Hi. I am trying to understand some features related to the order of a phase transition. It is known that there are finite size effects in a finite system. The finite size scaling theory provides relations between some quantities with the lenght of the system L. At second order phase transitions, finite size effects are due to the fact that the correlation lenght ε is about the same size of the system L in the transition point T_c of the control parameter as, for example, the temperature in a Ising-like system. However, it is not clear for me the relation between the correlation lenght and the system size at first order phase transitions. I have read that at first order phase transitions there are no critical exponents because the correlation lenght never increase up to the size of the system. In contrast, I read today that there are some kind of weakly first order phase transitions in finite systems where critical exponents can be defined and metastable states are seen with power law distributed domains. To sum up, if there is a weakly first order phase transition, can one use the cumulant method to find the transition point T_c? Thanks.