• Canonical ensmble
• Grand canonical ensmble
• Nose-Hoover method.

Closed systems:

A closed system can exchange heat with its surroundings.
 

Chacateristics:
 

Physical description:
 

• Total number of particles fixed.
• Volume of system is fixed.
• Total energy fluctuating.
• Average energy (temperature) fixed at equilibrium.

 

Mathematical description:
 

sum P(x) = 1

< E > = sum E(x) P(x)
          = sum H(x) P(x)

where H(x) is the Hamiltonian of the system.
 

 

Canonical ensemble is a statistical description of a closed system in equilibrium:

 
 

Entropy is maximum with the condition that the number of particles and average energy is constant:

 

d  int dx [ a P(x) + b H(x) P(x) - k_B P(x) ln c^N P(x) ] = 0

 
 

From this equation, we can obtain:

 
 

P(x) = exp{-H(x)/k_BT} / c^N Z
 

Z = int dx exp{ -H(x)/k_BT }  / c^N

 

Z is the partition function. All physical quantities can be derived from it:

 
 
Free energy:
 

F = - k_BT ln Z

Entropy:
 

S = - dF / dT  (with V and N fixed).

Internal energy:
 

< E > = F + S T

Pressure:
 

P = dF / dV  (with T and N fixed)

Heat Capacity:
 

C_V = d< E > / dT = T dS / dT
                                 (with V and N fixed)

 
 
 
 
 

 
 

Open systems:

 

An open system can exchange both heat and matter with its surroundings
 

Characteristics:
 
 

Physical description:
 
 
• Total energy and particle number fluctuating.
• Average energy (temperature), pressure and everage particle number fixed at equilibrium.
 
 

           Mathematical description:
 

               < N > = sum N P(x) 

               < E > = sum E(x) P(x)
                         = sum H(x) P(x)

               where H(x) is the Hamiltonian of the system.
 

 
Grand canonical ensemble is a statistical description of a closed system in equilibrium:

 

Entropy is maximum with the condition of fixed pressure, average number of particles and average energy:
 
 

    d  int dx [ a P(x) + b H(x) P(x)  + c N P(x)
                  - k_B P(x) ln c^N P(x) ] = 0

 

        From this equation, we can obtain:
 
 

           P(x) = exp{-[H(x) - mu N]/k_BT} / c^NZ'

           Z' = int dx exp{ -[H(x) - mu N]/k_BT }  / c^N

Z' is the grand partition function, and mu is the chemical potential. All physical quantities can be derived from Z':

 
 

Free energy:
 

G = - k_BT ln Z'

Entropy:
 

S = - dG / dT  (with P and mu fixed).

Internal energy:
 

< E > = F + S T

Volume:
 

P =  -dG / dP (with T and N fixed)

Heat Capacity:
 

C_p =  - T d²G /dT²
                                 (with P and N fixed)

 
 
 
 

 

General principles:

In statistical mechanics, we deal most conveniently with canonical ensemble, which has a fixed number of particles N, fixed volume V, and fixed temperature T.

It is also possible to simulate such systems by molecular dynamics. Such algorithms introduce addition forces to the Newton's equations. The forces are introduced purely for the sake of producing correct ensemble, they are not real forces. Thus these systems are somewhat artificial and less fundamental.

There are a number of methods which realizes canonical system, one of them is to rescale velocities at each time step. Here we'll just introduce the elegant method of Nose-Hoover
 

Algorithm for isothermal molecular dynamics (canonical ensemble)

The new dynamical equation proposed by Nose and Hoover in 1984 is the following:
 
dq_i/dt = p_i/m

dp_i/dt = F(q) - z p_i

dz/dt = sum _i [ p_i²/mk_BT - 1 ] / N t'd
 

The notations p and q stand for the vector (p₁, ..., p_N) and (q₁, ..., q_N). The frictional coefficient z is a scalar variable introduced to realize the canonical ensemble. The parameter t' is an arbitrary constant, d is dimension, and N is the number of particles.

Note that if we set z = 0, we reduce the equations to Hamilton's equations of motion. The temperature T appears in the equation, which sets the equilibrium temperature of the system.

It can be shown that the canonical distribution augmented by a Gaussian distribution in the frictional coefficient:
 
 

P(p, q, z) ~ exp[ - H(p,q)/kBT - 1/2 z2 t'² ]
 
is a steady state distribution of the Nose-Hoover dynamics. This guarantees that the time average over the trajectories of the system is equal to the average over the canonical distribution.

We can rewrite the Nose-Hoover dynamics in a more familiar Newtonian form:
 

m dv_i/dt = F_i - z m v_i

dz/dt = [K/K₀ - 1] / t'²

K = sum _i 1/2 mv_i²

K0 = 1/2 NkBTd

Here K is total kinetic energy, and K₀ is average kinetic energy. This set of equations can be solved with an algorithm similar to that of Verlet as
 
r_i(t+h) = 2r_i(t) - r_i(t-h) + h² F_i(t) / m
              - z(t)/2h [r_i(t+h) - r_i(t-h)]

z(t+h) = z(t) + h/t'² [ K(t+h/2)/K₀ - 1 ]

K(t+h/2) = m/2h² sum _i ( r_i(t+h) - r_i(t) )²

The difference scheme here is also exactly time-reversible:

If you start from ri(-h), ri(0), and z(0), the system develops to ri((n-1)h), ri(nh), and z(nh), after n steps with positive h.

Now if you run the system backwards, namely, starting from ri(nh) as the previous position, ri((n-1)h) as the current position and z((n-1)h), applying the recursion relation after n steps with h negative, you get back exactly the starting values.

This cannot be true if a fourth-order Runge-Kutta method were used. This time-reversal symmetry is the key ingredient for long-time stability of the algorithm. In the usual microcanonical molecular dynamics, total energy is a constant. This is no longer so in the canonical dynamics. Nevertheless, there exists an analogous quantity which is a constant of the motion:
 
 

C = K + V + K0(t'z)2 + 2K0 int z(t'') dt''
 
This equation can be used to check the stability of integration schemes.
 
 
 

 

An isothermal-isobaric ensemble is a one with fixed number of particles N, temperature T, and pressure P₀.

Most experimental conditions are of this type.

In an isothermal-isobaric ensemble volume is a variable. The canonical distribution is replaced by:
 

P(p,q,V) ~ exp{ -[ H(p,q) + P₀ V ]/ kBT }
 

where H=K+V is total energy, P₀ is the constant pressure. In this ensemble, the logarithm of the (isothermal-isobaric) grand partition function is -G/kB T, G is Gibbs free energy.

The generalization of the Newton's equations of motion is

d² xi /dt² = F_i/mL - (z+2e) dx_i/dt

de/dt = V(P - P₀)/kBTa_V

e = 1/L dL/dt

dz/dt = [K/K₀ - 1] / a_T

K₀ = 1/2 N k_B T d
 
 

where x_i = r_i/L is the coordinate vector normalized by the length of the box; the volume of the box is V = L^d; a_V and a_T are two parameters; P and K are the instantaneous pressure and kinetic energy, defined by
 

PV = 2/d sum _i 1/2 m v_i² + 1/d sum _i<j r_ij * F_ij

K = sum _i 1/2 m v_i²

 
 
 
 

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