2. Specification of macrostates and microstates:

Macrostate: The macroscopic state of a thermodynamic system at equilibrium is specified by the values of a set of measurable thermodynamic parameters. For example, the macrostate of a fluid system can be specified by pressure, temperature and volume, . For an isolated system for which there is no exchange of energy or mass with the surroundings, the macrostate is specified by the internal energy E, number of particles N and volume V; (E,N,V). In case of a closed system which exchanges only energy with the surroundings (and no particle) and is in thermodynamic equilibrium with a heat bath at temperature T, the macrostate is given by (N,V,T). For an open system, exchange of both energy and particle with the surroundings can take place. For such systems, at equilibrium with a heat bath at temperature T and a pressure bath at pressure P (or a particle bath of chemical potential μ), the macrostate is specified by (N,P,T) (or ).

The equilibrium of an isolated system corresponds to maximum entropy , for a closed system it corresponds to minimum Helmholtz free energy , and for an open system the equilibrium corresponds to minimum of Gibb's free energy (or minimum of the grand potential ).

Microstate: A microstate of a system is obtained by specifying the states of all of its constituent elements. However, it depends on the nature of the constituent elements (or particles) of the system. Specification of microstates are made differently for classical and quantum particles.

Microstates of classical particles: In order to specify the microstates of a system of classical particles, one needs to specify the position q and the conjugate momentum p of each and every constituent particle of the system. In a classical system, the time evolution of q and p is governed by the classical Hamiltonian and Hamilton's equation of motion

For a system of N particles in 3-dimension. The state of a single particle at any time is then given by the pair of conjugate variables a point in the phase space. Each single particle then constitutes a 6-dimensional phase space ( 3-coordinate and 3-momentum). For N particles, the state of the system is then completely and uniquely defined by 3N canonical coordinates and 3N canonical momenta . These variables constitute a -dimensional Γ-space or phase space of the system and each point of the phase space represents a microstate of the system. The locus of all the points in Γ-space satisfying the condition , total energy of the system, defines the energy surface.

(a)	(b)

Figure 1.1: ( a) Accessible region of phase space for a free particle of mass m and energy E in a one dimensional box of length L. ( b) Region of phase space for a one dimensional harmonic oscillator with energy E, mass m and spring constant k.