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Session 4: Diffusion part 1, random walks

Goal

be able to explain characteristics of Gaussian random walk processes without drift, with constant drift, and with origin-directed drift.

  • Preview: again the single molecule force spectroscopy measurement again; this time rather than binning into open/closed, consider the whole continuous reaction coordinate x.

  • Simplest case of diffusion with no “potential” – e.g. observation of pollen grains under a microscope, or of microbeads…

    • Problem setup: consider the position of a particle, x(t), over multiple discrete timesteps.

    • present simple case of +1/-1 random walk; long times is Gaussian

    • then generalize to +a/-a, time interval dt, N steps with T = N dt. Notation is x(t + dt) = x(t) + e where e = a or -a.

    • generalize to other step size distributions, note the universality.

    • Draw some example trajectories and histograms. Note that can think of two ways, as stochastic trajectory x(t), or as a probability distribution evolving over time, P_t(x).

    • Derive P_{1 dt}(x) = Normal(0, a); P_{N dt}(x) = Normal(0, Na^2).

    • generalize to higher dimensions.

  • Now add a constant drift term: v dt, so that each time step you move as x(t + dt) = x(t) + v dt + e. Then P_{N dt}(x) = Normal(v N dt, N a^2). Plot this, draw some example trajectories.

  • How to keep the variance from exploding? Add a drift term that brings you back to the origin. Call it - c x dt. This is like a spring, or optical trap, or what not. Then x(t + dt) = x(t) - c x dt + e. (note this is the overdamped case where velocity = mobility times force). Next time we will derive the stationary distribution of this process

  • Final note: an N-dimensional random walk model of a polymer! Doesn't account for self-avoiding effects…

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