Question

Consider the two-sided room, (a) Which is more likely to have an imbalance of five particles (ie., \(N_{\mathrm{k}}=\frac{1}{2} N+5\) ): a room with \(N=20\) or a room with \(N=60\) ? (Note: The total number of ways of distributing particles, the sum of \(W_{N_{k}}^{N}\) from \(O\) to \(N,\) is \(2^{N}\).) (b) Which is more likely to have an imbalance of \(5 \%\) (i.e., \(\left.N_{R}=\frac{1}{2} N+0.05 N\right) ?\) (c) An average-size room is quite likely to have a trillion more air molecules on one side than on the other. Why may we say that precisely half will be on each side?

Short answer

For an absolute imbalance, the room with fewer particles (N=20) is more likely to have an imbalance of 5 particles. For a relative imbalance, the room with more particles (N=60) is more likely to have an imbalance of 5% particles. Finally, we say that precisely half of the particles will be on each side due to the principle of maximum entropy, which states that in equilibrium, a system will assume the configuration with the maximum number of possibilities.

Step-by-step solution

Evaluate the likelihood of an absolute imbalance

We use the formula for binomial probability. The imbalance is defined as \(N_{k}=N/2 + 5\). For \(N = 20\), and \(N = 60\), compute the probability \(P = \frac{W_{N_{k}}^{N}}{2^{N}}\) to find which room this imbalance is more likely.

Evaluate the likelihood of a relative imbalance

The imbalance is given as a percentage. Hence compute \(N_{k}=N/2 + 0.05N\). Calculate the probability as in step 1 and compare the likelihood for rooms with \(N = 20\) and \(N = 60\).

Explain why half the particles are likely on each side

In an average room, the sheer number of particles and possible configurations means that the most likely scenario is an equal distribution. This can be explained using the concept of entropy and the principle of maximum entropy. Despite the possibility of a trillion more particles on one side, the probability is miniscule compared to the total number of configurations.

Key concepts

Binomial Probability

Binomial probability is a fundamental concept used to determine the likelihood of obtaining a specific number of successes in a fixed number of independent and identically distributed trials. In this exercise, we are considering the probability of finding an imbalance in the distribution of particles between two rooms.
To compute binomial probabilities, we often use the formula:\[P(x; N, p) = \binom{N}{x} p^x (1-p)^{N-x}\]where:

• \( N \) is the total number of particles or trials.
• \( x \) is the number of successes (specific state of interest, e.g., imbalance of particles).
• \( p \) is the probability of success on an individual trial (usually 0.5 for equal distribution).

In our case, the success represents particles gathering on one side, leading to an imbalance. The goal here is to compute the likelihood that a given imbalance of particles will occur in different-sized rooms. As more particles are involved, smaller fluctuations become less probable as the variance decreases, as explored under maximum entropy.

Particle Distribution

Particle distribution refers to how particles are spread across available space, such as in different rooms or compartments. This exercise examines the probability of finding a specific particle distribution pattern, focusing on two rooms of varying size where particles can shift.
When calculating the likelihood of an imbalance, we consider the possible ways to arrange particles. Each arrangement or configuration corresponds to a way to distribute particles. The symmetry of the situation (equal probability of being on either side of a room) simplifies calculations. The total number of configurations is given by \( 2^N \), where \( N \) is the number of particles.
For instance, the imbalance of five particles can be conceptualized as having five more particles on one side than the other. Larger rooms provide more micro-states or potential arrangements, thereby reducing the probability of any particular imbalance, highlighting the nature of particle distribution in statistical mechanics.

Maximum Entropy

In statistical mechanics, the principle of maximum entropy provides insight into how systems naturally evolve towards the most probable state. It suggests that, given multiple possible distributions of particles, the configuration with the highest entropy (or disorder) is the most likely.
Entropy here is a measure of the number of ways particles can be distributed. In practical terms, this means that even with a trillion more particles on one side, the most common configuration would be a near-equal distribution simply because there are many more ways for particles to be spread evenly than to gather disproportionally on one side.
The concept of maximum entropy explains why, despite occasional fluctuations, most systems settle into states of equilibrium. Entropy reaches its peak when there is uniformity, making any deviation (like a significant imbalance) less probable on a larger scale. Hence, understanding entropy in the context of particle distribution allows students to predict and rationalize observed physical phenomena.