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 (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.equation 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, equation
  • The equilibrium fluctuations are measured by the current density correlation function:

Green-Kubo Formulas

  • The equation relatingo sub alpha beta to c sub apha betacomes from linear response theory, and is an example of a Green-Kubo formula:
    equation equation
    but we consider theomega = 0 (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:equation
    • Time-stationarity is usually assumed:
      equation

Time Average vs. Ensemble Average Correlation Functions

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

  • Ensemble Average (many orbits):
    equation
  • Time Average (single orbit):
    equation
  • For sufficiently ergodic systems (i.e. strongly stochastic) CTyy = CEyy, but this system is not strongly ergodic [Chen, 1992].
  • We will study these two methods of computing Cyy in order to determine which method is most relevant to this problem.

Previous Work (for Space Plasmas)

  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-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:
    equation
    where bn = BZO/BXO .
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Results

Time Average Correlations

  • We have computed CTyy for several orbits at different values.
  • We find the following results:
    1. For regular orbits CTyy is oscillatory and does not decay.
    2. For "sticky" orbits, i.e. orbits near the boundary between the chaotic and regular regions, CTyy is oscillatory, often with some variation in amplitude but may not decay.
    3. For chaotic orbits CTyy has an amplitude which decays as a power law, as shown by a log-log plot.
  • Progression from Periodic to Chaotic: bn=0.3, Py=0.0, H=0.5
    Regular
    graph
    Borderline ("sticky" orbit)
    graph
    Chaotic
    graph
  • Progression from Periodic to Chaotic: bn=0.1, Py=0.0, H=0.5
    Regular
    graph
    Borderline ("sticky" orbit)
    graph
    Chaotic
    graph

Ensemble Average Correlations

  • We have computed CEyy for several ensembles of orbits at different values of bn.
  • We find the following results:
    1. For regular orbits, CEyy is relatively insensitive to the number of initial conditions sampled.
    2. For chaotic orbits, CEyy becomes less "noisy" as more initial conditions are sampled.
    3. For ensemble of regular orbits, CEyy is oscillatory with power law decay in amplitude.
    4. For mixed ensembles of chaotic and regular orbits, the amplitude of CEyy decays more rapidly.
    5. For ensembles of chaotic orbits, the decay of CEyy is consistent with an exponential decay rate.
  • Chaotic and Regular: bn=0.1, Py=0.0, H=0.5
    Chaotic: 100 orbits
    graph
    Chaotic: 100,000 orbits
    graph
    Regular: 100 orbits
    graph
    Regular: 100,000 orbits
    graph
  • Progression from Periodic to Chaotic: bn=0.1, Py=0.0, H=0.5
    Regular
    graph
    Mostly regular
    graph
    Mostly chaotic
    graph
    Chaotic
    graph

Comparison of CTyy and CEyy

  • The decay rate for CEyy is generally faster that that for CTyy. This is consistent with phase mixing in the ensemble average.
  • CTyy 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 Cyy on model parameters: field ratio bn 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.