Question

I follow the notation used in stochastic processes in physics and chemistry(p.245) by Van Kampen.

Left hand side of master equation is
∂P(n,t)∂t=⋯.
We split n
into 2 elemnts
n=Ωϕ(t)+Ω1/2ξ
and rewrite the first to
∂P(n,t)∂t=∂∂tΠ(ξ,t)∣∣∣∂ξ∂n∣∣∣=Ω−1/2[∂Π∂t+∂ξ∂t∂Π∂ξ]=Ω−1/2[∂Π∂t−Ω1/2dϕdt∂Π∂ξ].
Here my question. Why is n held constant?

Answer 1

A student inquired about the manipulation step in the derivation of a master equation in the context of statistical physics. Specifically, they were confused about why the variable 'n' is held constant during a differentiation step. The answer explains it as a result of applying the chain rule in a transformed coordinate system where we observe from a moving frame associated with the deterministic part of 'n'.

Step-by-step explanation:

The question involves the derivation of a stochastic master equation in the context of statistical mechanics, specifically within the realm of stochastic processes in physical systems. The student is referring to a step in the mathematical formalism where the variable n is broken down into a deterministic part and a fluctuating part, n = Ωφ(t) + Ω1/2ξ. Then, the master equation's left-hand side is rewritten to account for this new representation in terms of Π(ξ, t).

The key point of confusion for the student is the reasoning behind keeping n constant while differentiating, which stems from applying a chain rule for partial differentiation in a transformed coordinate system where ξ is the variable of interest rather than n. Here, the assumption is made that we are observing the system from a frame moving with the deterministic part of n, and how the probability distribution Π(ξ, t) evolves in this moving frame is of prime interest.