# How to compare scales in a thermodynamic system

Suppose we have a generic system with hamiltonian

$$$$H= JH_1 +\mu H_2$$$$ where $$J$$ and $$\mu$$ are couplings and $$H_1$$ and $$H_2$$ are just two dimensionless parts of the hamiltonian. How can we estimate which term is going to dominate over the other? My first approach was to say "if $$\mu$$ is much bigger than $$J$$ then we can ignore $$H_1$$. But then I thought that, actually, the important parameters are not $$J$$ and $$\mu$$ but $$\beta J$$ and $$\beta \mu$$ so at $$T\rightarrow 0$$ both terms would be equally important, even if $$\mu>>J$$.

What is the right approach to think about these things? does it make sense to think about it in this generic example or further information is relevant to make a statement?