Relationship between trace and phase space

[sunrah]

So I noticed we can define entropy in two very different ways:
1) quantum mechanically
$S = -k Tr(\hat{\rho}\ln{(\hat{\rho})})$
2) classically
$S = -k \int \rho \ln{(\rho)} d\Gamma$


where Tr is the trace and $d\Gamma = \frac{1}{h^{3N}N!}\prod_{i}^{N} d^{3N}q_{i}d^{3N}p_{i}$ is the phase space volume element.

My question is how summing over the diagonal of an operator is like integrating the same quantity over phase space.

Edit: I realize that we are working out a mean value.

 

[jvicens]

Hello,

Thank you for bringing up this interesting topic. I am always fascinated by the different ways in which we can define and understand physical quantities. In this case, entropy, which is a measure of the disorder or randomness in a system, can indeed be defined in two different ways - quantum mechanically and classically.

To answer your question, let's first understand the concept of entropy in both quantum and classical mechanics. In quantum mechanics, entropy is defined as the von Neumann entropy, which is a measure of the uncertainty or lack of knowledge about the state of a system. It is given by the trace of the density operator, which represents the state of the system. On the other hand, in classical mechanics, entropy is defined as the Boltzmann entropy, which is a measure of the number of microstates (i.e. possible arrangements of particles) that correspond to a given macrostate (i.e. the state of the system as a whole).

Now, coming to your question, the reason why summing over the diagonal of an operator is equivalent to integrating the same quantity over phase space is because both methods are essentially calculating the average value of a quantity. In quantum mechanics, the trace of the density operator gives the average value of the quantity over all possible states of the system. Similarly, in classical mechanics, integrating over phase space gives the average value of the quantity over all possible microstates that correspond to a given macrostate.

To understand this better, let's consider an example. Imagine a system of particles confined to a box. In quantum mechanics, the density operator would represent the probability of finding a particle at a particular position in the box. Summing over the diagonal of this operator would give the average probability of finding a particle at any position in the box. Similarly, in classical mechanics, integrating over phase space would give the average number of particles at any position in the box.

In summary, while the mathematical expressions for entropy may be different in quantum and classical mechanics, the underlying concept of averaging over all possible states or microstates remains the same. I hope this helps clarify your question.