Proof of Liouville's Theorem: Phase Volume Constant

[Petar Mali]

Phase volume is constant.

$$\int_{G_0}dx^0=\int_{G_t}dx^t$$

$$x=(x_1,...,x_{6N})$$

$$\int_{G_0}dx^0=\int_{G_t}dx^t=\int_{G_0}Jdx^0$$

We must prove that $$J=1$$

$$J=\frac{\partial (x_1^t,...,x_{6N}^t)}{\partial (x_1^0,...,x_{6N}^0)}$$

$$J$$ is determinant with elements

$$a_{ik}=\frac{\partial x_i^t}{\partial x_k^0}$$

Minor for $$a_{ik}$$ is

$$D_{ik}=\frac{\partial J}{\partial a_{ik}}$$

And now $$J$$ is define like

$$J=\sum_{k}D_{ik}a_{ik}$$

Why is define like this? Why not

$$J=\sum_{i,k}D_{ik}a_{ik}$$ ?

 

[Valerie Park]

The reason is that we need to prove that J=1, so we need that sum of all elements in matrix D_{ik} will be 1. But if we have J=\sum_{i,k}D_{ik}a_{ik} then sum of all elements in matrix D_{ik} will not be 1 and we will not prove that J=1. So we define J=\sum_{k}D_{ik}a_{ik}, because it guarantee that sum of all elements in matrix D_{ik} will be 1 and we will prove that J=1.

 

[charng]

I would like to provide a response to the content provided. Liouville's Theorem is a fundamental concept in the field of statistical mechanics, which states that the phase volume of a system remains constant over time. This theorem has been proven to hold true for classical as well as quantum mechanical systems.

The provided content presents a mathematical proof of this theorem, which involves the concept of phase volume and its constancy. Phase volume is defined as the volume in phase space, which is a multi-dimensional space where each point represents the state of a system. In this case, the phase space is described by the variables x_1^0,...,x_{6N}^0 and x_1^t,...,x_{6N}^t, representing the initial and final states of the system, respectively.

The proof uses the concept of a Jacobian, denoted as J, which represents the change in volume between the initial and final states of the system. The content correctly states that J is a determinant with elements a_{ik} representing the partial derivatives of the final state variables with respect to the initial state variables. The content then introduces the minors of J, denoted as D_{ik}, which represent the partial derivatives of J with respect to its elements.

The content then goes on to define J as the sum of the products of the minors and the elements of J. This is a valid definition, as it represents the overall change in volume between the initial and final states of the system. However, the content then questions why J is not defined as the sum of the products of all the minors and elements, rather than just the sum over one index (k). This is a valid question, but it does not affect the proof of Liouville's Theorem.

In conclusion, the content provides a clear and correct mathematical proof of Liouville's Theorem, which states that the phase volume of a system remains constant over time. The mention of the alternative definition of J does not affect the validity of the proof and is simply a matter of mathematical convention. As scientists, it is important to understand and question all aspects of a proof, but it is also important to recognize when a question may not have a significant impact on the overall concept being proven.