Diffusion in double well potential

Many problems involve motion in a potential with more than one equilibrium point. In this problem you'll consider motion in a "double well" potential. A recent example is the study of molecular unfolding by applying an external force to a single molecule. This kind of problem is analyzed in the attached code.

Equilibrium

1. Take the Langevin equation in this homework problem, and replace $- k x$ $-kx$ by a more general force $f (x)$ $f(x)$ .

How do we choose $f (x)$ $f(x)$ ? We'll derive it from the double well potential in the function $V (x)$ $V(x)$ .

2. Complete the function f(x) as shown in the hints file hw5/nonlinear2_hints.py. It's very important that you get the algebra of this correctly and also remember the relationship (in 1d) between potential and force.

We also know that in thermal equilibrium, the probability of finding the particle at position $x$ $x$ is given by the Gibbs distribution. The rest of the code compares the simulation results with the exact result from the Gibbs distribution.

3. Run the code and compare the analytic and numerical solution.

4. Now try lowering the temperature to 0.01 and 0.007. What happens now? Do the analytical and numerical results agree? If not, why not?

Escape time over a barrier

In systems with metastable or bistable states, how long does it take to get from on state to the other? We will utilize the code developed above to calculate the time it takes on average to hop from one state to the other.

1. In the source file, hw5/hopping_time2_hints.py, we have modified the code in the double well problem so that we can start the particle in one well, at x=0 and measure how long it takes to get to the other well. The way this is done is to update the position inside a function "escape_from_hole", which you can complete using the code from nonlinear2_hints.py. It runs until the particle crosses a threshold value and then it returns the time t, that this occurred.

This code is called NumAve times and a histogram of escape times is generated as well as the mean time to escape.

2. Run the code for different values of the temperature. If the temperature gets too low, it'll never escape. But it's most interesting when the temperature is low as the time scale for many processes are controlled by degrees of freedom in this regime. Explain your results qualitatively.

3. From this, come up with a mathematical formula relating the this time to temperature, and compare it with your numerical results.

Compare this with the discussion from the Cavendish Lab Links to an external site. on page 13.

Applications

1. Give at least one example of a chemical reaction where the barrier height is so large that equilibrium is not reached on reasonable time scales.
2. What biochemical process can speed up equilibration? Explain briefly how this works by altering the the picture of a particle diffusing over a barrier of fixed height.
3 Can you see how diffusion over a barrier is related to the motion of molecular motors such as myosin?