Group problem solutions

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Group problem solutions

• (a)
• (b)

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• 2. In order to be reversible we need
• or equivalently
• Now divide by h and let h go to 0.
• 3. Assuming (as in Holgate, Math. Gazette, 1967)
  that the intensity of amalgamation is
  proportional to the number of herds (perhaps it
  should be proportional to the number of pairs of
  herds) and the intensity of splitting a herd of
  size xi is proportional to xi-1, the overall
  intensity of splitting will be the sum of these.
  Hence the G-matrix has

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• so the equations for the stationary distribution
  are (using Holgates model)
• whence the stationary distribution is
  1Bin(K-1,???????).

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Browns motion

• Richard Brown (1773-1858)
• Scottish botanist
• First systematic description of Australian fauna
• Named the cell nucleus
• Observed irregular pollen motion in liquid (first
  described by Ingenhousz in 1784)

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Observed motion
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Interpretation

• The basic unit of living matter (molecule)
• Same motion with dust particles (or carbon on
  oil)
• Ongoing motion in closed box forever
• Explained by fluctuating temperature due to
  heating by incident light electrical forces
  temperature difference in the liquid.
• Delsaux (1877) impact of molecules of liquid on
  immersed particleslots of small bumps

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Einsteins contribution

• 1905 Annus mirabilis
• Photoelectric effect
• Special relativity
• Emc2
• On the Motion Required by the Molecular Kinetic
  Theory of Heat of Small Particles
  Suspended in a Stationary Liquid

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Some physics

• Two forces act on suspended particles
• osmotic pressure (tendency towards equal
  concentration)
• diffusion (particles hit by heat-induced movement
  of liquid molecules)
• If q(x) is the density of particles per unit
  volume, the osmotic force is
• (R gas constant, T abs temp, N Avogadros
  number)
• This force causes flux

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More physics

• The flux due to diffusion with diffusion
  coefficient D is
• These two fluxes must be equal in equilibrium, so
• If we know D and r we can compute N (which was
  not known in 1905).

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Einsteins stochastic model

• Over a short time period ?, a particle is
  displaced by a random quantity Y. It should have
  mean 0 in equilibrium. Assume for simplicity
  with probability 1/2 each. Let x be a point. Only
  particles in (x-y,x) can pass x from left to
  right in a time interval ?.
• Only half of those particles will in fact move
  from left to right, i.e. about particles.
• Similarly, about particles move from
  right to left.

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• Net flux is then
• and the quantity diffusing in unit time is
• so Dy2/2?.
• The displacement over unit time will be the sum
  of 1/? iid (why?) random variables, so has
  variance ?2Var(Y)/?.
• Since Var(Y)y2 we have D?2/2.
• Perrin used this to estimate Avogadros number.
  Nobel prize in physics, 1926.

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The distribution of Brownian motion

• A particle starting at the origin jumps Y units
  in time ?. The pgf of Y is
• In time t, the particle takes t/??independent
  steps. The total displacement has pgf
• with mean 0 and variance ty2/?.
• In order to have the variance converge to ?2t we
  need y??1/2. Then

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An excursion

• Let XN(???2). Then

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• Let se?. This yields the moment generating
  function. We get
• Exponentiating back we get the mgf for a
  N(0,?2t)-random variable.
• One can use this type of limit to calculate the
  forward equation for the process, which turns out
  to be the heat equation.

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A general definition

• A Brownian motion process is a stochastic process
  having
• Independent increments
• Stationary increments
• Continuity for any ???

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Some Brownian motion process paths
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Properties of Brownian motion process

• X(t)N(?t,?2t)
• Continuous paths
• Finite squared variation
• Not bounded variation
• Not differentiable paths
• Derivative of location is velocity, so for small
  time intervals this is not defined (not a very
  accurate model!)
• Is it Markov?

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More realistic assumption

• Stationarity is the probability counterpart to
  conservation of momentum (mass x velocity)
• Instead of independent increments of location, we
  could consider independent increments of momentum
• Velocity change in particle would be twice the
  speed of the molecule times the ratio of molecule
  to particle mass

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The Ornstein-Uhlenbeck approach

• Let U(t) be the velocity of a particle.
  Langevins equation says
• where K(t) is a stochastic process with mean
  zero and very quickly decreasing covariance
  (collisions from independent molecules)
• Divide through by m to get ??/m and
  K(t)K(t)/m. Multiply by exp(?t), so

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A formal solution

• The integral makes sense if K(t) is continuous,
  but it probably is not.
• Write formally K(t)dB(t)/dt to get
• The problem is now to make sense of the
  stochastic integral on the rhs (492). The result
  is the Ornstein-Uhlenbeck process.

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Properties of the O-U process

• It is the only stationary Gaussian Markov process
  in continuous time and space.

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The displacement

• is another Gaussian process with mean X(0) and
  variance
• For large t this behaves like a constant times t,
  as Einstein found, but for small t this behaves
  like t2. It has a derivative (velocity). Why?