1. Stochastic Acceleration in Turbulence:
L. A. Fisk
University of Michigan

3. Evolution of low-energy ion energy spectrum across TS
Increasing intensities j(40 keV) ≈103 f/u
Spectral fluctuations markedly decreased by
Spectrum of protons keV evolving to single power-law with index =1.5

4. Different Expressions for the Observed Power Law
When expressed as a distribution function in velocity space:
When expressed as differential intensity [T is kinetic energy per nucleon]:
When expressed as differential number density:

6. The Transport Equations
In the frame of the solar wind with random convective motions.
Continuity equation is
ST is the differential stream; term on right is the change in differential number density due to the turbulence doing work on the particles.
The differential pressure is

7. Transport Equations
Multiply equation by kinetic energy T and manipulate,
The differential energy density is
The spatial transport of energy is
The third term is the flow of energy in energy space, with
The term on the right is the work done on the particles by the turbulent motions.

8. Equilibrium In a steady state: Time derivative must be zero
Flow of energy into and out of the core must be zero.
The work done on the particles by the turbulence must be balanced by the flow of energy [heat flux] into and out of the volume.
The particles receive energy from the turbulence and do an equal amount of work on the turbulence, and the heat flux distributes the energy.

9. Equilibrium Spectrum
The flow of energy into and out of the core must be zero, which requires that
Which is satisfied by U  T ‑2 as is observed

10. Summary Statements
The tail is formed by a cascade in energy, in which the tail particles are in equilibrium receiving and performing an equal amount of work on the turbulence.
The tail pressure is proportional to the pressure in the core particles; the proportionality constant depends on the amplitude of the fluctuations in pressure.
The tails start where the particles are sufficiently mobile to undergo stochastic acceleration. The tails stop when the particle gyro-radii exceed the scale size of the turbulence.
The proportionality of the tail and core particles can be used to specify the composition of the tail particles.