Understanding chemical potential in AdS/CFT

Question

I always find it very difficult to understand the notion of chemical potential physically/intuitively unlike pressure and temperature in statistical mechanics. Can some one suggest some nice references or briefly discuss about it intuitively. Actually I came across it again in the context of AdS/CFT. It will be really helpful if it is discussed in holographic context.

Answer 1

$\mu =\frac{\partial G}{\partial N} $ i.e. it is free energy per particle in an ensemble, or energy needed for a particle to add it to the system so that the system will stay in equilibrium. (Some call it Gibbs free energy/enthalpy. Free energy is usually the name for helmoltz free energy). Formal definition -

$$ G(T,P,N)=E(T,P,N)+P\cdot V(T,P,N)-T\cdot S(T,P,N) $$ so the equilibrium state is given for minimal value of $ G $. If you put in contact 2 closed systems with $ G_1(T_1,P_1,N_1) $ and $ G_2(T_2,P_2,N_2) $ , the combined system system will seek a new equlibrium with maximal entropy and minimal $ G $ - $$ G^*(T^*,P^*,N^*)=G_1^*(T^*,P^*,N_1^*)+G_2^*(T^*,P^*,N_2^*) $$

Because of the maximal entropy principle - one of the ways the system tries to minimise $ G $ /maximize entropy is by flow of particles from one system to another. The particles will flow from system with high value of $ \mu $ to the system with low value - that according to the principle of minimizing free energy in the system.