Vocaloid:

Random Walk Problem [Statistical Mechanics]

3 days ago
Vocaloid:

I understand some of the basic principles, just having trouble with some of the intermediate steps :S

3 days ago
Vocaloid:

we have N objects (in this case, fleas) and m move forward while N - m move backward p = m/N gives us the probability of forward motion q = (N-m)/N gives us the probability of backwards motion

3 days ago
Vocaloid:

for (p+q)^N we get the binomial expansion W(m) = N!/(m!(N-m)! * q^(N-m) * p^m (derived from the combination formula) for probability of m steps forward and N-m steps backward

3 days ago
Vocaloid:

The mean displacement is where I start getting a little lost

3 days ago
Vocaloid:

\[<m>^2 = \sum_{m=0}^{N}m^2W(m)\]

3 days ago
Vocaloid:

for step length L, we get displacement = (steps forwards - step backwards) *L = (m - [N - m])l = (2m-N) * L

3 days ago
Vocaloid:

doing a substitution for displacement <m> we get <m> = m * N!/(m!(N-m)! * q^(N-m) * p^m

3 days ago
Vocaloid:

if we let Q = N!/(m!(N-m)! * q^(N-m) * p^m and take the partial derivative of Q wrt p we get:

3 days ago
Vocaloid:

\[\frac{ ∂Q }{ ∂P } = m \frac{ N! }{ m!(N-m)! }q^(N-m)*p^(m-1)\]

3 days ago
Vocaloid:

and we multiply this by p to convert it to a partition function

3 days ago
Vocaloid:

which basically just turns that p^(m-1) back to a p^m

3 days ago
Vocaloid:

\[<m> = p(∂Q/∂p)\]

3 days ago
Vocaloid:

if Q also equals (p+q)^N we have: \[(∂Q/∂p) = N(q+p)^{N-1} \]

3 days ago
Vocaloid:

\[<m> = p(∂Q/∂p) = Np(q+p)^{N-1} = Np\]

3 days ago
Vocaloid:

for a true random unbiased walk, p = 1/2 for forward step

3 days ago