Does Irreversibly/no detailed balance implies there is no thermal equilibrium? Consider the following transition matrix
$$
   T=
  \left[ {\begin{array}{cccc}
   \frac{1}{3}  & \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\\
   \frac{1}{3} & \frac{1}{4} & \frac{1}{5} &  \frac{2}{6}\\
   \frac{1}{3} & \frac{1}{4} & \frac{3}{5}&  0\\
   0 & \frac{1}{4} & 0& \frac{3}{6}\\
  \end{array} } \right]
$$
of a Markov chain process
$$P_{t+1}=TP_t$$
The steady state is given by
$$
   P^*=\frac{1}{137}
  \left[ {\begin{array}{c}
   33  \\
   36 \\
   50 \\
   18 \\
  \end{array} } \right]
$$
The detailed balance condition implies that
$$\sum_i T_{ij}P_i^*=\sum_j T_{ji}P_j^*$$
since for example $T_{24}P^*_2\neq T_{42}P_4^*$, the chain process does not satisfy the detailed balance condition.
I want to understand if the system is in thermal equilibrium. Obviously the system is not reversible, since there is no detailed balance, but is it a sufficient condition? 
can a system be irreversible and in thermal equilibrium? 
If so, what is the mathematical condition that needs to be satisfied in order to be in thermal equilibrium?
 A: According to the Wikipedia entry on detailed balance:

The detailed balance condition is stronger than that required merely for a stationary distribution; that is, there are Markov processes with stationary distributions that do not have detailed balance. Detailed balance implies that, around any closed cycle of states, there is no net flow of probability. For example, it implies that, for all a, b and c, $$P(a,b)P(b,c)P(c,a)=P(a,c)P(c,b)P(b,a)$$

If you mess around with your system, you will see that this does not hold for closed cycles between states 1, 2, and 4 and closed cycles involving all four states. Therefore, we have an equilibrium distribution without detailed balance.
This is not my area of expertise, but all we have here is that the probability distributions approach an "equilibrium value", but I am not sure if this necessarily means that the system itself needs to be at thermodynamic equilibrium. From the first page on this site https://www.stat.auckland.ac.nz/~fewster/325/notes/ch9.pdf

Note: Equilibrium does not mean that the value of $X_{t+1}$ equals the value of $X_t$
  .
  It means that the distribution of $X_{t+1}$ is the same as the distribution of $X_t$

I imagine a circular body of water where there is a uniform whirlpool in the center. Even though we have an equilibrium probability distribution of where a water molecule will be, the system itself is not in equilibrium. This is not a perfect way to view this (as pointed out in the comments) since sometimes the system might not converge to the equilibrium distribution, but I still think it is a good picture here.
