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