Consider a system that consists of two standard playing...

Consider a system that consists of two standard playing dice, with the state of the system defined by the sum of the values shown on the top faces. (a) The two arrangements of top faces shown here can be viewed as two possible microstates of the system. Explain. (b) To which state does each microstate correspond? (c) How many possible states are there for the system?(d) Which state or states have the highest entropy? Explain. (e) Which state or states have the lowest entropy? Explain.

Key Concepts

Microstate

A microstate is a detailed specification of the entire system at a microscopic level, describing the exact configuration or arrangement of all individual components. In statistical physics, each microstate represents one possible outcome of the system’s configuration without any coarse-graining or aggregation.

Macrostate

A macrostate is a coarse-grained description of a system defined by macroscopic observable properties, such as energy, temperature, or, in probability problems, aggregated outcomes. It does not detail the specific underlying configurations, meaning that many microstates can correspond to the same macrostate.

Degeneracy

Degeneracy refers to the number of microstates that correspond to a single macrostate. In other words, it is the measure of how many distinct configurations have the same macroscopic appearance. Higher degeneracy implies that a particular macrostate can be realized in many ways.

Entropy

Entropy is a measure of disorder or uncertainty in a system, quantifying the number of microstates corresponding to a particular macrostate. From a statistical mechanics point of view, a macrostate with higher degeneracy has higher entropy because there are more configurations (microstates) that the system can take on to achieve that state.

Combinatorics

Combinatorics involves the mathematical techniques used for counting and arranging different configurations of a system. In the context of systems composed of multiple components, combinatorial analysis is crucial for determining the total number of possible microstates and understanding how they contribute to the overall macroscopic behavior of the system.

Transcript

00:01 Let's see how micro -states and entropy and dice can all be interrelated.

00:08 And if you've ever played the game of craps, that's pretty much what this problem is going to be about.

00:18 And let's say we've got two dice that we're going to use and we're going to throw.

00:23 And how do we know what's going on with the arrangements of the dice? well, what we can think about is having our two dice and having, when we throw that dice, having a combination of microstates.

00:41 And so each die is six -sided.

00:45 And so it can have a value of one to six.

00:49 And the other die would have a value of one to six.

00:53 And so how many different combinations or how many different microstates can there be? now, it could turn out that like we can roll a three where there's a two on the die on the left and a one on the right or vice versa.

01:07 One on the left and two on the right.

01:09 Each die is independent.

01:12 So the first one has one of six possible combinations.

01:16 The second one has one of six possible combinations, six combinations, six combinations to die.

01:22 We have a total of 36 possible microstates that the time, die when we have two of them and we roll them could be in.

01:35 Or 36 possible arrangements that those dice would be in.

01:41 And if you take a look at the pair on the left, if we wanted to know what state that is in, well, what did we roll? we rolled the three.

01:55 So that would be called state three.

02:00 On the right, we have a four and a three.

02:02 We roll the seven.

02:04 That is going to wind up being state seven.

02:08 And we can roll those in a number of different combinations.