Classical ensemble theory

Consider a system consisting of N classical particles which is in thermodynamic equilibrium with its environment. The generalized coordinate and velocity of each particle are (q1,q2,q3) and (p1,p2,p3), respectively. We denote the coordinates and momenta of all the particles by

2.1. Brief review of thermodynamics and statistical mechanics 15

q1,q2, ...,q3N and p1,p2, ...,p3N, respectively. The indexes run from 1 to 3N because the par-

ticles are described by three dimensional coordinates and velocities. Thus, these parameters ({qi,pi}) evolve with time within a 6N dimensional space called phase space. A point in the

phase space represents a state of the entire N-particle system, and is referred to as the repre-

sentative point. In other words, a representative point corresponds to a specific set of{qi,pi}

called amicrostate. Anensembleconsists of all possible microstates available to the system. The phase space may conveniently be described by a density functionρ(p,q,t), where (p,q) is an abbreviation for (q1,· · · ,q3N;p1,· · · ,p3N), so that

ρ(p,q,t)d3Nq d3Np

is the number of representative points that at timetare contained in the infinitesimal volume el- ementd3N_{q d}3N_{pof the phase space centered about the point (p},_{q). An ensemble is completely}

specified byρ(p,q,t).

Given ρ(p,q,t) at any time t, its subsequent values are determined by the dynamics of molecular motion. Let the Hamiltonian of a system in the ensemble beH (q1,· · · ,qN;p1,· · · ,pN).

The equations of motion for a system are given by ˙ pi = − ∂H ∂qi (i=1, ...,3N), (2.7) ˙ qi = ∂H ∂pi (i= 1, ...,3N). (2.8)

These will tell us how a representative point moves in phase space as time evolves.

Liouvilles’s Theorem Liouville’s theorem states that if we follow the motion of a repre- sentative point in the phase space, we find that the density of representative points in its neighborhood is constant. This implies that the volume element in generalized coordinates, i.e d3Nq d3Npis invariant under a Hamiltonian or canonical transformation. Mathematically, Liouville’s theorem is dρ dt = ∂ρ ∂t + 3N X i=1 (dρ d pi ˙ pi+ dρ dqi ˙ qi)= 0. (2.9)

The observed value of a dynamical quantityOof the system, which is generally a function of the coordinates and momenta, is supposed to be its averaged value taken over a suitably chosen ensemble: hOi= R d3Nq d3Np O(p,q)ρ(p,q,t) R d3N_{q d}3N_pρ_(p,_q,_t) (2.10)

This is called the ensemble average ofO. Its time dependence comes from that of ρ, which is governed by Liouville’s theorem.

There are three common ensembles in statistical mechanics namely thecanonical, micro-

canonicalandgrand canonicalensemble.

In the Microcanonical ensemble, the macrostate of a system is defined by the number of particlesN, volumeV, and energyE. This ensemble describes an isolated system that does not exchange energy/matter with its environment. For this ensemble, if we calculate the number of distinct microstatesΩ(N,V,E), then the phase space densityρ(q,p) can be obtained as

ρ(q,p)= 1

Ωδ(H (q,p)−E), (2.11) whereδis the Dirac delta function andH is the Hamiltonian. The Microcanonical ensemble is the most natural ensemble for MD, since in integrating Newton’s equation of motion the energy is conserved.

In physical experiments, controlling the energy E of a system is hard and we never deal with a completely isolated system. Therefore, the canonical ensemble is introduced which describes a closed system in contact with a heat bath at a constant temperature T, thus being able to exchange energy with the environment. The macrostate of the system is defined by the number of particlesN, volumeV, and temperatureT. The equilibrium phase space density can be obtained as

ρ(q,p)= 1

QN(V,T)

exp[−H(q,p)/kBT], (2.12) where QN(V,T) is the canonical partition function, and kB is the Boltzmann constant. The average value of a physical quantity hOi (averaged over the ensembles not over the time) is then hOi=hOi_ens = R O e−H(q,p)/kBT _d3N_{q d}3N_p R e−H(q,p)/kBT d3Nq d3Np (2.13) whereOens is the ensemble average. The canonical ensemble is the natural ensemble for the

Monte Carlomethod since it generates ensembles according to a canonical distribution.

A physical system represented by a grand-canonical ensemble is in equilibrium with an external reservoir with respect to both particle and energy exchange. This is an extension of the canonical ensemble, but instead the grand canonical ensemble is allowed to exchange energy and particles with its environment. The chemical potential is introduced to specify the fluctuation of the number of particles, just as temperature is introduced into the canonical ensemble to specify the fluctuation of energy.

2.1. Brief review of thermodynamics and statistical mechanics 17

Figure 2.1: (le f t) A non-ergodic system in which the trajectory does not pass through all possible states in the phase space. Theredline encases the sub-phase space of the non-ergodic trajectory. (right) An ergodic system in which the trajectory passes through all possible states in phase space.