Elements of Ensemble Theory

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Elements of Ensemble Theory

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Ensemble: An ensemble is a large collection of systems in different microstates for the same macrostate (N,V,E) of the given system.

• An ensemble element has the same macrostate as the original system (N,V,E), but is in one of all possible microstates.
• A statistical system is in a given macrostate (N,V,E), at any time t, is equally likely to be in any one of a distinct microstate.

Ensemble theory: the ensemble-averaged behavior of a given system is identical with the time-averaged behavior.

2.1 Phase space of a classical system

• Consider a classical system consisting of N-particles, each described by (x_i,v_i) at time t.
• A microstate at time t is

(x₁,x₂,…,x_N; v₁,v₂,…,v_N), or

(q₁,q₂,…,q_3N; p₁,p₂,…,p_3N), or

(q_i, p_i) - position and momentum, i=1,2,…..,3N

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• Phase space: 6N-dimension space of (q_i, p_i).

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Representative point

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• Representative point: a microstate (q_i,p_i) of the given system is represented as a point in phase space.

• An ensemble is a very large collection of points in phase space Ω. The probability that the microstate is found in region A is the ratio of the number of ensemble points in A to the total number of points in the ensemble Ω.

Hamilton’s equations

• The system undergoes a continuous change in phase space as time passes by

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• Trajectory evolution and velocity vector v
• Hamiltonian

i=1,2,…..,3N

Hypersurface

• Hypersurface is the trajectory region of phase space if the total energy of the system is E, or (E-Δ/2, E+Δ/2).

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H(q_i,p_i) = E

• e.g. One dimensional harmonic oscillator

Hypershell (E-Δ/2, E+Δ/2).

H(q_i,p_i) = (½)kq² + (1/2m)p² =E

Ensemble average

• For a given physical quantity f(q, p), which may be different for systems in different microstates,

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where

d^3Nq d^3Np – volume element in phase space

ρ(q,p;t) – density function of microstates

Microstate probability density

• The number of representative points in the volume element (d^3Nq d^3Np) around point (q,p) is given by

ρ(q,p;t) d^3Nqd^3Np

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• Microstate probability density:

(1/C)ρ(q,p;t)

• Stationary ensemble system: ρ(q,p) does not explicitly depend on time t. <f> will be independent of time.

2.2 Liouville’s theorem and its consequences

• The equation of continuity

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At any point in phase space, the density function ρ(q_i,p_i;t) satisfies

So,

Liouville’s theorem

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From above

Use Hamilton’s equations

where

Consequences

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For thermal equilibrium

• One solution of stationary ensemble

where

(Uniform distribution over all possible microstates)

Volume element on phase space

Consequences-cont.

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• Another solution of stationary ensemble

A natural choice in Canonical ensemble is

satisfying

2.3 The microcanonical ensemble

• Microcanonical ensemble is a collection of systems for which the density function r is, at all time, given by

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If E-Δ/2 ≤ H(q,p) ≤ E+Δ/2

otherwise

• In phase space, the representative points of the microcanonical ensemble have a choice to lie anywhere within a “hypershell” defined by the condition

E-Δ/2 ≤ H(q,p) ≤ E+Δ/2

Microcanonical ensemble and thermodynamics

• Γ – the number of microstates accessible;

ω – allowed region in phase space;

ω₀ – fundamental volume equivalent to one microstate

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• Microcanonical ensemble describes isolated sysstems of known energy. The system does not exchange energy with any external system so that (N,V,E) are fixed.

Example – one particle in 3-D motion

• Ηamilton: H(q,p) = (p_x²+p_y²+p_z²)/(2m)

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• Microcanonical ensemble

If E-Δ/2 ≤ H(q,p) ≤ E+Δ/2

otherwise

• Fundamental volume, ω₀~h³

• Accessible volume

2.4 Examples

1. Classical ideal gas of N particles

a) particles are confined in physical volume V;

b) total energy of the system lies between E-Δ/2 and E+Δ/2.

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2. Single particle

a) particles are confined in physical volume V;

b) total energy of the system lies between E-Δ/2 and E+Δ/2.

3. One-dimensional harmonic oscillator

2.4 Examples

1. Classical ideal gas of N particles

• Particles are confined in physical volume V
• The total energy of system lies between (E-Δ/2, E+Δ/2)

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Hamiltonian

Volume ω of phase space accessible to representative points of microstates

Examples-ideal gas

The fundamental volume:

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• The multiplicity Γ (microstate number)

* A representative point (q,p) in phase space has a volume of uncertainty , for N particle, we have 3N (qi,pi) so,

and

Example-single free particle

2. Classical ideal gas of 1 particles

• Particle confined in physical volume V
• The total energy lies between (E-Δ/2, E+Δ/2)

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Hamiltonian

Volume of phase space with p< P=sqrt(2mE) for a given energy E

Examples-single particle

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• The number of microstates with momentum lying btw p and p+dp,

• The number of microstates of a free particle with energy lying btw ε and ε+dε,

where

Example-One-dimensional simple harmonic oscillator

3. Harmonic oscillator

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Hamiltonian

Solution for space coordinate and momentum coordinate

Where k – spring constant

m – mass of oscillating particle

Example-One-dimensional simple harmonic oscillator

• The phase space trajectory of representative point (q,p) is determined by

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With restriction of E to

E-Δ/2 ≤ H(q,p) ≤ E+Δ/2

• The “volume” of accessible in phase space

Example-One-dimensional simple harmonic oscillator

• If the area of one microstate is ω₀~h

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The number of microstates (eigenstates) for a harmonic oscillator with energy btw E-Δ/2 and E+Δ/2 is given by

So, entropy