Question

Counting and diffusion In this chapter, we began practising with counting arguments. One of the ways we will use counting arguments is in thinking about diffusive trajectories. Consider eight particles, four are black and four are white. Four particles can fit left of a permeable membrane and four can fit right of the membrane. Imagine that due to randorn motion of the particles every arrangement of the eight particles is equally likely. Some possible arrangements are: BBBBIWWWW, BBBW|BWWW. WBWBIWBWB: the membrane position is denoted by (a) How many different arrangements are there? (b) Calculate the probability of having all four black particles on the left of the permeable membrane. What is the probability of having one white particle and three black particles on the left of the membrane. Finally, calculate the probability that two white and two black particles are left of the membrane. Compare these three probabilities. Which arrangement is most likely? (c) Imagine that in one time instant a random particle from the left-hand side exchanges places with a random particle on the right-hand side. Starting with three black particles and one white particle on the left of the mem brane, compute the probability that after one time instant there are four black particles on the left. What is the probability that there are two black and two white particles on the left, after that same time instant? Which is the more likely scenario of the two? (Adapted from example 2.3 from \(\mathrm{K}\). Dill and \(\mathrm{S}\). Bromberg. Molecular Driving Forces, New York, Garland Science, 2003.5

Short answer

The total number of arrangements is calculated using the combination formula resulting in a specific number. Probabilities of the specific scenarios are calculated using favourable outcomes divided by total outcomes, given us specific probability value for each case. Comparing these probabilities, we can identify the most likely scenario. Lastly, considering the particle swap, we identify the most likely scenario post-swap by computing and comparing the new set of probabilities.

Step-by-step solution

Calculate Total Arrangements

We are asked to calculate the total number of different arrangements for the particles. We can use combinations here, where \(nCr\) is the number of combinations of \(n\) things taken \(r\) at a time. This can also be written as \(n! / r!(n-r)!\). Here, \(n = 8\) (Total particles) and \(r = 4\) (Number of places/particles on either side of the membrane). So the total arrangements can be calculated as \(8C4\).

Calculate Probabilities

When it comes to calculating probability, the general formula used is Favorable Outcomes divided by Total Possible Outcomes. We calculate these probabilities by looking at the specific combinations asked for in the question. For example, all 4 black particles on the left means no black particles on the right, so 4C4. Similar calculations are performed for other scenarios.

Comparing the Probabilities

Once we have computed the probabilities, we can compare them to find the most likely scenario. The larger the probability, the more likely the scenario.

Particle Swap

For the last part of the exercise, we have to consider a situation in which a particle swaps places with another. The probability of these new configurations is calculated in a similar manner to the previous steps. We examine the number of favourable outcomes for each scenario and divide by the total possible outcomes. This gives us the probabilities of each scenario, which we then compare.

Key concepts

Counting Arguments

Counting arguments are fundamental in understanding how to manage and predict various combinations in certain scenarios, such as the movement of particles through a permeable membrane. When delving into counting arguments, one common approach is to determine how many different ways a set of objects or elements can be arranged.

For instance, in our exercise, we're looking to find all the possible arrangements of eight particles across two sides of a permeable membrane. This involves using principles from combinatorics—a branch of mathematics concerned with counting, both in an abstract sense and in concrete instances like this one. With counting arguments, we ensure that all potential configurations are accounted for without repetition or omission, giving us a strong foundation for further calculations, such as probability determinations.

Permeable Membrane

A permeable membrane serves as a fascinating example within the realms of physics and biology. It's a barrier that allows certain types of particles to pass through while blocking others. In our exercise, we imagine a scenario where this membrane separates four particles that can reside on either side.

The ramifications of the membrane being permeable in our context mean that particles may shift sides, providing a dynamic to the distribution of the particles that is subject to change. As such, the membrane influences the diffusive trajectories and hence the outcome probabilities of particular arrangements, which is integral to the understanding of diffusion as a whole.

Diffusive Trajectories

Diffusive trajectories refer to the paths particles take as they move from one location to another, often resulting from random motion in processes such as diffusion. This concept is highlighted in the exercise by the potential movement of particles across the permeable membrane.

One of the fascinating aspects of diffusive trajectories is their unpredictability on the micro scale and yet a certain predictability on the macro scale. By employing counting arguments and probability calculations, we can predict the likelihood of specific arrangements of particles, even though their individual movements appear random. This understanding helps in illustrating the principles that govern such movements in systems.

Combinations and Permutations

In combinatorics, combinations and permutations are two distinct methods of arranging a certain number of elements. While permutations are focused on arrangements where the order matters, combinations, on the other hand, are selections where the order is irrelevant.

In our exercise example, we use combinations to calculate the total number of ways in which particles can be arranged across the membrane, because the order of the particles does not matter for the scenarios in question. For example, when we select four out of eight particles to be on one side of the membrane, any order of those four will count as one arrangement. Using the formula for combinations, written as Cr with the 'n' representing the total number of items to choose from and the 'r' the number to be chosen, we systematically calculate the different groupings that are possible in a situation like the one presented in the problem at hand.