A collection of three noninteracting particles shares 3 units of energy. Each particle is restri

A collection of three noninteracting particles shares 3...

Key Concepts

Macrostate

A macrostate is defined by the overall, coarse-grained description of a system in terms of macroscopic variables, such as the total energy or the number of particles in specific energy ranges. In statistical mechanics, different macrostates represent different ways the system's energy or other conserved quantities can be overall distributed, regardless of the detailed arrangement at the microscopic level.

Microstate

A microstate is a detailed, specific configuration of a system that describes the exact state of every component, such as the particular energy level of each particle. Each macrostate can correspond to many microstates, and the number of microstates is directly related to the probability and entropy associated with that macrostate.

Energy Quantization

Energy quantization refers to the principle that in many systems, especially at atomic or molecular scales, energy can only take on discrete, integral values rather than a continuous range. This concept is crucial when counting the ways energy can be distributed among particles, as it imposes restrictions on the allowed energy levels each particle can adopt.

Combinatorial Methods in Statistical Mechanics

Combinatorial methods involve using techniques from combinatorics to count the number of possible configurations (microstates) that correspond to a given overall state (macrostate). These methods are essential for calculating the statistical properties of a system, such as entropy and probabilities, by accounting for all the ways energy or particles can be arranged under given constraints.

Transcript

00:01 In this problem, we're going to be taking a look at a system where three classical non -interacting particles share three units of energy.

00:07 We're going to calculate the number of macros, the macro states, the number of microstates associated with those macro states, and some probabilities of finding particles in certain energy levels.

00:18 All right.

00:19 So we know that there are three particles which share three units of energy.

00:24 So we know the sum of all their energies has to be equal to three.

00:27 So the macro states then are the macro state where each particle is in the first energy level.

00:37 They each have one unit energy.

00:39 There is the macro state where one particle has two units of energy.

00:44 One has one and one has zero.

00:49 And then there is the macro state where one particle has three units and the others have zero.

00:55 We're just seeing that these are all the possible macro states where the sum of the energies are three.

01:01 All right.

01:02 So we now want to calculate the number of micro states for each of these macro states, otherwise known as the multiplicity.

01:08 And that's just the number of unique ways you can find a permutation of these orders of numbers, right? so how many different ways can you write these sequences of numbers such that all of those numbers show up? so for this first one, there's only one microstate.

01:26 That's because there is no unique arrangement besides this one of one, it's all going to be one -one -one -one.

01:35 For the 210 macro -state, you have three choices of where to put the two, and then after you've chosen where to put the two, you have two choices of where to put the one, and that means you only have one choice each for where to put the zero particle.

01:54 So there's going to be six total microstates for that.

01:57 That's three times two, three choices for where to put the two, two choices for where to put the one.

02:02 That gives you six microstates.

02:06 Finally, for the 3 -00 macro -state, you're going to have three total micro -states for that...

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