PHAS0061 Problem Sheet 3
Please submit by 4pm on 8th March.
1. How can it be argued that irreversible macroscopic behaviour, characterised by the second law
of thermodynamics and uncertain predictions of future behaviour, can arise in a world governed
by reversible deterministic Newtonian mechanics? Use diagrams in your answer.
2. Construct the probabilities of reaching points m = 0,±1,±2 in a symmetric random walk of
8 steps starting from the origin where a particle becomes stuck at m = ±2 upon its first visit.
[Hint, this can most easily be done with simple arithmetic or a probability branching diagram].
Show that the probability of reaching one of these sticking points after precisely n steps is
P (n) = 2−n/2 for n = 2, 4, 6, 8 and zero otherwise. Show that the mean number of steps taken
up to sticking is 13/4. Confirm that if this pattern holds for all n then the probability of
sticking is unity if we wait long enough and once again determine the mean number of steps
taken up to sticking.
3. In the ‘Ehrenfest Urn’ problem, a particle moves randomly on a grid of positions x = ma,
with m an integer in the range −L ≤ m ≤ L, and with timestep τ . The probability, when at
position m, of a step to the right m → m + 1 is T+(m) = 12
(
1− m
L
)
and the probability of a
step to the left m→ m− 1 is T−(m) = 12
(
1 + m
L
)
.
(a) Evaluate the coefficientsM1−4 of the Kramers–Moyal equation for this process.
(b) Take the continuum limit a → 0, τ → 0, L → ∞ such that a2/τ → 2D and La2 → 2σ2,
where D and σ are constants, to show that the Fokker–Planck equation describing the
evolution of the pdf p(x, t) is
∂p
∂t
= D
σ2
∂ (xp)
∂x
+D∂
2p
∂x2
.
(c) At large times, such anOrnstein–Uhlenbeck process may be described by the time-independent
pdf p(x,∞) = (2piσ2)−1/2 exp(−x2/2σ2). Verify that this expression satisfies the Fokker-
Planck equation.
(d) Roughly sketch the time evolution of the Gibbs entropy of a system undergoing an Orn-
stein–Uhlenbeck process with initial condition p(x, 0) = (2∆)−1 for −∆ ≤ x ≤ ∆ with
∆ σ.
4. At t = 0 a clumsy professor disturbs a beehive. A very angry bee immediately emerges from
the hive and starts performing a realisation of a 1d Wiener process characterised by a diffusion
coefficient D. The professor runs away at speed v.
(a) Where is the professor situated in the time period t→ t+ dt?
(b) Write down the probability that the bee meets the professor (and therefore stings him) in
the time period t→ t+ dt.
(c) Evaluate the total probability that the professor is stung at least once. Would you advise
him to run faster to avoid getting stung entirely?
