Lec17_Distributed_Systems_Time

Lec17_Distributed_Systems_Time - Lecture 17 Distributed...

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Lecture 17 i t ib t d S t Ti V i bl Distributed Systems: Time Variable
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Mr. Fick Again! Fick’s Second Law 2 CC E  2 t x 2 ) 4 exp( 2 ) , ( Dt x Dt m t x C p This solution is for the case of instantaneous r “impulse” spill All mass was concentrated or impulse spill. All mass was concentrated at one point initially (at x =0 for t =0)
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One-dimensional Random Walk n ant starts at x=0 and begins to crawl along a line An ant starts at x=0 and begins to crawl along a line You flip a coin. The ant moves one step to the right if you land heads. Otherwise, it moves to the left. x t x p ) exp( 1 ) , ( 2 t x D Dt Dt 2 4 p( 2 2 Concentration Probability ) 4 exp( 2 ) , ( 2 Dt x Dt m t x C p Pascal’s m p : total mass of particles Triangle Time
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Connection between Random Walks and Diffusion onsider the probability that our ant is at position fter me steps Consider the probability, p ( m , n ), that our ant is at position m after n time steps. To get to position m , the ant must be at position m -1 or position m +1 at the previous step (i.e., n-1 ). In either case, the probability that the walker moves to position m is 0.5. Hence 1 Subtracting p ( m , n -1) from both sides gives:   ) 1 , 1 ( ) 1 , 1 ( 2 ) , ( n m p n m p n m p 1 The left hand side is the change in p over one time step and the right hand side has the form of a second derivative with respect to the position (the step size is unity).   ) 1 , ( 2 ) 1 , 1 ( ) 1 , 1 ( 2 ) 1 , ( ) , ( n m p n m p n m p n m p n m p Letting the time and space steps become infinitesimally small leads to a diffusion equation 2 p D p where D is the diffusion coefficient. 2 x t Note that the probability of finding the ant at a point is the concentration C! Why?
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Diffusion with Reaction
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Advection –Diffusion: Random Walk
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Representation of the Random Walk by a normal distribution ick’s Second Law: 2 2 x C E t C Fick s Second Law: Et 2 Why does the distribution have higher density at the origin? The spreading results in a movement of particles from high to low concentration The Gaussian distribution can be used to predict how much tracer is within a certain region Use the 4 rule-of-thumb: 95% of tracer is contained in the range 2
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Observations Et x Et M C 4 exp 4 2
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Lec17_Distributed_Systems_Time - Lecture 17 Distributed...

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