Department of Physics

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# Decay of Correlations

## By Philip Valek, Robert J. Moore, and R.F. Martin,Jr.

Illinois State University Physics Department

## Introduction

- The problem is motivated by studies of the low density (1-10 cm
^{-3}) high temperature (10^{7}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.

### Fluctuation-Dissipation Theorem

- 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:

### Green-Kubo Formulas

- The equation relating to comes from linear response theory, and is an example of a Green-Kubo formula:

but we consider the (static fields) case only. - Assumptions
- The system is near equilibrium, i.e. it will spontaneously relax to equilibrium upon the removal of the constraint. \
- The response is linear in the applied force, i.e. Ohms Law is valid:
- Time-stationarity is usually assumed:

## Time Average vs. Ensemble Average Correlation Functions

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.

## Previous Work (for Space Plasmas)

- Martin [1986] suggested chaotic diffusion leads to resistivity, estimated with Lyapunov (orbit separation) timescale.
- Horton and co-workers [1990; 1991 a,b,c; 1992] have done extensive studies using the time average correlations for single orbits.
- Martin and Speiser [1992] have compared conductivities based on chaos to other measures.
- 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-periodi
- 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

### Time Average Correlations

- We have computed C
^{T}_{yy}for several orbits at different values. - We find the following results:
- For regular orbits C
^{T}_{yy}is oscillatory and does not decay. - 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. - For chaotic orbits C
^{T}_{yy}has an amplitude which decays as a power law, as shown by a log-log plot.

- For regular orbits C
- 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

### Ensemble Average Correlations

- We have computed C
^{E}_{yy}for several ensembles of orbits at different values of b_{n}. - We find the following results:
- For regular orbits, C
^{E}_{yy}is relatively insensitive to the number of initial conditions sampled. - For chaotic orbits, C
^{E}_{yy}becomes less "noisy" as more initial conditions are sampled. - For ensemble of regular orbits, C
^{E}_{yy}is oscillatory with power law decay in amplitude. - For mixed ensembles of chaotic and regular orbits, the amplitude
of C
^{E}_{yy}decays more rapidly. - For ensembles of chaotic orbits, the decay of
C
^{E}_{yy}is consistent with an exponential decay rate.

- For regular orbits, C
- 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

### Comparison of C^{T}_{yy} and C^{E}_{yy}

- 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

### Which Method is Best?

- 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.

### What is the next step?

- 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.