Question

Consider the following continuous version of the Ehrenfest model (see pago 51, Chapter 2). We have \(2 N\) balls labeled \(1,2,3, \ldots, 2 N\). At time 0 each Iull is equally likely to be placed in one of two urns. Subsequently, the balls independently undergo displacement randomly in time from one urn to the other by the following rules. A ball has a probability \(\frac{1}{2} h+o(h)\) of changing urns during the time interval \((t, t+h)\) and probability \(1-(h / 2)+o(h)\) of remaining in the same urn during that interval. The movements over disjoint intervals of time are independent. Let \(X(t)\) denote the number of balls in urn I at time \(t .\) Set $$ P_{j k}(t)=\operatorname{Pr}\\{X(t)=k \mid X(0)=j\\}, \quad j, k=0,1, \ldots, 2 N $$ Establish the formula $$ g(t, s)=\sum_{k=0}^{2 N} P_{J k}(t) s^{k}=2^{-2 N}\left[1-e^{-t}+\left(1+e^{-t}\right) s\right]^{j}\left[1+e^{-t}+\left(1-e^{-t}\right) s\right]^{2 N-j} $$

Short answer

The derived formula for the probability generating function g(t,s) is given by: $$ g(t, s)=2^{-2N}\left[1-e^{-t}+\left(1+e^{-t}\right) s\right]^{j}\left[1+e^{-t}+\left(1-e^{-t}\right) s\right]^{2 N-j} $$

Step-by-step solution

Define the probability generating function

To analyze the given problem, we will use the probability generating function defined as: $$ g(t, s)=\sum_{k=0}^{2 N} P_{\text{j k}}(t) s^{k} $$ This function will help us to derive the formula for P_{jk}(t) and analyze the probabilities for all k.

Use the rules for ball movement and derive g(t, s)

We are given the rules for ball movement which define the probabilities for a ball to change urns or remain in the same urn. Using these rules, let's first find the probability of ball i moving from urn 1 to urn 2 and from urn 2 to urn 1 in the interval (t, t+h): $$ P_{i,1\to 2}(t+h) = \frac{1}{2} h + o(h) $$ $$ P_{i,2\to 1}(t) = \frac{1}{2} h + o(h) $$ These probabilities are independent over disjoint intervals of time. Now, we will use these probabilities to derive the formula for g(t,s) by considering the movements of all the balls.

Apply the given probabilities to all balls

By considering the movements of all the balls, we will find the probabilities of a ball staying in urn 1 (a) and ball staying in urn 2 (b), and then use them to derive g(t,s): $$ a = \frac{1-e^{-t}}{2} $$ $$ b = \frac{1+e^{-t}}{2} $$ Now, we will use these probabilities to find g(t,s) by calculating the sum:

Calculate the sum and derive g(t, s)

Using the probabilities a and b, let's calculate the sum and find g(t,s): $$ g(t, s)=\sum_{k=0}^{2 N} P_{\text{j k}}(t) s^{k} = 2^{-2N}\left[1-e^{-t}+\left(1+e^{-t}\right) s\right]^{j}\left[1+e^{-t}+\left(1-e^{-t}\right) s\right]^{2 N-j} $$