# What is the answer to Problem 1 part c in Statistical Mechanics 6.1?

• ehrenfest
In summary, two variables are needed to specify the orientation of a line segment. However, if you want to determine the thermodynamic/macroscopic state of the system, you need to provide all of the z-components.
ehrenfest
[SOLVED] statistical mechanics 6.1

## Homework Statement

The answer to Problem 1 part c is 2N. I disagree. I think it should be N because if you specify the z component of the system then you know both of the macroscopic quantities M and E.

## The Attempt at a Solution

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anyone?

If you specify only the z component, you have lost information about the number of equivalent states that would give the same M and E. This is key stat mech information. There is only one state with z = $\mu$, but many with z = 0.5$\mu$, and you need to differentiate them. Providing another distance coordinate (or alternatively, an angle) accomplishes this.

Looking at it another way: two variables are sufficient to describe a line's orientation. With N lines/dipoles, you need 2N variables to describe all the states.

Mapes said:
Looking at it another way: two variables are sufficient to describe a line's orientation. With N lines/dipoles, you need 2N variables to describe all the states.

What line are you talking about?

A dipole is like a line segment, in that rotation around the axis of its length is undetectable and does not constitute a degree of freedom.

I don't understand. What "key stat mech information" can you not get if you have all of the z components. If you have all of the z-components, then you know M, E, and can calculate $\Omega$ with the given equation. What else do you want?

You want the number of possible microstates that would result in those macrostate values of M and E. This is the fundamental idea of stat mech: We want to know the probability distribution of microstates that are compatible with our macrostate constraints.

This doesn't seem to be sinking in, so let's go back to the original question: "How many microscopic variables are necessary to completely specify the state of the system?" You give me a z value for each of the N dipoles. But you can't quit there. You haven't completely specified the microstate yet, since the dipoles can rotate in three dimensions and I don't know any of the x values.

I understand that you cannot completely specify the microstate of the system by providing only the z components. It is clear that there are 2N variables needed to specify the microstate of the system.

I think I see the flaw in my thinking now. By "state" in the question they really mean "microscopic state" not "thermodynamic/macroscopic state". Specifying only the z components of the system WILL determine the thermodynamic/macroscopic state of the system by the equations they provide, however, the microscopic state will still be ambiguous. Please confirm that this is correct.

Sounds good.

## 1. What is statistical mechanics 6.1?

Statistical mechanics 6.1 is a branch of physics that studies the behavior of systems composed of a large number of particles. It uses statistical methods to understand the macroscopic properties of these systems by analyzing the microscopic behavior of individual particles.

## 2. What is the difference between statistical mechanics 6.1 and classical mechanics?

Classical mechanics describes the behavior of macroscopic objects while statistical mechanics 6.1 focuses on the behavior of microscopic particles. Statistical mechanics takes into account the probabilistic nature of particles and their interactions, while classical mechanics assumes deterministic behavior.

## 3. What are the main principles of statistical mechanics 6.1?

The main principles of statistical mechanics 6.1 are the laws of thermodynamics, the concept of entropy, and the principle of equipartition of energy. These principles help explain the behavior of systems at the macroscopic level based on the behavior of individual particles at the microscopic level.

## 4. How is statistical mechanics 6.1 used in research and practical applications?

Statistical mechanics 6.1 is used in various fields such as physics, chemistry, biology, and engineering to study the behavior of complex systems and predict their properties. It is also used in the development of new materials, understanding phase transitions, and modeling biological systems.

## 5. What are some common models used in statistical mechanics 6.1?

Some common models used in statistical mechanics 6.1 include the ideal gas model, the Ising model, and the lattice gas model. These models simplify complex systems to make calculations and predictions more manageable and provide insights into the behavior of real-world systems.

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