Question

Suppose a large number of particles are bouncing back and forth between \(x=0\) and \(x=1,\) except that at each endpoint some escape. Let \(r\) be the fraction reflected each time; then \((1-r)\) is the fraction escaping. Suppose the particles start at \(x=0\) heading toward \(x=1 ;\) eventually all particles will escape. Write an infinite series for the fraction which escape at \(x=1\) and similarly for the fraction which escape at \(x=0 .\) Sum both the series. What is the largest fraction of the particles which can escape at \(x=0 ?\) (Remember that \(r\) must be between 0 and \(1 .\) )

Short answer

The largest fraction of particles escaping at \(x=0\) approaches 1 when \(r \to 0\). The overall escape fraction is 1.

Step-by-step solution

Define the Problem

Particles bounce back and forth between the points \( x=0 \) and \( x=1 \). At each endpoint, a certain fraction \( r \) is reflected, and \( 1-r \) escapes. We need to find an expression for particles escaping at both endpoints and calculate the largest possible fraction escaping at \( x=0 \).

Define the Initial Escape at \( x=1 \)

Initially, 100% of particles start at \( x=0 \) and move toward \( x=1 \). When these particles hit \( x=1 \) for the first time, a fraction \(1-r\) escapes. Therefore, the initial escape fraction at \( x=1 \) is \((1-r)\).

Consider Reflected Particles from \( x=1 \) to \( x=0 \)

The fraction of particles reflected back to \( x=0 \) is \( r \). These particles return to \( x=0 \) and a fraction \(1-r\) escapes from \( x=0 \). Thus, the second escape fraction at \( x=0 \) is \(r(1-r)\).

Particles Bouncing Back and Forth

Continuing this process indefinitely, the particles keep reflecting and escaping at \( x=1 \) and \( x=0 \).

Series for Fraction Escaping at \( x=1 \)

The total fraction escaping at \( x=1 \) can be written as an infinite series: \[(1-r) + r^2(1-r) + r^4(1-r) + r^6(1-r) + \, \text{...} \] This series sums the particles escaping at \( x=1 \) after each reflection.

Sum Infinite Geometric Series at \( x=1 \)

Identify and sum the geometric series: \[ S_1 = (1-r) \ S_2 = (1-r)(r^2 + r^4 + r^6 + \, \text{...}) \] The second series is a geometric series with common ratio \( r^2 \). Sum the series: \ \[ S_2 = \frac{r^2 + r^4 + r^6 + \, \text{...}} {1-r^2} \] Substituting back, we get: \ \[ S_1 = (1-r) \left( 1+ \frac{r^2} {1-r^2} \right) = \frac{1-r}{1-r^2} \]

Series for Fraction Escaping at \( x=0 \)

Similarly, the total fraction of particles escaping at \( x=0 \) over time can be written as an infinite series: \ \[ (1-r)r + (1-r)r^3+ (1-r)r^5 + \, \text{...} \] This sums the particles escaping at \( x=0 \).

Sum Infinite Geometric Series at \( x=0 \)

Identify and sum the geometric series: \ \[ S = (1-r)(r + r^3 + r^5 + \, \text{...}) \] Sum the series: \ \[ S = (1-r) \cdot r \cdot \frac{1} {1-r^2} \] Simplifying, we get: \ \[ S = \frac{r(1-r)}{1-r^2} \]

Combine and Find the Largest Fraction Escaping at \( x=0 \)

Combine both fractions and find their sum: \ \[ \frac{1-r}{1-r^2} + \frac{r(1-r)}{1-r^2} = \frac{1-r + r(1-r)}{1-r^2} = \frac{1-r + r - r^2}{1-r^2} = \frac{1-r^2}{1-r^2} = 1 \] Therefore, the largest fraction of particles that can escape at \( x=0 \) is determined by finding the minimum direct escaping fraction: \( 1-r \). This occurs when \( r \) is smallest, near zero. Thus, the largest fraction escaping possible is close to 1 when \( r \to 0 \).

Key concepts

Reflection Coefficient

The reflection coefficient, often denoted by \( r \), is a key concept in the study of particle escape fractions. It represents the fraction of particles that are reflected back into the system at each endpoint instead of escaping. In mathematical terms:
- \( r \): the fraction of particles reflected
- \( 1 - r \): the fraction of particles that escape
Consequently, if all particles start at one point, they will keep bouncing back and forth, with some escaping at each reflection. The reflection coefficient determines how much of the initial population remains within the boundary after each bounce. This coefficient must always lie between 0 and 1 for physical realizability.
For example, if you have particles bouncing between \(x=0\) and \(x=1\), at \(x=1\), a fraction \(1-r\) escapes and the rest, \(r\), gets reflected back to \(x=0\). This process repeats, creating a series of reflections and escapes at both endpoints.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. When dealing with particle escape fractions, we use infinite series to express the total fraction of particles escaping at each endpoint. Each term in the series represents an escape event after a specific number of reflections.
For instance, when particles start at \(x=0\) and move toward \(x=1\), a fraction \(1-r\) escapes initially. Those that are reflected, \(r\), head back to \(x=0\) and the process continues. The infinite series for the particles escaping at \(x=1\) is expressed as:
\[ (1-r) + r^2(1-r) + r^4(1-r) + \text{...} \]
Each term here considers the particles that escape after two additional reflections. This sum can be expressed compactly and understood better by identifying it as a geometric series, which brings us to our next concept.

Geometric Series

A geometric series is a series with a constant ratio between successive terms. Geometric series are vital to solving the escape fraction problem because they allow us to sum the infinite series for particle reflections succinctly.
The basic structure of a geometric series is:
\[ a + ar + ar^2 + ar^3 + \text{...} \]
where \(a\) is the first term and \(r\) is the common ratio. For it to converge, \(|r|\) must be less than 1.
Using the example from the infinite series for particles escaping at \(x=1\), we identify the common ratio \(r^2\). So, the infinite series:
\[ (1-r)(1 + r^2 + r^4 + \text{...}) \]
can be summed using the geometric series sum formula:
\[ S = \frac{a}{1 - r}, \]
where \(a\) is the first term, \(1-r\), and \(r^2\) is the common ratio.
Applying this, the infinite sum for the fraction escaping at \(x=1\) becomes:
\[ S = (1-r) \frac{1}{1-r^2} = \frac{1-r}{1-r^2}. \]
Similarly, for the particles escaping at \(x=0\), the process is mirrored, and we find:
\[ \frac{r(1-r)}{1-r^2}. \]
By understanding geometric series, we can easily sum complex, infinite sums to reveal simple expressions, helping to solve particle escape problems.