# A statistical definition of temperature

In order to define temperature in terms of statistics, "Concepts of Thermal Physics" by Blundell states:

Let us assume that the first system can be in any one of $$Ω_1(E_1)$$ microstates and the second system can be in any one of $$Ω_2(E_2)$$ microstates. Thus the whole system can be in any one of $$Ω_1(E_1)Ω_2(E_2)$$ microstates.

as expected from the probabilities.

However, when is considered that when the two systems can exchange energy with each other, it is stated that:

For our problem of two connected systems, the most probable division of energy between the two systems is the one which maximizes $$Ω_1(E_1)Ω_2(E_2)$$, because this will correspond to the greatest number of possible microstates.

My question is: Given the fact that occurs an exchange of energy between the two systems, I was expected that we will have:

$$\Omega_{1+2} \geq \Omega_{1}\Omega_{2}$$ as the number of microstates for the whole system

Could anyone explain me what is wrong above in my argument?´

• What is your argument exactly? Why does energy exchange imply that the number of states in the joint system is not the product? – kaylimekay Jan 21 at 9:22
• @kaylimekay Exactly – JoseAf Jan 21 at 9:24

Your argument is correct. What is not told explicitly here is that before the systems come in contact you can consider $$E_1,E_2$$ fixed. After you bring them in contact they are no longer fixed but the quantity $$E=E_1+E_2$$ is fixed. This leaves only $$E_1$$ as the free parameter (or $$E_2$$, just pick one). This means that implicitly $$\Omega_{1+2}(E_1)=\Omega_1(E_1)\Omega_2(E_2)=\Omega_1(E_1)\Omega_2(E-E_1).$$ After the systems have reached thermal equilibrium $$E_1$$ is again fixed because it has reached the unique value that maximizes $$\Omega_{1+2}(E_1)$$.
I should add that initially $$\Omega_{1+2}=\Omega_1\Omega_2$$ but after thermalization $$\Omega_{1+2}\geq\Omega_1\Omega_2$$.