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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.

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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.

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  • $\begingroup$ Are you sure $T_1 + T_2 = Constant$? $\endgroup$
    – user115350
    Aug 25, 2016 at 16:21
  • $\begingroup$ For a ideal system(like that of an ideal gas) E is proportional to T. Here I've used that. Of course the question asks for intuitive explanation. So, simplicity is mandatory. $\endgroup$
    – Ari
    Aug 27, 2016 at 5:00

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