Stochastic acceleration of charged particle in nonlinear wave fields

He Kaifen

Inst. Low Ener. Nucl. Phys.

Beijing Normal Univ., Beijing 100875, China

kfhe@bnu.edu.cn

Stochastic acceleration

Stochastic acceleration is an interesting problem in many research areas;Fermi acceleration is an idea proposed to explain the origin of high speed particles in cosmic rays;In Laser Wake Field Acceleration, an electron can ‘ride’ on the wave if it moves to a correct phase and be accelerated.

Fermi’s acceleration

Stochastic acceleration is an idea that Fermi put forward to explain high-speed particles in cosmic rays. Charged particles collide with magnetic clouds that move chaotically in the interstellar space. The particle will more frequently collide with bodies moving towards it, so it will more often gain energy than loss it. Therefore they are accelerated, on the average.

Stochastic acceleration and heating

For a certain symmetry of the problem, the particle velocity remains unchanged on the average, only the mean square of the velocity, i.e., the particle energy increases .In this case we are dealing with stochastic heating, not acceleration.So we have to answer the question as why the stochastically moving of particle shows a prior direction.

Intrinsic stochasticity arising from nonlinearity

Stochastical acceleration and heating often occur even when there is no random force.As well known, stochasticity may arise from nonlinearity of a system, caused by various instabilities (bifurcations). Some authors study the effect of intrinsic nonlinearity on the particle motion.

Effect of nonlinearity

Fermi (Ulam?) considered a particle bouncing between two walls, one is fixed, the other oscillates periodically.If the colliding position is assumed to depend on the particle phase nonlinearly, it is found that the particle can gain energy from the oscillating wall. However, the particle moves up and down, it is not accelerated directionally.

Ulam’s model:Particle bouncing dbetween two plates,one is oscillating periodically

Nonlinear wave field

The aim of the present work is to test the possibility of particle acceleration by nonlinear wave field.A nonlinear wave has certain group velocity, can a particle gain the velocity by wave-particle interaction? Nonlinear waves may display different types of patterns.Can a particle be accelerated by any type of wave fields? Or only by certain type of wave?Does particle behavior depend on the wave dynamics?

Newtonian mechanics

1-d dimensional Newtonian equation of a single particle:

The wave solutions solved from a nonlinear wave equation are adopted as the fields in which the particle moves.

x

txq

dt

dv

vdt

dx

),(

Nonlinear wave equation

A nonlinear drift wave equation is used to produce a wave field.

Pseudospectral method is used to solve the equation.

Different types of wave patterns can be observed depending on parameter regimes of .

)sin(

2

3

txx

fx

cxt

at

),(

Spatially regular and spatiotemporal chaotic wave

A spatially regular (SR) but temporally non-steady wave solution and spatiotemporally chaotic (STC) solution are used as the wave field. Can a particle be accelerated in such nonlinear wave field? What is the characteristic motion of a particle in SR and STC fields respectively?

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A spatial regular and temporal chaotic wave

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X

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A spatiotemporal wave

t

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-1.075

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Spatiotemporally chaotic wave

k-spectrum :

Exponential law in SR wave;

Power law in STC wave.

Steady wave and its instabilities

The system has steady wave (SW) solution, in the form of , which has group velocity :

In the present work it is unstable under perturbation wave (PW) due to saddle instability.

The above SR and STC wave patterns are developed from the saddle SW respectively.

)(0 tx

),( tx

Particle motion in spatially regular nonlinear wave

In a SR wave field, the averaged velocity of the particle is about the same as the unstable SW, i.e.

the particle is finally trapped into one wave trough.The particle makes cyclic motion in the reference frame following the SW.

0.65 6475.0v

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t

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Particle motion in a SR wave field of temporally periodic

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v

x-t

7.021014E-01 (7.611827E-01)

Phase plot of particle in the frame with speed

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Particle motion in a spatially regular and time chaotic field

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x-t

6.910985E-01 (7.297304E-01)

Phase plot in the frame of velocity

Particle orbit trapped in wave trough

In the frame following the SW, one can see clearly that the particle orbit is trapped into a trough of the wave.The triangles give the unstable SW solution, bullets give the realized wave solutions at several instants.

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),- 0(

)

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Particle motion in the wave trough

Particle motion in spatiotemporally chaotic field

In STC field the particle displays trapped phase and free phase, it can be kicked out of wave trough frequently.In STC the averaged particle velocity can be larger than the group velocity of SW:

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Particle motion in the STC field

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8.988767E-01 (6.227521E-01)

Phase plot of particle in the frame of speed

Effect of particle charge sign

Averaged velocity of the accelerated particle depends on the charge sign.In the SR case, , it has changes little when the charge sign is changed due to its trapped orbit.In the STC, or depending on the charge sign, it can be much larger or smaller than the SW velocity.However, in lab frame the acceleration direction is always along the wave direction.

v

v v

Particle positions in the STC field

In the STC in the moving frame the evolution of particle position displays jumping-phases and random oscillation-phases, corresponding to free and trapped phase of the particle respectively.Forward jumpings are more than backward ones, so statistically the particle moves forward.This feature is very similar to that of motor protein. This comparison helps us to understand the acceleration mechanism.

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Evolution of particle position in moving frame

T/s

d/nm

Experiment data of motion of motor protein [ 《 Nature 》 v233,533 (19771)]

Particle motion in the unstable steady wave

Unstable is a solitary wave-like solution. It can not be realized due to saddle instability, so it is a virtual coherent wave packet.If formally taking the solution as a field, particle can NOT be accelerated by such field, .So itself can not accelerate particles, the PW plays negligible effect in the acceleration.

0v

)(0 ),(

)(0 tx

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5.0x10-7

v

t

B

Particle motion in (unstable) SW )(0 tx

What can we learn from motor protein?

For motor protein people studies how biological organism gains directional velocity from thermal energy. Langevin equation is solved by assuming, e.g. asymmetric potentials.The second law forbids the particle from any net drift speed, however, it is found that if there is external force with time correlations, detailed balance is lost, the particle can gain net drift speed. Two sufficient ingredients are: asymmetric potential and time correlated force.

Schematic diagram of motor protein

‘ratchet’ and time-correlated force

Discussion of the mechanism in our case

Our non-steady wave solution has intrinsically the two ingredients : it can be written as the sum of the SW with its PW in the frame following the SW.

PW is governed by ttx

,

),()(),( 0

)()]([ 0

NL

Simplified model wave field

Solving and , using a few key modes of the result to construct the simplified model field:

are the modes of are the modes of k=2 is the highest mode, besides, it is crucial for asymmetric potential.

)()()(1,2

11

k

kkkk bkA coscos

},{ kkA )(0

},{ kkb ),(

)(0 ),(

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Phase plot of particle in the moving framein simplified SR field

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Trapped state of particle in the SR wave

Trapped-Free states of particle in the simplified STC field

In the STC field a particle motion can be in trapped -free state.

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Trapped-free state ofparticle phase plotin the STC wave field

Acceleration mechanism

Two ingredients for the acceleration in our case:SW----Potential with broken symmetry (‘ratchet’)

PW----Time correlated force

)(0

),(

What difference compared to motor protein?

Our time-correlated force is a natural result of the deterministic nonlinear wave equation.Our ‘ratchet’ moves in the lab frame, so even in the SR field where the particle is trapped, the particle is carried along by the wave, and is accelerated.

Behavior of the key mode of PW in SR state

In SR state, the k=1 mode of PW is:the amplitude and phase are small perturbation to the SW;

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(b)

k=1

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Evolution of k=1 mode of SR wave

Behavior of the key mode of PW wave in STC state

In STC state, k=1 PW mode is:(1) its amplitude can be comparable or even larger than that of the SW;(2) the phase can go ever . That is it can be in-phase as well as anti-phased respect to the SW.When they are in-phase, the waves add up, the particle can be strongly accelerated if it moves to correct positions, and can be kicked out of the potential if then the barrier is weak enough.

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Motion of k=1 mode when transiting to STC

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Summary

Possibility of charged particle acceleration in nonlinear electric wave field is investigated. in a spatially regular nonlinear field the particle motion is eventually trapped into a wave trough. in a spatiotemporal chaotic wave field the particle can be alternatively in trapped and free phase. In both cases averagely the particle is accelerated along the wave direction.

Summary(continue)

The averaged accelerated velocity depends on the sign of the particle charge.The two ingredients for acceleration are:the SW ---- ‘ratchet’;the PW ---- time-correlated force.Whether the acceleration depends on the type of instability of SW is still open.

Thank you !