Question

We claim that the famous exponential decrease of probability with energy is natural, the vastly most probable and disordered state given the constraints on total energy and number of particles. It should be a state of maximum entropy! The proof involves mathematical techniques beyond the scope of the text, but finding support is good exercise and not difficult. Consider a system of 11 oscillators sharing a total energy of just \(5 h \omega_{0}\). In the symbols of Section \(9.3, N=11\) and \(M=5\). (a) Using equation \((9-9),\) calculate the probabilities of \(n\) being \(0.1,2 .\) and \(3 .\) (b) How many particles. \(N_{n}\), would be expected in each level? Round each other nearest integer. (Happily. the number is still 11 . and the energy still \(\operatorname{she}_{0}\) ) What you have is a distribution of the energy that is as close to expectations is possible, given that numbers at each level in a real case are integers. (c) Entropy is related to the number of microscopic ways the macrostate can be obtained. and the number of ways of permuting particle labels with \(N_{0}\). \(N_{1}, N_{2},\) an \(d N_{1}\) fixed and totaling 11 is \(11 ! /\left(N_{0} ! N_{1} !\right.\) \(\left.N_{2} ! N_{3}^{\prime}\right) .\) (See Appendix J for the proof.) Calculate the number of ways for your distribution. (d) Calculate the number of ways if there were 6 particles in \(n=0.5\) in \(n=1\), and none higher. Note that this also has the same total energy. (e) Find at least one other distribution in which the 11 oscillators share the same energy, and calculate the number of ways. (f) What do your findings suggest?

Short answer

The conclusions would depend on the calculations that have been made in the steps. The goal is to see whether the most probable distributions yield the most disordered state, a state with maximum entropy. The short answer could therefore be 'the most probable distribution of particles across energy levels is the one which maximises the entropy of the system.'

Step-by-step solution

Calculate the probabilities for each energy level

The probability is calculated using equation 9-9 described in the textbook: \( P(n)=\frac{\omega(n)}{\Omega(M,N)} \) where \(\omega(n)\) represents the number of microstates corresponding to an energy level (n) and \(\Omega(M,N)\) is the total number of microstates of the system. The values for \(M=5\) and \(N=11\) can be substituted in. This equation can be used to find the probabilities for \(n=0\), \(n=1\), \(n=2\), and \(n=3\).

Calculate the expected number of particles at each level

The expected number of particles at each energy level, or \(N_n\), can be calculated as \(N_n = N*P(n)\), where N is the total number of particles (11) and P(n) is the probability for that energy level. Again, this must be calculated for \(n=0\), \(n=1\), \(n=2\), and \(n=3\). The results can be rounded to the nearest integer.

Calculate the number of ways the distribution can be achieved

The number of ways of achieving this distribution can be calculated using permutations. The number of permutable ways equals \(N! / (N_0! N_1! N_2! N_3!)\), where \(N = 11\), and are calculated from Step 2. This can be calculated by substituting in the numerical values.

Recalculate with different distribution

Now we recalculate the number of ways for a different distribution where we have 6 particles in n=0, 5 in n=1, with none higher. This can be calculated using a similar process to Step 3 with their respective scenarios.

Find and evaluate different distribution

We need to find at least one more distribution in which the 11 oscillators share the same energy, and calculate the number of ways for this distribution using similar process.

Conclude findings

Finally, we examine and summarize the results of the calculations, and consider what implications these might have for the distribution of particles across energy levels in the oscillators system.

Key concepts

Entropy

Entropy in statistical mechanics is a measure of the disorder or randomness of a system. It's connected to the number of ways we can arrange a system without altering its overall energy state. In simpler terms, entropy gives us an idea of the degree of chaos in a system. Higher entropy corresponds to more disorder, where systems have many possible arrangements or microstates that result in the same macroscopic condition.

When we say that a state has maximum entropy, we're referring to the fact that it's the state with the greatest number of microstates, making it the most probable due to its many possible arrangements. For our 11-oscillator system, each arrangement where energy is distributed across different energy levels results in a different microstate. However, some distributions have more microstates than others, resulting in higher entropy. By calculating these for different energy distributions, we can determine the state of maximum entropy.

Microstates

Microstates represent the specific arrangements of particles or energy in a system, while producing the same macroscopic state. Each possible arrangement that fits within the constraints of the system's total energy and particles count is a microstate.

In the exercise, we look at a system of 11 oscillators sharing a fixed total energy, where different microstates reflect varying distributions of energy among the oscillators. For each potential energy level—say one oscillator having more energy while another has less—the pattern creates unique microstates.

The total number of microstates informs us about how numerous and diverse energy distributions can be, all maintaining the same total energy. A system's macrostate, such as having a specific total energy level or particle number, arises from summing these microstates, demonstrating their relevance in calculating probabilities and understanding entropy.

Energy Distribution

Energy distribution encompasses how energy is allotted among particles in a system, in this case, oscillators. We need to determine how energy levels are populated in terms of number of particles at different energy levels.

Calculating energy distribution involves using probabilities determined from microstate counts. By examining various energy levels, we consider how likely each configuration occurs in terms of particle distribution. This information helps us understand what typical or probable configurations look like under specified conditions.

In the exercise, energy distribution calculations reveal how 11 oscillators share a specific amount of energy, applying probabilities to anticipate the number of particles at each energy level. This analysis supports insights into why certain distributions are more probable, owing to their higher entropy and more extensive microstate count.

Oscillators

Oscillators in statistical mechanics refer to systems—often simplified as particles or units—that have specific energy levels they may occupy. These can be thought of as vibrating particles, where quantized energy states allow these oscillators to reside in different energy levels.

Our problem involves 11 such oscillators distributing a quantized energy amount, typically depicted in physics by the term \(h\omega_0\). The exercise asks us to consider how these oscillators, given a set quantum of energy, might distribute themselves across allowable energy levels.

These oscillators behave under the constraints of statistical mechanics, where their arrangement across energy levels becomes significantly important. The task involves calculating how these particles distribute, understanding their probabilities, and what configuration results in maximized entropy. By studying this distribution through computation of probabilities, microstates, and entropic states, we unlock insights into the state behavior of collective oscillators.