First of all, \(\Omega_1(E_1)\delta E_1\) and \(\Omega_2(E_2)\delta E_2\) are the phase space volume for these two subsystems. Since most of the volume in phase space is near the surface, we can instead write \(\Omega_1(E_1)\) and \(\Omega_2(E_2)\) to denote the two volumes in phase space. Second, it’s clear that the probability density of subsystem 1 having energy \(E_1\) is proportional to the ’allowed’ volume product of the two subsystems.

\[\rho_1(E_1) \propto \Omega_1(E_1)\Omega_2(E_2)\]

where \(E_2 = E - E_1\). And finally, we impose the normalization condition:

\[\int \rho_1(E_1)dE_1 = A \int \Omega_1(E_1)\Omega_2(E-E_1)dE_1 = A \Omega(E) = 1\]

A is the proportional constant and equal to \(1/\Omega(E)\). Therefore,

\[\rho_1(E_1) = \frac{\Omega_1(E_1)\Omega_2(E_2)}{\Omega(E)}\]

If we maximize the probability at \(E_1\):

\[\frac{d\rho_1}{dE_1} = 0\]

\[\frac{1}{\Omega_1}\frac{d\Omega_1}{dE_1} - \frac{1}{\Omega_2}\frac{d\Omega_2}{dE_2} = 0\]

By plugging the definition of entropy \(S = k_B log(\Omega)\),

\[\frac{1}{k_B}\frac{dS_1}{dE_1} - \frac{1}{k_B}\frac{dS_2}{dE_2} = 0\]

\[\frac{dS_1}{dE_1} + \frac{dS_2}{dE_1} = \frac{d}{dE_1}(S(E_1)+S(E_2)) = 0\]

,which implies the maximization of the sum of entropies of both subsystems.

For (6), we see that if the probability density \(\rho(E_1)\) is maximized,

\[\frac{1}{T_1} = \frac{1}{T_2}\]

The chemical potential is defined as

\[\mu = \left( \frac{\partial E}{\partial N} \right) _{S,V}\]

The chemical potential is the energy cost to add in a particle. It is also associated with intermolecular potentials. As one particle is introduced, the chemical potential is the average of all the possible interactions with other particles originally in the system. As the number of introduced particles increases, the energy of the system also increases due to the chemical potential, which is also thought of as the population pressure for the molecules. For two subsystems of different chemical potential, i.e. one is more populated than the others, the particles will flow from the high chemical potential (or high density) one to the other.

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