# How to solve this matrix differential equation?

## 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