Introduction

• The problem is motivated by studies of the low density (1-10 cm^-3) high temperature (10⁷ K) plasma in the earth's "magnetotail" a region 10-200 earth radii behind the earth.
• This region appears to be responsible for energizing ions and electrons responsible for the auroral displays, during magnetic storms and substorms.
• The standard model for this energization mechanism requires some electrical resistivity in the plasma. But his collisionless plasma has essentially zero collisional resistivity.
• Many investigators have searched for this "anomalous" resistivity without much success. We are interested in determining whether chaotic dynamics can produce the required resistivity.

Collisionless Resistivity

• Resistance is usually caused by collisions, e.g. electron-lattice collisions in a conducting material.
• The resistivity of the conducting material is related to the mean free path between collisions for particles. m.f.p
  
  
• However, many plasmas (in space e.g.) are effectively collisionless systems because the mean free path is greater than the entire system.
• Martin [1986] suggested that the resistance of such plasmas could be due to chaotic diffusion of the particles in phase space.
• We may use the Fluctuation-Dissipation Theorem from non-equilibrium statistical mechanics to determine this resistance.

• The connection between resistivity and dynamics is given by the fluctuation-dissipation theorem:
  
  The dissipative response of a system near equilibrium to an external force is related to the average fluctuation of the system from equilibrium.
• The response to an external force is given by the generalized susceptibility: in the case of electrical systems, the conductivity,



• The equilibrium fluctuations are measured by the current density correlation function:

Correlation functions can be calculated from particle dynamics in two ways:

• Ensemble Average (many orbits):



• Time Average (single orbit):

For sufficiently ergodic systems (i.e. strongly stochastic) C^T_yy = C^E_yy, but this system is not strongly ergodic [Chen, 1992].

We will study these two methods of computing C_yy in order to determine which method is most relevant to this problem.

1. Martin [1986] suggested chaotic diffusion leads to resistivity, estimated with Lyapunov (orbit separation) timescale.
2. Horton and co-workers [1990; 1991 a,b,c; 1992] have done extensive studies using the time average correlations for single orbits.
3. Martin and Speiser [1992] have compared conductivities based on chaos to other measures.
4. Holland and Chen [1992] have criticized this approach, but there are still regions where it should be valid.

Particle Dynamics and Correlation Functions

• Two dimensional Hamiltonian systems are characterized by three types of motion:
• periodic or quasi-periodic
• transient
• chaotic
• Particles undergoing chaotic motion have the following characteristics:
  • orbits of particles with nearby initial conditions will diverge exponentially
  • motion appears to be random
  • ensembles of orbits diffuse throughout phase space
• We use a magnetic field model with parabolic field lines:
  where b_n = B_ZO/B_XO .

Results

• We have computed C^T_yy for several orbits at different values.
• We find the following results:
1. For regular orbits C^T_yy is oscillatory and does not decay.
2. For "sticky" orbits, i.e. orbits near the boundary between the chaotic and regular regions, C^T_yy is oscillatory, often with some variation in amplitude but may not decay.
3. For chaotic orbits C^T_yy has an amplitude which decays as a power law, as shown by a log-log plot.
• Progression from Periodic to Chaotic: b_n=0.3, P_y=0.0, H=0.5
  

  Regular

  Borderline ("sticky" orbit)

  Chaotic

• Progression from Periodic to Chaotic: b_n=0.1, P_y=0.0, H=0.5
  

  Regular

  Borderline ("sticky" orbit)

  Chaotic

• We have computed C^E_yy for several ensembles of orbits at different values of b_n.
• We find the following results:
1. For regular orbits, C^E_yy is relatively insensitive to the number of initial conditions sampled.
2. For chaotic orbits, C^E_yy becomes less "noisy" as more initial conditions are sampled.
3. For ensemble of regular orbits, C^E_yy is oscillatory with power law decay in amplitude.
4. For mixed ensembles of chaotic and regular orbits, the amplitude of C^E_yy decays more rapidly.
5. For ensembles of chaotic orbits, the decay of C^E_yy is consistent with an exponential decay rate.
• Chaotic and Regular: b_n=0.1, P_y=0.0, H=0.5
  

  Chaotic: 100 orbits	Chaotic: 100,000 orbits
  Regular: 100 orbits	Regular: 100,000 orbits

• Progression from Periodic to Chaotic: b_n=0.1, P_y=0.0, H=0.5
  

  Regular	Mostly regular
  Mostly chaotic	Chaotic

• The decay rate for C^E_yy is generally faster that that for C^T_yy. This is consistent with phase mixing in the ensemble average.
• C^T_yy varies for individual orbits, and is sensitive to fine structure within a single orbit.

Conclusions

• Real Measurements sample both finite time periods and finite sets of orbits. So they are actually a combination time average and ensemble average.
• The sensitivity of the time average to structure within a single orbit makes it less useful: how can a restivity be defined which may vay with time even for the same orbit?
• We conclude that the ensemble average is the more relevant method for determining correlation decay.

• We need now to concentrate on ensemble average computations, studying the dependence of C_yy on model parameters: field ratio b_n and energy H.
• We will then compute the conductivity integral, and compare results with conductivities computed with other methods to determine the relative importance of the chaotic conductivity.