Decay of Correlations
By Philip Valek, Robert J. Moore, and R.F. Martin,Jr.
Illinois State University Physics Department
- 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.
- 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  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:
- 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.
- 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) 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)
- Martin  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  have compared conductivities based on chaos to other measures.
- Holland and Chen  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
- Particles undergoing chaotic motion have the following
- orbits of particles with nearby initial conditions will diverge
- motion appears to be random
- ensembles of orbits diffuse throughout phase space
- We use a magnetic field model with parabolic field lines:
bn = BZO/BXO .
Time Average Correlations
- We have computed CTyy for several orbits at different values.
- We find the following results:
- For regular orbits CTyy is oscillatory and does not decay.
- 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.
- 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,
| Borderline ("sticky" orbit)
- Progression from Periodic to Chaotic: bn=0.1,
|Borderline ("sticky" orbit)
Ensemble Average Correlations
- We have computed CEyy for several ensembles
of orbits at different values of bn.
- We find the following results:
- For regular orbits, CEyy is relatively
insensitive to the number of initial conditions sampled.
- For chaotic orbits, CEyy becomes less
"noisy" as more initial conditions are sampled.
- For ensemble of regular orbits, CEyy is
oscillatory with power law decay in amplitude.
- For mixed ensembles of chaotic and regular orbits, the amplitude
of CEyy decays more rapidly.
- For ensembles of chaotic orbits, the decay of
CEyy is consistent with an exponential decay
- Chaotic and Regular: bn=0.1, Py=0.0,
|Chaotic: 100 orbits
|Chaotic: 100,000 orbits
|Regular: 100 orbits
|Regular: 100,000 orbits
- Progression from
Periodic to Chaotic: bn=0.1,
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
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.