# Meaning of density of microstates in phase space

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• Kaguro
In summary, the conversation discusses the concept of microstates and ensembles in statistical mechanics. The system is completely described by the positions and momenta of all particles, represented by a single point in 6N-D gamma space. When restricting the system, there is an allowed region where each point is a different state. The distribution of states is like the density of real numbers, and even with Planck's constant, it can be either a finite constant or 0. An ensemble is a large number of fictitious copies of the system, and the density (probability distribution) is not the number of microstates per phase volume, but the number of members of the ensemble per phase volume. In equilibrium, the density of representative points is
Kaguro
TL;DR Summary
A microstate is one of possible states of the system common to a macrostate. In 6N dimensional phase space every point is a possible state, then what is the meaning of density of states?
Hello all. I am studying stat mech from Pathria's book.
It says a system is completely described by all positions and momenta of all the N particles. This maybe represented by a single point in 6N-D gamma space. So, each point is a (micro)state.

Now if we restrict the system (N,V,E to E+ΔE), the there is an allowed region in which microstates may exist.
In this region, every single point is a different state (in classical mech., there's no Planck's constant and unlimited precission.) So what's even the meaning of asking the distribution of states? It's like asking density of real numbers!

For every point in this space, either it is one of the allowed state or it is not. Even if there's a Planck's constant and limited precision, still this density can be either a finite constant or 0. It sounds digital to me.

Okay, I think I got it:
It doesn't matter if you consider a small volume of phase as a single microstate. So ([q0,q0+dq],[p0,p0+dp]) is a small patch and belongs to a single microstate. It doesn't matter exactly how small the dq and dp are. Planck's constant doesn't appear in Classical statistics.

Secondly, an ensemble is not the set of all the microstates. It is a set of a very large number of fictitious copies of the system. Large enough to cover all the microstates at least several times. Several members of the ensemble may be in a single microstate.

This density rho(probability distribution) is not the number of microstates per phase volume. That's always constant. It is the number of members of ensemble per phase volume. This density may change in non-equilibrium systems.

Each member of the ensemble has a representative point. These move around according to the Hamilton's equations. This causes shift in density. If the system is in equilibrium, then the number of such pts moving in is the same as the number moving out.

The density of these pts is constant if you move along with them. This is Liouville's Theorem.

If system is in equilibrium, then the density at a single region in phase space is also constant. That is the distribution of representative pts doesn't move around. This is a stronger condition than the previous one.

I understood all this from an excellent article:
https://web.stanford.edu/~peastman/statmech/statisticaldescription.html

berkeman

## 1. What is the meaning of density of microstates in phase space?

The density of microstates in phase space refers to the distribution of possible states of a system in its phase space. It represents the number of microstates per unit volume in phase space and helps to understand the probability of a system being in a particular state.

## 2. How is the density of microstates related to entropy?

The density of microstates is directly related to entropy, as it is a measure of the number of possible arrangements or configurations of a system. The higher the density of microstates, the higher the entropy and vice versa.

## 3. Can the density of microstates be calculated?

Yes, the density of microstates can be calculated using statistical mechanics and the fundamental equation of thermodynamics. It involves considering all possible microstates of a system and assigning probabilities to each state.

## 4. How does the density of microstates change with temperature?

The density of microstates increases with temperature, as the higher energy states become more accessible at higher temperatures. This means that the system has more possible microstates and therefore a higher entropy.

## 5. What is the significance of understanding the density of microstates in phase space?

Understanding the density of microstates in phase space is crucial in understanding the behavior and properties of a system. It helps to explain the relationship between entropy and temperature, and provides insights into the thermodynamic behavior of a system.

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