  $M_n(\mathbb{C})$ denotes the set of $n$ by $n$ complex matrices. Consider
continuous time quantum semigroups $\mathcal{P}_t= e^{t\, \mathcal{L}}$, $t
\geq 0$, where $\mathcal{L}:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ is the
infinitesimal generator. If we assume that $\mathcal{L}(I)=0$, we will call
$e^{t\, \mathcal{L}}$, $t \geq 0$ a quantum Markov semigroup. Given a
stationary density matrix $\rho= \rho_{\mathcal{L}}$, for the quantum Markov
semigroup $\mathcal{P}_t$, $t \geq 0$, we can define a continuous time
stationary quantum Markov process, denoted by $X_t$, $t \geq 0.$ Given an {\it
a priori} Laplacian operator $\mathcal{L}_0:M_n(\mathbb{C}) \to
M_n(\mathbb{C})$, we will present a natural concept of entropy for a class of
density matrices on $M_n(\mathbb{C})$. Given an Hermitian operator
$A:\mathbb{C}^n\to \mathbb{C}^n$ (which plays the role of an Hamiltonian), we
will study a version of the variational principle of pressure for $A$. A
density matrix $\rho_A$ maximizing pressure will be called an equilibrium
density matrix. From $\rho_A$ we will derive a new infinitesimal generator
$\mathcal{L}_A$. Finally, the continuous time quantum Markov process defined by
the semigroup $\mathcal{P}_t= e^{t\, \mathcal{L}_A}$, $t \geq 0$, and an
initial stationary density matrix, will be called the continuous time
equilibrium quantum Markov process for the Hamiltonian $A$. It corresponds to
the quantum thermodynamical equilibrium for the action of the Hamiltonian $A$.
