# Understanding chemical potential in AdS/CFT

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.

## 1 Answer

$\mu =\frac{\partial G}{\partial N}$ 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.

• Thanks for the answer. Actually I was looking for a more intuitive way of understanding chemical potential and specially in the context of AdS/CFT. – Physics Moron Jun 5 '15 at 15:26