# Has the von Neumann entropy ever been used in classical mechanics?

After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience , it occurred to me that this entropy might have applications to classical mechanics as it appears to be generally useful for dimensionality reduction.

## Motivation:

Let's suppose we have an ergodic dynamical system with $$N$$ observables $$x_i(t) \in \mathbb{R}$$ which are sampled using a sequence of $$n$$ measurements so we have a dataset $$X \in \mathbb{R}^{N \times n}$$. Given $$X$$, we may compute the statistics $$X_i = x_i - \langle x_i \rangle$$ and $$\sigma_i^2 = \langle X_i^2 \rangle$$ which allows us to define the Pearson correlation matrix with entries:

$$\begin{equation} R_{i,j} = \frac{X_i \cdot X_j}{\sigma_i \cdot \sigma_j} \tag{1} \end{equation}$$

Given $$R \in \mathbb{R}^{N \times N}$$, we may define the density matrix $$\rho = \frac{R}{N}$$ which is positive semi-definite, Hermitian and has unit-trace. Thus, we may calculate the entropy of $$R$$ using the von Neumann entropy:

$$\begin{equation} S(\rho) = -\text{tr}(\rho \cdot \ln \rho) = - \sum_{i=1}^N \lambda_i \cdot \ln \lambda_i \tag{2} \end{equation}$$

where $$\lambda_i$$ are the eigenvalues of $$\rho$$.

If $$N$$ is large, in order to compress the dataset $$X$$ so that we keep the dimensions that contain $$95\%$$ of the statistical information it is sufficient to find the discrete subset $$S \subset [1,N]$$ that maximises:

$$\begin{equation} - \sum_{i \in S} \lambda_i \cdot \ln \lambda_i \tag{3} \end{equation}$$

subject to the constraint $$\frac{- \sum_{i\in S} \lambda_i \cdot \ln \lambda_i}{- \sum_{i=1}^N \lambda_i \cdot \ln \lambda_i} \leq \frac{95}{100}$$ which may be done using sorting algorithms such as Quick Sort. Moreover, the cardinality $$\lvert S \rvert$$ provides us with an approximate upper-bound on the intrinsic dimension of an ergodic dynamical system.

## Question:

Has the von Neumann entropy ever been used in classical mechanics?

## Note:

It is worth noting that the idea of viewing the Pearson correlation matrix as a density matrix is quite natural and not new .

## References:

1. von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik (Mathematical foundations of quantum mechanics) Princeton University Press., . ISBN 978-0-691-02893-4.

2. H. Felippe et al. The von Neumann entropy for the Pearson correlation matrix: A test of the entropic brain hypothesis. Arxiv. 2021.

3. E.T. Jaynes. Information theory and statistical mechanics. The Physical Review. 1957.

4. linello (https://physics.stackexchange.com/users/10941/linello), Can a correlation matrix be regarded as a quantum density matrix?, URL (version: 2016-09-28): https://physics.stackexchange.com/q/282904