# Statistical mechanics - N distinguishable particles

• fuselage
In summary, for a system with N identical "boxes" with energy levels E0 and E1, there are (N-1)/2 possible microstates corresponding to the macrostate with total energy Mε.
fuselage

## Homework Statement

"A model system consists of N identical ”boxes” (e.g. quantum wells, atoms), each box with only two quantum levels, energies E0 = 0 and E1 = ε What is the number of microstates corresponding to the macrostate with total energy Mε?"

## The Attempt at a Solution

I've done questions like this in the past with a small number of quantum wells, but then I've simply counted the number of possible arrangements. For example, if you have 3 quantum wells with one particle in each, and each well has 4 energy levels (including E0=0), there are 10 different microstates if the macrostate energy is 3ε. But I can't find a general equation.

Does anyone know the general equation(s) that are used to solve a question like this?

Well aware of the binomial coefficients, but I still can't figure it out.

You're trying to figure out how to put M units of energy into N boxes.

But that doesn't work.

Take the case I described in my attempt at a solution: If you simply count by hand, you will find 10 ways to get 3 units of energy out of a system of three particles each in its own quantum well with four energy levels (Ranging from E0=0 to E3=3). But nCr(3,3) ≠ 10, nor does nCr(4,3).

fuselage said:

## Homework Statement

"A model system consists of N identical ”boxes” (e.g. quantum wells, atoms), each box with only two quantum levels, energies E0 = 0 and E1 = ε What is the number of microstates corresponding to the macrostate with total energy Mε?"

## The Attempt at a Solution

I've done questions like this in the past with a small number of quantum wells, but then I've simply counted the number of possible arrangements. For example, if you have 3 quantum wells with one particle in each, and each well has 4 energy levels (including E0=0), there are 10 different microstates if the macrostate energy is 3ε. But I can't find a general equation.

Does anyone know the general equation(s) that are used to solve a question like this?

Think of tossing a coin N times On each toss, 'heads' ↔ energy level ε, 'tails' ↔ energy level 0. You want to know how many sequences of tosses have M heads and (N-M) tails.

Note: that is essentially "classical" counting. If, instead, your N have integer spins you need to use the so-called "Bose-Einstein" statistics, which gives a radically different result.

fuselage said:
But that doesn't work.

Take the case I described in my attempt at a solution: If you simply count by hand, you will find 10 ways to get 3 units of energy out of a system of three particles each in its own quantum well with four energy levels (Ranging from E0=0 to E3=3). But nCr(3,3) ≠ 10, nor does nCr(4,3).
But that's not the case you have here. Each box can be in one of two states, which makes it much easier, and you can use nCr.

For the case where each system can be in more than one state, see http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disbol.html

## 1. What is statistical mechanics?

Statistical mechanics is a branch of physics that uses statistical methods to explain the behavior of large systems of particles. It is used to understand the properties of matter and the laws of thermodynamics at the microscopic level.

## 2. What are N distinguishable particles?

N distinguishable particles refer to a system of particles that can be differentiated from one another based on their physical properties, such as mass, charge, or position. These particles can be atoms, molecules, or even larger objects.

## 3. What is the difference between classical and quantum statistical mechanics?

Classical statistical mechanics deals with macroscopic systems where particles are treated as classical objects with known positions and momenta. Quantum statistical mechanics, on the other hand, takes into account the quantum nature of particles and their wave-like behavior.

## 4. How do you calculate the entropy of a system using statistical mechanics?

The entropy of a system can be calculated using the Boltzmann formula: S = kBlnW, where kB is the Boltzmann constant and W is the number of microscopic states corresponding to a given macroscopic state.

## 5. What is the significance of statistical mechanics in thermodynamics?

Statistical mechanics provides a microscopic explanation for the laws of thermodynamics. It helps us understand how the macroscopic properties of a system, such as temperature and pressure, arise from the behavior of individual particles. It also allows us to predict the behavior of a system under different conditions.

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