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TRANSCRIPT
Chapter 4 Lecture
Biological PhysicsNelson
Updated 1st Edition
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Random Walks, Friction & Diffusion (part I)
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Announcements
• Study methods – Read each chapter BEFORE class– Pair study groups (if class big enough)
• Grading format– Participation (inc. exercises) 30%– Midterm report/presentation 35% – Final report/presentation 35%
• Those of you taking Advanced Physics 1 (AP1) will be encouraged to take AP2, sorry.
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Important Dates
• Extra class– Wednesday May 6th (3rd Period)
• Midterm report presentation– Tuesday May 26th (5th Period)– Presentation on a chapter topic in book
• Final (written/presentation?) Report– Topic of you choice based on research
papers
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Summary Chapter 1
• The First Law of thermodynamics:DEtherm= Q - Wext
can be rephrased for different cases as:
• Just depends on the conditions you need• We’ll include μ the chemical potential later
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Enthalpy to Helmholtz to Gibbs
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Summary Chapter 3
• Don’t forget:-– Probability determines the motions and hence
pressure etc. of gases– Gaussian’s are ubiquitous in math/science– Maxwell Distribution– The more general Boltzmann Distribution– Activation Barriers
• Friction as slowing down due to molecular collisions
– The carrier of genetic information (in class discussion after homework problems)
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Chapter 3 Homework
1. Make 2D/3D Gaussians with plots for different σ using Wolfram α or similar software.
2. Derive Your Turn 3F
3. Do Problem 3.1 (The Dodgy Bakery)
4. Suppose you role 3 fair dice. What is the probability that you will get a 5 on at least one dice? (cf. your turn 3C)
Slide 1-8©1961. Used by permission of Dover Publications.
Random Walks?
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Outline
• Brownian motion• Random walks• Diffusion• Friction
• Three important equations, leading to the Fluctuation-Dissipation relation
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Brownian Motion
• Provides a proof of the existence of atoms /molecules – without ever observing them
• Discovered by Robert Brown, in 1828(A botanist)
• Motion of pollen grains in water• Very random and nonstop• More vigorous at high temperature
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Cause and a question?
Cause:• Collision between pollen and water molecules?
Questions arise:• Molecules are tiny, so how can a collision give a
visible movement of pollen?• The collision rate is 1012/sec, so how can we
resolve it with the naked eye (~30s-1)?
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Einstein’s answer
• These two questions cancel out if we use a Random Walk
• By considering the nature of the collision, it lead Einstein to one of his papers in 1905
• The tool is statistical mechanics
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Random Walks
• Total displacement has structure on all length scales in a random walk
• Every single step is random, but we can still effectively predict the motion of a collection of many molecules
“More is different!” P.W. Anderson
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2D Random Walk Simulations
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Random Walks
• Net displacement rather than return to starting point – Probability problem
• Idea:– Flip a coin and move left or right – Probability to return to the origin, the average
displacement and the average of mean square displacement
• Try “Your Turn 4a”
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Binomial Distributions: “Your Turn 4a”
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Example: Three step random walk
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Exercise
Exercise:
Make a diagram for 1D case of two steps?
Homework:-
Make a diagram for 1D case of four steps
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Random Walk
• 1D case, for N steps• How to find an equation for the displacement?
Assumption:1.Each step has length L2.The direction decided by kj with value ±1 3.The initial position x0 = 04.Collision occurs once per Δt, e.g., once per
second
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Mean and Variance
• Three step case (see Fig 4.4):-• In general case (Homework for 4 step case) is
– where last term is just L2 cf.– middle term is actually zero (see book) and
or – where D the diffusion parameter is D=L2/2Δt
and we have used N=t/Δt. In 3D we have
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Diffusion Law?
• Although the derivation is simplified, the law itself is universal. (Book Page 120)
• We can find D by experiment. The law is experimentally testable!
• Knowing D is not enough for finding the microscopic value L and Δt, and find the size of a molecule
• The Random Walk is model independent, see Section 4.1.3.
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Friction
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Friction
• 1D case, find the equation of terminal speed. • Assumption:1.Collisions occur once per Δt for some force f2.After collision, the initial speed of the object is
randomly determined from Newton’s law to be• Then we have
where v=Δx/Δt
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Diffusion and Friction
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1D Random Walks
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Homework
1. Read 4.1.3:- Understand statement: “Random Walk is model independent!”
2. Read 4.2:- What Einstein didn’t do3. Make a diagram for 1D case of four steps4. Extra:- Are two elevator shafts better when
stopping at odd and even floors only?• Assume the cost of the elevator is only to
start and stop ~ 50 Yen per ride
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Next Time
• Sections 4.3 – 4.6
• The arrow of time from diffusion and friction?
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Other Random Walks
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Polymer Diffusion
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Figure 4.8 (Schematic; experimental data; photomicrograph.) Caption: See text.
©1999. Used by permission of the American Physical Society.
Polymer Random Walks
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Random Walks on Wall Street
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Diffusion equation
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Functions and Derivatives
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Concentration c(x,t) plots
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Membrane Diffusion
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Nernst relation
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Discrete binomial vs. Gaussian
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Diffusion plot in time