Stochastic Problems in Physics and Astronomy

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Stochastic Problems in Physics and Astronomy

• S. Chandrasekhar
• Rev. Mod. Phys. 15, 1 (1943)

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Number of citations per area(ISI Web of Science)
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OUTLINE

• Random walks (random flights)
• Brownian motion
• Probability after-effects
• Stellar dynamics
• Planet formation
• Sunspots
• Cosmic rays
• Conclusions

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INTRODUCTION

• Application of probability methods to problems as
  diverse as colloid chemistry and stellar dynamics
• Common characteristic large number of variables
  governed by probability laws
• Specification of distribution function
  of a quantity which is the result of a large
  number of other quantities, with distributions
  over a range of values

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RANDOM FLIGHTS
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RANDOM FLIGHTS (ii)
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RANDOM FLIGHTS (iii)

• There are ways of
  arriving at m . Therefore, the probability that
  the
• particle arrives at m is
• For large n,

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RANDOM FLIGHTS (iv)

• Introducing the variable xml for the
  displacement, where l is the length of a step,
  and
• assuming that the particle suffers N
  displacements per unit time, we can write
• where

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RANDOM FLIGHTS (v)
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RANDOM FLIGHTS (vi)Three-dimensional case
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RANDOM FLIGHTS (vii)Spherical distribution of
displacements
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RANDOM FLIGHTS (viii)General solution for large N

• Assuming that all individual displacements are
  governed by the same
• distribution function t, we can write
• After performing a rotation of the coordinate
  system, the distribution function
• for the position of the particle is
• This is an ellipsoidal distribution centered at

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RANDOM FLIGHTS (ix)Formulation in terms of a
differential equation

• It is possible to make a description of the
  problem through the equation
• the solution of which is

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BROWNIAN MOTION

• Motions of Brownian particles are maintained by
  fluctuations in the collisions with molecules

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BROWNIAN MOTION (ii)Free particle

• The equation of motion for a free particle is
  Langevins equation
• Solving this equation has to be understood as
  specifying a probability
• distribution

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BROWNIAN MOTION (iii)Free particle
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BROWNIAN MOTION (iv)Free particle

• For long enough times (such that a Brownian
  particle will suffer a large
• number of displacements) the resulting motion can
  be regarded
• as a random flight, and consequently as a
  diffusive process.

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BROWNIAN MOTION (v)Harmonically bound particle

• When an external force field is present, the
  Langevin equation becomes
• Considering a one-dimensional harmonic
  oscillator, this can be written as
• What is sought is the probability distribution
  

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BROWNIAN MOTION (vi)General characteristics

• The increment in the velocity of a Brownian
  particle can be written in the form

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PROBABILITY AFTER-EFFECTS

• Consider a small element of volume dV of a
  solution containing Brownian
• particles in diffusion equilibrium. We perform
  observations of the system at constant time
  intervals .
• For finite time intervals, one can enquire about
  the transition probability W(nm) that m
  particles will be counted inside dV at a time
  , at the beginning of which n particles were
  counted. In particular, we can ask about W(nn).

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PROBABILITY AFTER-EFFECTS (ii)

• The mean life of a fluctuation state n (once we
  count n particles, and after having counted n
  particles successively thereafter, how much time
  elapses until we count a number of particles
  different from n) is
• The time of recurrence of a fluctuation state n
  is
• where is the
  frequency with which different numbers of
• particles will be counted in dV, and is the
  mean number of particles in dV

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STELLAR DYNAMICS

• Analogy with Brownian motion
• Encounters with small impact parameters (which
  cause appreciable deflections) are rare
• Encounters with large impact parameters (which
  are more frequent) are ineffective

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STELLAR DYNAMICS (ii)

• Difference with Brownian motion
• Stars influence each other, whereas colloidal
  particles are primarily influenced by the
  molecules of the fluid
• In both cases, even though star-star and
  molecule-colloidal particle encounters hardly
  affects the motion, what is important is the
  cumulative effect of a large number of separate
  events
• The force per unit mass acting on a single star
  is due to the rest of the system as a whole. But
  fluctuations in the complexion of the local
  stellar distribution introduce fluctuations in
  this force

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STELLAR DYNAMICS (iii)

• Statistical character of the force acting on a
  star

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STELLAR DYNAMICS (iv)Black hole dynamics in a
stellar system

• Three forces acting on the black hole (Chatterjee
  et al 2002)
• Restoring force of the stellar potential (typical
  cluster length agtgtr)
• Dynamical friction
• Random force F(t) due to encounters with stars
• The equation of motion for the BH is

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STELLAR DYNAMICS (v)Black hole dynamics in a
stellar system
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PLANET FORMATION

• Long term behavior of an ensemble of
  planetesimals revolving around the Sun, until
• they are captured by a protoplanet
• (Hayashi, C., Nakazawa, K., and Adachi, I., Publ.
  Astron. Soc. Japan 29, 163)

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PLANET FORMATION (ii)

• Gravitational encounters between particles are
  stochastic processes
• Rates of change of semi-major axis, eccentricity,
  and inclination are calculated
• By evaluating mean-square deviation of semi-major
  axis, a diffusion coefficient in a-space can be
  calculated as
• (compare with the diffusion coefficient for
  random flights,
• where n is the number of displacements per unit
  time)

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PLANET FORMATION (iii)

• Simulations of dust growth in Brownian stage
  (Kempf, S. et al 1999)
• The dynamics of a particle of mass m and
  friction time are modeled by the Langevin
  equation
• where the friction time is given by
  (time needed for particle to dissipate its
  
• kinetic energy)
• The evolution of the friction time is determined
  by the aggregate structure
• The aggregates are self-similar the dependence
  of their mass on their radius is assumed to be of
  the form , where D is the fractal
  dimension of the particle structure

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PLANET FORMATION (iv)

• (Meakin,
  P. and Donn, B. 1988, ApJ, 329, L39)
• In the cold part of a protoplanetary disk, the
  time scales for pure Brownian growth are too
  large to explain the formation of planets within
  the life time of the disk (Brownian growth is
  significant only during early stages)

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SUNSPOTS
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SUNSPOTS (ii)

• Associated with buoyancy of magnetic flux tubes

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SUNSPOTS (iii)

• A sunspot is a region of enhanced heat transport
  from the convective zone (Parker 1974)
• The convective transport of heat is a stochastic
  process that can be described by an equation for
  the ensemble average temperature, in
  statistically steady turbulence

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SUNSPOTS (iv)
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COSMIC RAY PROPAGATION

• High energy protons and nuclei, the precise
• origin of which is not known.
• Suggested origins SN I,II novae formation
• of giant molecular clouds galaxy formation
• Energy range eV

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COSMIC RAY PROPAGATION (ii)

• Relevant works
• Motion of charged particles in a spatially random
  magnetic field (Jokipii 1966)
• Transport of protons in supernova shells (Harding
  et al. 1991, ApJ, 378,163)
• Mixing of cosmic rays does not extend far into
  the shell, so protons diffuse only in inner
  regions
• A region of thickness h is formed outside
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  the pulsar wind cavity,
  containing cosmic
• 
  rays, tangled magnetic
  fields and matter.
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  Not all cosmic rays in
  that region take
• 
  part in high energy
  interactions some
• 
  diffuse out of the
  envelope. To calculate
• 
  the interaction rate of
  protons, cosmic ray
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  diffusion is simulated as
  a 1D random
• 
  walk, with a step
  . If the
• 
  proton interacts, a new
  energy between 0
• 
  and the energy before
  interaction is assigned to it
• 
  if it does not, its
  energy is appropriately reduced
• 
  to account for adiabatic
  losses.

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CONCLUSIONS

• Stochastic phenomena are ubiquitous in
  astrophysical systems, spanning a large range of
  length scales
• Many diffusive processes can be modeled as random
  walks, Brownian motion
• In order to be able to treat a force as
  stochastic, the time scale of fluctuations has to
  be shorter than the typical time scale in which
  other dynamic variables change