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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 $$\mu_2$$.