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Question

I'm looking for an analytic way (if possible) to solve the matrix differential equation or at least have some example solutions?

The matrix differential equation is given by:

$$ \frac{N}{l_0} \text{Tr}(\rho(x_i - x_{i+1})) - 1 = \alpha \Big(\frac{\partial \text{Tr}(\rho \ln\rho) }{ \partial \text{Tr}( \rho H)}\Big )_{N,V}$$

where $\rho$ is the density matrix, $H$ is the Hamiltonian (which includes operator of $x_1$ to $x_N$), $N$ is the particle number, $l_o$ and $\alpha$ are constants.

Motivation

https://physics.stackexchange.com/questions/440417/an-idea-to-model-a-one-dimensional-thermometer

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