 =0.
Figure 9: Diffusion coefficient for case (C), with ¯α = 1, C¯ = 100 and variable W 2.
timestep
D
Α = 10
W^2 = 20
W^2 = 10
W^2 = 5
W^2 = 1
W^2 =0.
W^2 =0.
W^2 =0.
W^2 =0.
Figure 10: Diffusion coefficient for case (C), with ¯α = 10, C¯ = 100 and variable W 2.
timestep
MSD
xyz
W= 20
W= 10
W= 5
W= 1
W=0.
W=0.
W=0.
W=0.
Figure 13: Mean square displacement of the charged particle undergoing stochastic motion in a constant magnetic field (D), for D¯ = 100.
timestep
D
xyz
W= 20
W= 10
W= 5
W= 1
W=0.
W=0.
W=0.
W=0.
Figure 14: Diffusion coefficient for case (D) (msd shown in Fig. 13).
4 Discussion
The present paper introduces a general approach to calculating instantaneous diffusion coefficients for some particular configurations; the directing idea is that diffusion coefficients are easily calculated numerically if the equation of motion of the charged particle is properly set up. This approach bypasses the usual difficulties appearing in the analytical calculation of diffusion coef- ficients. The diffusion of a particle in a given medium is connected to the mean square of the distance a random walker starting at x 0 reaches in n steps (see e.g. (Mahnke et al., 2009, Chapter 6)), 〈x^2 (n)〉x 0 , i.e.: the charged par- ticle is a random walker, undergoing an infinite number of walks, with the same initial conditions x 0 , and in identical settings. If this walker is com- pletely unconstrained and freely (and randomly) chooses his next step, than 〈x^2 (n)〉x 0 ∼ n. However, if the medium in which the walk occurs somehow biases the walk, say by increasing the probability that the walker chooses one direc- tion over the other, the quantity 〈x^2 (n)〉x 0 will no longer exhibit a simple behaviour (see, e.g, Klafter & Sokolov (2011)). In some of the cases presented in this paper, the mean square displace- ments depart from the simple ∼ n law, and we infer that the background physics is set up such that the walker is biased. Even more, since the diffu- sion coefficient is the first derivative of the mean square displacement with respect to time, it will also depart from its theoretical value. Let us analyse each case in turn, recalling that in our dimensionless ap- proach, q ∼ δx of x = ¯x+δx, i.e., we are analysing departures from the mean trajectory, not the entire trajectory.
4.1 Mean square displacement
For a charged particle undergoing stochastic motion in a constant electric field, (A), the stochastic part of the motion does not couple to the electric field, the friction is unbiased and thus we obtained the expected result that 〈q^2 (n)〉 ∼ n (Figure 1). For a charged particle undergoing stochastic motion in an external har- monic potential, characterised by a dimensionless frequency W , our results show that the quantity 〈q^2 (n)〉 roughly follows the general description (Fig. 3)
is not affected by the presence of the field, such that 〈Z^2 (n)〉 ∼ n; in the perpendicular plane the particle becomes trapped rather fast and is not al- lowed to diffuse beyond a radius imposed by the magnetic field, thus making a constant contribution to 〈q^2 (n)〉 0.
4.2 Diffusion coefficient
The instantaneous diffusion coefficient is the derivative with respect to time of the msd. In the long time limit this derivative is expected to be constant. This is recovered for cases (A) and (D), as expected based on the previous discussion. For the other cases, the long time limit also produces a con- stant; however, in the intermediate regime the diffusion coefficient varies in a manner worth investigating. For the particle in a harmonic potential, one can see in Figure 4 that for intermediate times the diffusion coefficient has a peculiar behavior, most notably, for certain W 2 , it surpasses the value of the long time limit by one order of magnitude. In view of using this in astrophysical applications, in which large distances may be reached due to low collision rate, this interme- diate time regime may prove to be very important, especially if the diffusion coefficient is larger than its expected value. Figures 9 - 12 give an indication of the complex situation appearing in case (C). The long time limit approaches a constant. But in this case as well the intermediary regime shows an interesting behavior. Fortunately, from both a qualitative and a quantitative point of view, the behavior of the dif- fusion coefficient depends clearly on which combination of parameters was used (as curves for different configurations do not superimpose) and as such it may serve for both diagnosis and prognosis. Both the maximum value of the diffusion coefficient and the timestep at which this maximum appears are decreasing functions of W 2 ; so, as expected, a stronger external potential is more efficient at trapping the particle. For constant W 2 , the maximum diffu- sion coefficient and the timestep at which it appears are decreasing functions of ¯α; as expected, a larger friction coefficient is more efficient at reducing diffusion.
4.3 General conclusions
It is worth the effort to perform numerical simulations for the diffusion coeffi- cient in specific astrophysical configurations, as they sometimes depart from
analytical results or that these results do not even exist. Although there are ways to analytically tackle this problem with the assumption of a stationary regime, more often than not, interesting astro- physical phenomena are transitory (high energy astrophysics); working only with a stationary system makes diagnosis and prognosis difficult. The approach presented in this paper does not aim to be exhaustive, but shows that, based on the diffusion properties manifested by charged particles in stochastic motion, differential diagnosis on the physics of different astrophysical configurations may be performed. Acknowledgements This work was supported by a grant of the Ro- manian Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-PD-2016-0215, within PNCDI III.
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