# How can we show the increase of number of microstates intuitively?

After the thermal exchange of two bodies with different temperatures $T_1$ and $T_2$ reaching a equilibrium temperature $T_2$< $T_3$ < $T_1$, how can we prove the number of microstates is increased intuitively? Don't use the entropy explanation, since the entropy is defined on the number of microstates.

To get an intuitive idea we start by assuming that no. of microstates$(N)$ are monotonically increasing(or simply linear) function of temperature. This is fairly intuitive since more temp usually allows the system to access more of its energy levels. Thus $N$ $\alpha$ $T$.
Now one of the bodies are at temperature $T_1$. For that number of microstates are $N_1(T_1)$. And for the other body $N_2(T_2)$. Now since the microstates of each body are independent of the other, the total number of microstates for the whole system is given by $N = N_1*N_2$. Now we can maximise this number $N$. Energy conservation will give $T_1 + T_2 = Constant.$ $=>$ $N_1+N_2= Const.$. Use this to maximise N and you will get that it is maximum when both temp. are equal. Thus at thermal equilibrium number of microstates are maximised.
• Are you sure $T_1 + T_2 = Constant$? Aug 25, 2016 at 16:21