Equilibrium

Based on a system's independent variables, the concept of equilibrium can be divided into three categories: isolated systems, systems in contact with a reservoir of constant temperature, and systems in contact with a reservoir of constant temperature and pressure [Rei, §8.1, §8.2 and §8.3].

Common misconceptions about equilibrium.
Let v_{1} be the number of particles going from the system to the reservoir per unit time and v_{2} be the number of particles going from the reservoir to the system per unit time.
 Misconception #1. When v_{1}=v_{2}, the system and the reservoir are in equilibrium.
Why is the above statement incorrect?
Equilibrium must be described in terms of states.
 Misconception #2. When the system and the reservoir are in equilibrium, v_{1}=v_{2}.
Why is the above statement incorrect?
Like an eigenvalue in quantum mechanics, the measurement of a system's particle number is legitimate only if the state of the system is specified. The net rate v_{1}=v_{2} depends on the system's initial state and final state during the measurement.
Correct statement: Most probably (the sample space is the state space of the system), v_{1}=v_{2}. However, there is a tiny probability that v_{1}>>v_{2} and also a tiny probability that v_{2}>>v_{1}.
 Misconception #3. When an isolated system is in equilibrium, the particle number of each chemical species in the system is fixed.
Similiar to (ii), the correct statement should be as follows:
Most probably, the particle number of each chemical species in the system is
fixed. However, there is a tiny probability that the particle number of a
certain chemical species in the system changes significantly.
 To derive [Kit, p.267, (30)], Kittel establishes dG=0 [Kit, p.266, l.−1] first. However, Kittel bases on his argument on Misconception #3 to prove dG=0, see [Kit, p.262, l.−2]. At best, all Kittel can conclude is that most probably, dG=0. For a correct proof of dG=0, see [Rei, p.296, (8.3.6)].
 The contrast between an isolated system and a system in a reservoir is not
limited to thermodynamics. It can be applied to electrostatics also [Wangs,
p.104, l.10p.106, l.10].