1. Lecture 34: Diffusion and Chemical Potential

2. Key concepts from last time
The probability distribution of a random walker is a binomial distribution
After many steps, the probability distribution is well-approximated by a Gaussian
The variance of this Gaussian distribution, 2Dt, increases linearly with time
Through the derivation, we practiced using Stirling’s approximation, Taylor expansions, and normalization via a Gaussian integral formula.

3. Time evolution of p(x,t)
Here I have used D=1.
What do we expect this probability distribution to look like as t -> 0? Introduce Dirac delta distribution.

4. What happens as t -> 0? Here I have used D=1.
What do we expect this probability distribution to look like as t -> 0? Introduce Dirac delta distribution.

5. What happens as t -> 0?
The distribution stays normalized, but becomes infinitely narrow. This is a bit like keeping the total area of a rectangle = 1, while shrinking its width.
Here I have used D=1.
What do we expect this probability distribution to look like as t -> 0? Introduce Dirac delta distribution.
In the limit of infinitesimal width, the distribution has an infinite value at the origin but is zero everywhere else.

6. What happens as t -> 0?
This odd distribution (which comes up often in science) is called the Dirac delta distribution and denoted d.
It allows us to describe that, with absolute certainty, the particle was at position x=0 at time t=0.
Here I have used D=1.
What do we expect this probability distribution to look like as t -> 0? Introduce Dirac delta distribution.
Compare how this looked when our distribution was discrete:

7. How “fast” does the random walker move?
Let’s return to our solution and try to gain some intuition for how “fast” things diffuse.

8. How “fast” does the random walker move?
What is the mean position?
What is the most likely position?
x=0 for all t
x=0 for all t

9. How “fast” does the random walker move?
What is the mean position?
What is the most likely position?
What is the mean square position?
x=0 for all t
x=0 for all t

10. How “fast” does the random walker move?
What is the mean position?
What is the most likely position?
What is the mean square position? <x2> = 2Dt
We can estimate a “typical net distance traveled” using the root mean square distance:
x=0 for all t
x=0 for all t
Notice that this returns us to units of length. It now becomes clear that the units of this coefficient D are [L]^2 / [T]
In order for the RMS distance to increase two-fold, we’d need to wait four times as long.

11. Typical values of D
Small molecules in air: Small molecules in water: Heat through water:
20 mm2/s
0.002 mm2/s
0.143 mm2/s
Estimate: How long until the turkey’s done?

12. Typical values of D
Small molecules in air: Small molecules in water: Heat through water:
20 mm2/s
0.002 mm2/s
0.143 mm2/s
Estimate: How long until he who dealt it smelt it?
What does this tell you about how smells reach your olfactory receptors?
How long would it take for a small molecule to diffuse from one end of a 1 m axon to the other? About 100^2 = 10,000 times as long!

13. We have learned how a particle’s probability distribution evolves from a known initial position. How will p(x,t) change with time in general?

14. How does a particle reach position x at time t?
To reach x = 0 at time t, the particle must have been at either x=-1 or x=1 at time t - Dt
Only half of the particles previously at x=1 and x=-1 will wind up at x=0 in the next step:

15. How does a particle reach position x at time t?
More generally, but with the same logic,
Reason for expanding to different orders is clarified when we plug in: the Dx terms will cancel.

16. How does a particle reach position x at time t?
This result is called Fick’s second law of diffusion. You may recognize it from this week’s problem set question on Bicoid.
A worthwhile time investment: convince yourself that the Gaussian found earlier satisfies the equality.
“Diffusion equation” also often used, though there is a more general case with D not constant.
In the case of Bicoid, we aren’t following the probability distribution for one particle, but rather the concentration profile for many particles (which is essentially just the sum of their individual p(x,t)s).

17. What have we learned?
A particle diffusing from a known initial position has a Gaussian probability density:
A handy measure for the “likely net distance traveled” is given by the RMS distance:
Probability densities for diffusing particles obey Fick’s second law

18. Diffusion lecture plan
Diffusion of single particles as a random walk
Parallels first lac operon lecture
Adds introduction to the diffusion equation
Diffusion as a consequence of chemical potential differences
Alternative derivation of Fick’s second law
Description of the diffusion coefficient
Gaussian integrals and FRAP
Example applications
Diffusion to detection (Berg & Purcell)

19. Diffusion and Chemical Potential
Statistical mechanics helps us understand the average behavior of systems as a consequence of the behavior of individual particles.
We have studied how a single particle behaves under diffusion. How will a collection of particles behave?
Statistical mechanics helps us understand the average behavior of
Two major approaches to tackling this problem:
Start from first principles
Superposition of the results for single particles

20. What is a potential? Potential energy can be used to drive motion
Total Energy = Kinetic Energy + Potential Energy
The potential energy can be described by a function F called a potential
Example: gravitational potential for an object of mass m at height h near Earth’s surface
We have been referring to the “chemical potential” so far without mentioning that potential is a loaded word
Convince that Phi has units of energy (kg * m/s^2 * m = J)

21. Potentials have associated physical forces
Forces accelerate the objects on which they act
The force associated with a potential function F(x) is
This force accelerates objects from regions of high F toward regions of low F
Notice that when calculating the force, it doesn’t matter how we defined “zero potential energy”
The acceleration from the force is how potentials “cause” changes in motion
Another thing to notice: it does not matter at all, in terms of calculating the force, what we define to be “zero energy” for our potential.

22. Common examples of potentials and their associated forces
Gravitational potential and gravitational force:
Compressed or extended spring
These examples will be more evocative for students who have taken general physics in high school.
Sign for gravitational force indicates that the force points toward decreasing h (downward, just as we’d expect).
Draw the spring. Introduce that x is a deviation from the rest length and k is a spring constant indicating, in some sense, how stiff the spring is. The force result is Hooke’s Law.
For the spring, remind students that

23. Is there a force associated with chemical potential?
Recall that the chemical potential for a chemical species is given by …where c is the chemical’s concentration (which may vary with position) and m0, c0 are constants.
Notice that the magnitude of the force is greater when the concentration gradient is steeper (or c is small).

24. There is a force associated with chemical potential
Which way will this force accelerate an object? From high c to low c This is the driving force of diffusion!
Notice that the magnitude of the force is greater when the concentration gradient is steeper (or c is small).

25. Is this the only force acting on diffusing particles?
A particle moving in a fluid encounters resistance as it collides with other molecules. These collisions tend to decelerate the particle. Like other changes in acceleration, we can represent it with a force, called drag. The faster the particle moves, the more collisions it experiences, and the faster it decelerates.
Ask whether this drag is any different from the friction they are used to (mu Fnormal), which in fact does not scale at all with velocity.
Sign indicates that the force will point in the opposite direction of the velocity: that is, it well act against its motion.

26. A brief discussion of drag
The drag coefficient depends on the viscosity h of the fluid
From

27. A brief discussion of drag
The drag coefficient also depends on the particle’s shape
The bus has the higher drag coefficient (both skin and form drag). Mention the spoiler as an aside.
They have used ksi instead of zeta to represent the drag coefficient – oh well.