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Consider the grand canonical Gibbs distribution where a system is in thermal and diffusive equilibrium with a large reservoir.

The system can take on different energies $E_i$ and contain different number of particles $N_i$ when it is in different accessible microstates.

The definition of chemical potential for the system is $$\mu = -T(\frac{\partial S}{\partial N}) $$$$\mu = -T\left(\frac{\partial S}{\partial N}\right) $$ where $T$, $S$, $N$ refers to the temperature, entropy and number of particles of the system.

If the system in the Gibbs distribution take on different $E_i$ and $N_i$, why is $\mu$ the same? For example if system is in $E_1$,  $N_1$ state it will have a corresponding $S_1$ and a calculated $\mu_1$ using the forumlaformula above. If system is now in $E_2$, $N_2$ state it will now have entropy $S_2$ and $\mu_2$.

Consider the grand canonical Gibbs distribution where a system is in thermal and diffusive equilibrium with a large reservoir.

The system can take on different energies $E_i$ and contain different number of particles $N_i$ when it is in different accessible microstates.

The definition of chemical potential for the system is $$\mu = -T(\frac{\partial S}{\partial N}) $$ where $T$, $S$, $N$ refers to the temperature, entropy and number of particles of the system.

If the system in the Gibbs distribution take on different $E_i$ and $N_i$, why is $\mu$ the same? For example if system is in $E_1$,$N_1$ state it will have a corresponding $S_1$ and a calculated $\mu_1$ using the forumla above. If system is now in $E_2$, $N_2$ state it will now have entropy $S_2$ and $\mu_2$.

Consider the grand canonical Gibbs distribution where a system is in thermal and diffusive equilibrium with a large reservoir.

The system can take on different energies $E_i$ and contain different number of particles $N_i$ when it is in different accessible microstates.

The definition of chemical potential for the system is $$\mu = -T\left(\frac{\partial S}{\partial N}\right) $$ where $T$, $S$, $N$ refers to the temperature, entropy and number of particles of the system.

If the system in the Gibbs distribution take on different $E_i$ and $N_i$, why is $\mu$ the same? For example if system is in $E_1$,  $N_1$ state it will have a corresponding $S_1$ and a calculated $\mu_1$ using the formula above. If system is now in $E_2$, $N_2$ state it will now have entropy $S_2$ and $\mu_2$.

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Why is chemical potential constant for the system in Gibbs distribution?

Consider the grand canonical Gibbs distribution where a system is in thermal and diffusive equilibrium with a large reservoir.

The system can take on different energies $E_i$ and contain different number of particles $N_i$ when it is in different accessible microstates.

The definition of chemical potential for the system is $$\mu = -T(\frac{\partial S}{\partial N}) $$ where $T$, $S$, $N$ refers to the temperature, entropy and number of particles of the system.

If the system in the Gibbs distribution take on different $E_i$ and $N_i$, why is $\mu$ the same? For example if system is in $E_1$,$N_1$ state it will have a corresponding $S_1$ and a calculated $\mu_1$ using the forumla above. If system is now in $E_2$, $N_2$ state it will now have entropy $S_2$ and $\mu_2$.