Dr. Fernanda Pinheiro
Dr. Simone Pompei
In this problem, we develop the phase-space description for the diffusive motion of a Brownian particle.
The particle has mass m and phase space coordinates x = ( r, p ); it is subject to a potential V ( r ) and to interactions with a heat bath.
(a) Hamilton’s equations of motion define a velocity field v ( x ) in phase space. Write down the resulting transport equation for a particle distribution in phase space, P ( x, t ) = P ( r, p, t ).
(b) Now add the diffusive dynamics of the particle’s momentum caused by interactions with the heat bath: (i) dissipative change described by a drift field v diss
( p ) = − Γ ∂H kin
/∂p , (ii) diffusive change described by a random force η p
( t ) with moments h η p
( t ) i = 0 and h η p
( t ) η p
( t ′ ) i = Dδ ( t − t ′ ).
You obtain the Fokker-Planck equation for the distribution P ( x, t ), which is called the Kramers equation .
(c) Verify that this equation has the equilibrium distribution
P eq
( x ) =
1
Z exp[ − βH ( x )] p 2 with H ( x ) =
2 m
+ V ( r ) and β =
D
.
(1)
(d) To understand how the Kramers equation is related to the ordinary diffusion equation for the particle’s position, we now assume that the spatial potential V ( r ) is sufficiently smooth so that we can approximate its gradient V ′ ( r ) as constant over typical relaxation distances of the momentum, ℓ = β − 1 / 2 m/ Γ. Now consider the restricted dynamics of p at constant V ′ . Show that this leads to a stationary distribution of the momentum,
P stat
( p | V ′ ) =
1
Z exp − β
( p − m Γ V ′ ) 2
2 m
.
(2)
(e) Use the equation of motion ˙ = p/m together with the results of (d) to compute mean and average of the particle’s displacement ∆ r over a time interval τ ≫ Γ /m . Write down the resulting
Fokker-Planck equation for the marginal distribution P ( r, t ). This is known as the Smoluchowski equation .
This problem describes how the frequency of a particular form of a gene – a so-called allele – varies in time due to randomness in reproduction. The resulting model, called the Wright-Fisher model , is one of the most popular models for the evolution of a population. The goal of this exercise is to study the Wright-Fisher model in the diffusion approximation. As will become clear, this is an example of a diffusion process in which the diffusion parameter is not constant.
Let us assume a large population of N individuals that can be either of type (allele) A or B . At a given generation, these alleles appear with frequencies x = n/N and 1 − x = 1 − n/N , respectively.
1

The next generation is drawn randomly from this initial pool; that is, each individual has allele A with probability x or allele B with probability 1 − x .
(a) Compute the expected value E( n
′
| n ) and the variance Var( n
′
| n ) for the number of individuals with allele A in this next generation. Hence, compute the conditional moments where ∆ x is the change in frequency between two subsequent generations.
h ∆ x i x and h (∆ x ) 2 i x
,
(b) Assume that the probability distribution P ( x, t ) describing the allele A frequency x obeys a Fokker-
Planck equation. Use the results of (a) to derive this equation.
Hint: This is a diffusion equation where the diffusion parameter is x -dependent, so the ordering of operators matters. Recalling the notation used in class, argue that the ordering α = 0 is appropriate for this system.
(c) Find the equilibrium distribution P eq
( x ). Sketch the form of this solution and explain it as a result of the x -dependent diffusion parameter.
(d) Let us now allow for mutations to happen in this model: individuals randomly change their allele at a rate µ ,
↽
µ
B.
(3)
| n )+ N µ (1 − 2 n
N field v ( x ) in the Fokker-Planck equation.
( n
′
Show that this process modifies the expected value of individuals in the next generation, E
µ
E( n
′
). Compute the resulting conditional moment h ∆ x i x
| n ) =
, which defines a velocity
(e) Write down the modified Fokker-Planck equation for this case and compute its equilibrium distribution. What is the mathematical difference to the solution in (c)?
Due date: Wednesday, May 6.
2