Chap 6. Statistical ensembles

ensemble ="a collection of all possible microscopic states available & a given system under certain conditions."

[eqn: [V:fixed]]

[eqn: left {pile { microcanonical & ; & i solated ~system & E, N ~fixed # ensemble & & & ## canonical & ;& system ~i n~ contact~ with & E~ vary # ensemble & & a~ heat~ reservoir & N~fixed# & & (energy~exchange) ## grancanonical & ;& energy ~\& particle & E vary # ensemble & & exchange & N~vary } right}]

☆ Equivalence of ensembles ; For example, one can use canonical or grandcanonical ensemble to study the isolated system just for convenience.

(Relative variances of E and/or N w. r. t.(with respect to) their mean values are negligibel in the thermodynamic limit)

[eqn: {sqrt { < (△E)^2 > } } over <E> {{―――->} atop {N, V―inf } } phi ... etc]

6-1. Microcanonical ensemble

(1) Microcanonical distribution

Consider the isolated system with energy between [eqn: E]and [eqn: E+delta E] [eqn: (E/N << delta E << E ) ~,~~where ~E/N ~is ~energy~ gap]

Constraints on ensemble : V, N, E < energy <[eqn: E + delta E]

(# of microscopic states in the ensemble ∼ [eqn: O(V^N )]

* Fundamental postulate:

An isolated system in equilibrium is equally likely to be in any of its accessible states.

∀ probability of finding the system in state r is independent of r in microcanonical ensemble.

[eqn: P_r = left { pile { constant & if ~ E< E_r <E + delta E # phi& otherwise }]

with normalization [eqn: sum from r P_r =1]

[eqn: (E < E_r < E + delta E )]

sum over all accessible states with

(2) Number of accessible states : [eqn: Omega (E)]

[eqn: Omega(E)]≡ # of accessible states with energy between E and [eqn: E + delta E]

∀	[eqn: P_r = left { pile { 1 over Omega(E) & if~ E < E_r <E + delta E # phi& other wise }]

with [eqn: Omega(E) = sum from r 1.]

(sum over all accessible states with [eqn: E < E_r < E + delta E])

* mean values

[eqn: y]: a parameter like magnetic moment, velocity, ...

[eqn: y_r]: the value of [eqn: y]in a state [eqn: r].

[eqn: bar y = sum from r y_r P_r = {sum from r y_r } over Omega(E) = <y> lpile { microcanonical # e nsemble~ average }]

[eqn: Omega(E)]: essential quantity in microcanonicla ensemble.

* thermodynamics

[eqn: S = k_B ln Omega]	[eqn: TdS = dE + pdV(or~ sum from alpha bar X_alpha dx_ alpha )]

[eqn: 1 over T = left ( ∂S over ∂E right )_V , ...(Chap~3.2.1)]     □

[eqn: {bar X _alpha } over T = left ( ∂S over ∂x_alpha right )_E , ... left ( lpile { x_alpha &:& external~parameter~like~V # bar X_alpha &:& (mean) ~g e n eralized~f o r c e# & &conjugate~ t o~ x_alpha~ l i k e~ p }right )]

□

(Chap 3.3.1)

(3) Application to Ideal gas

[eqn: E = sum from i=1 to N {{vec { p_i} }^2 } over 2m]

[eqn: lpile { Omega(E) & = 1 over {h _0 ^3N } int ... int 1 cdot d^3 {vec r}_1 d^3 {vec r}_2 ... d^3 {vec r}_N d^3 {vec p}_1 ... d^3 {vec p}_N # & ~~~~~~~~~ E< sum from i=1 to N {{ vec {p_i } } ^2 } over 2m < E + delta E # & = 1 over { h_0 ^3N } V^N chi (E) }]

[eqn: lpile {where~chi ( E ) & = int ... int d^3 {vec p}_1 ... d^3 {vec p}_N # &~~~~ E < sum from i=1 to N {{vec p_i } ^2 } over 2m < E + delta E #& ∼ E^ { 3 over 2 N } ~~~(Chap 2.1.4) }]

[eqn: lpile{ ∴S & = k_B &[ N ln V &+ ln chi(E) &+ const.] # & = N k_B &[ ln V &+ 3 over 2 ln E &+ const. ] }]

[eqn: 1 over T = left ( ∂S over ∂E right )_V = 3 over 2 N k_B cdot 1 over E ~~~ ∀E = 3 over 2 N k_B T]

[eqn: P over T = left ( ∂S over ∂V right )_E = N k_B 1 over V ~~~ ∀PV = N k_B T]

But, it is not trivial to calculate [eqn: chi(E)]exactly

6-2. Canonical ensemble

(1) Canonical distribution

Consider a system in contact with a hear reservoir.

(change of [eqn: T']due to the presence of [eqn: A]is negligible because [eqn: A'>>A])

In equilibrium, [eqn: T' = T].(temp. of A) (Chap 3.2.1)

Question :

In equilibrium, what is the probability [eqn: P_r]of finding the system [eqn: A]in a state [eqn: r]?

Assume weak interactions between [eqn: A]and [eqn: A']([eqn: H_{i n t.}]negligible)

Let	[eqn: A^(0) = A + A ']: total system (isolated)

[eqn: E^(0) = E_r + E']: constant.

If A is in one definite state [eqn: r,]

# of accessible states of [eqn: A^(0) =under { Omega' ( E^(0) - E_r )}]

# of accessible states of [eqn: A']

with energy [eqn: E^(0) - E_r]

Fundamental postulate tells us that

[eqn: P_r PROPTO Omega ' ( E^(0) - E_r )] ( # of accessible states of [eqn: A^(0)]when [eqn: A]is in a state [eqn: r])

Use the fact [eqn: E_r << E^(0)]and expand the slowly varying function [eqn: ln Omega'(E')]around [eqn: E' =E^(0)].

[eqn: lpile { ln Omega' (E^(0) - E_r ) & = ln Omega' (E^(0) ) + { ∂ln Omega ' (E' ) } over ∂E' right | _ {E' = E^(0) }#&~~ -(E_r ) + ... # & = ln Omega ' (E^(0) + beta ' (E ^(0) (-E_r ) + ... }]

where [eqn: beta ' (E^(0) )]= inverse temperature of [eqn: A'] when [eqn: E' = E^(0) . left ( = 1 over { k_B T' (E^(0) ) } right )]

·[eqn: beta ' (E^(0) ) approx beta ' ( E ^(0) - E_r ) = beta ' (E')]

change of [eqn: T']due to hear change is negligible

·[eqn: beta ' (E ') = beta (E)]: inverse temp. of [eqn: A]with energy [eqn: E.]

[eqn: left ( = 1 over { k_B T} right )]

[eqn: pile { ∴ ln ( Omega ' (E^(0) - E_r ))& approx &under { ln Omega ' (E ^(0) ) } - beta E_r # & & r-i n dependent # Omega ' ( E^(0) - E_r ) &propto& e ^ { - beta E_r } }]

∴ [eqn: P_r propto e ^ { - beta E_r }] (Boltzmann factor)

with normalization [eqn: sum from r P_r = 1]

( sum over all accessible states of [eqn: A] with arbitrary energy)

(2) Partition function : [eqn: Z]

[eqn: Z == sum from r e ^{ -beta E_r }]

(sum over all accessible states with arbitrary energy)

[eqn: ∀~P_r = { e^ { - beta E_r }} over Z]

* mean values

[eqn: bar y = sum from r y_r P_r = { sum from r y_r e^{-beta E_r }} over Z = <y>]  "[eqn: pile{canonical ~ensemble# aeverage}]"

[eqn: Z]: essential property in canonical ensemble.

(easier to calculate [eqn: Z]than [eqn: Omega]: no restricted summation)

* mean energy and energy fluctuations

[eqn: lpile { <E> & = { sum from r E_r e^{-beta E_r } } over Z & = 1 over Z left ( - ∂ over ∂beta right ) sum from r e^{-beta E_r } # & = - 1 over Z ∂ over ∂beta Z & = - ∂ over ∂beta (ln Z) }]

[eqn: <(△E)^2 > = < ( E - <E>)^2 > = <E^2 > - <E>^2]

[eqn: lpile { <E^2 > & = 1 over Z sum from r E_r ^2 e^{-beta E_r } = 1 overZ ∂^2 over ∂beta^2 Z # & = ∂ over ∂beta left ( 1 over Z ∂ over ∂beta Z right ) + 1 over Z^2 left ( ∂Z over ∂beta right )^2 # & = ∂^2 over ∂beta^2 ( ln Z ) + <E>^2 # }]

[eqn: ∴~~~<(△E)^2 > = ∂^2 over ∂beta^2 (ln Z)]

[eqn: lpile{<(△E)^2 >& = - ∂ over ∂beta <E> = k_B T^2 ∂<E> over ∂T # &= k_B T^2 C_V ~~~~(V:fixed~i n ~ our~case)}]

(Fluctuation-dissipation theorem)

relative variance

[eqn: {sqrt{<(△E)^2 > }} over <E> ∼ {sqrt{N}} over N = 1 over sqrt{N} {――-> }atop {N-> inf} phi] [eqn: left (lpile { &e q u i v a l e n c e~ t o #&microcanonical #&ensemble }right)]

* thermodynamics

Consider a single external parameter [eqn: x]

[eqn: □W = bar X dx] when [eqn: bar X = bar { - left ( ∂E_r over ∂x right )}] (Chap 2.2.3)

[eqn: lpile { bar X &= - { sum from r left ( ∂E_r over ∂x right ) e^{-beta E_r } } over Z= 1 over beta { ∂ over ∂x left ( sum from r e^{-beta E_r } right )} over Z # & = 1 over beta ∂ over ∂x (ln Z ) }]

{In general, [eqn: bar X_alpha = 1 over beta ∂ over ∂x_alpha (ln Z ) ~[ <P> = 1 over beta ∂over ∂V ( ln Z ) ]]}

[eqn: lpile { TdS & = □W + dE#&= 1 over beta ∂over ∂x (ln Z ) dx + dE }]

[eqn: lpile { dS &= k_B left [ ∂ over ∂x ( lnZ) dx + beta dE right ] # & = k_B left [ d(ln Z) under {- ∂ over ∂beta (ln Z ) } d beta + beta dE right ] }]

(Here, [eqn: E] means [eqn: <E>].)

[eqn: lpile { dS & = k_B [ d(lnZ) + d(beta E ) ]# & = k_B d( lnZ + beta E )}]

[eqn: ∴ ~~S = k_B ( ln Z + beta E )]

additional constant?

Consider [eqn: T -> 0^+]limit

[eqn: Z = sum from r e^{-beta E _r } {――-> } atop {( beta -> inf )} Omega_0 e^{-beta E_r }]

, when [eqn: E_0]= lowest possible value of [eqn: E_r]

(ground-state energy)

and [eqn: Omega_0]= # of ground states

(degeneracies)

[eqn: E = - ∂ over ∂beta (ln Z) -> E_0]

[eqn: ∴S = k_B [ ln Omega_0 - beta E_0 + beta E_0 ] = k_B ln Omega _0]

∀ there is no additional constant.

·easy derivation

[eqn: Z = sum from r e^{-beta E_r } = sum from E Omega(E) e^{-betaE}]

[eqn: ∴ Z approx Omega ( <E>) e^{-beta<E>}]

[eqn: ∀~~ln Z = ln Omega - beta <E>]

[eqn: ∀~~S = k_B ln Omega = k_B [ln Z + beta <E> ]]

[eqn: 1 over T = left ( ∂S over ∂E right )_V = k_B beta ~~~~correct!]

[eqn: {bar X} over T = left ( ∂S over ∂x right )_E = k_B ∂ over ∂x (ln Z)]

[eqn: ∀~~bar X = 1 over beta cdot ∂over ∂x (ln Z) ~~~~~~correct!]

. Helmholtz free energy

[eqn: F = E - TS= -k_B T ln Z = - 1 over beta ln Z]

[eqn: E = - ∂over ∂beta ln Z = ∂over ∂beta (beta F )]

[eqn: bar X = 1 over beta ∂ over ∂x ln Z = - ∂ over ∂x F~~left ( p = - ∂F over ∂V right )]

↑

[eqn: dF = -SdT - pdV]

(3) Applications

① Ideal gas

[eqn: lpile{ Z & = 1 over {h_0 ^3N } int cdot cdot cdot int e^{ - beta sum from i=1 to N {vec p_i}^2 over 2m } d^3 {vec r}_1 ...d^3 {vec r}_N d ^3 {vec p}_1 d^3 {vec p}_N # & = 1 over {h_0 ^3N } V^N int d ^3 {vec p}_1 e^{ - beta {vec p_1}^2 over 2m } ... int d^3 { vec p}_N e^{ - beta {vec p_N}^2 over 2m } # & = 1 over {h_0 ^3N } V^N left [ int d^3 vec p e^{ - beta {vec p}^2 over 2m } right ] ^N #& =V^N C^N (beta ) }]

[[eqn: C(beta)]will be calculated in next chapter]

[eqn: F = - 1 over beta ln Z = - 1 over beta [ N ln V + N ln C(beta) ]]

[eqn: p= - ∂F over ∂V = N over beta 1 over V~~∀~pV = Nk_B T]

[eqn: E \& S] can be obtained if [eqn: C(beta)]is known.

non-interacting paricles

[eqn: lpile {∀Z=(Z_a )^N~where~ Z_a &= 1 over {h_0 ^3} int int e^{- beta {{vec p} ^2} over 2m } d^3 vec r d^3 vec p # & =V cdot C(beta) }]

[eqn: ∀]Free energy per a particle

[eqn: F_a = - 1 over beta ln Z_a = - 1 over beta [ ln V + ln C(beta) ]]

[eqn: F = NF_a]

[eqn: cdot cdotcdot p, E, S, cdotcdotcdot( i n~the~same~way)]

② Paramagnetism

[eqn: N] magnetic atoms with magnetic field [eqn: H] applied.

magnetic atom ; spin [eqn: 1 over 2] and magnetic moment [eqn: vec mu]

magnetic energy of an atom ;

[eqn: epsilon = - vec mu cdot vec H]

[eqn: {atomic~ state } atop □ ; left { pile{ either & p arallel& t o& vec H &;& S &=& +1 # or & anti par a llel & t o & vec H &;& S &=& -1 }]

↑

state variable

[eqn: {∀~~epsilon =} atop ⇒ left{ lpile { -mu H & for~s=1 # +mu H & for ~s=-1 } right } = - mu Hs]

[Assume that there is no interactions between atoms]

Total energy

[eqn: E = sum from i=1 to N ( - mu H s_i )]

[eqn: lpile { Z & = sum from { lpile { all~ possible # states~ r }} e^{- beta E_r} = sum from { s_1 +- 1 } sum from { s_2 +- 1 } ... sum from { s_N +- 1 } e ^{beta mu H sum from i=1 to N s_i } # &= sum from {s_1 = +-1} e^{beta mu H s_1 } cdot sum from {s_2 = +-1} e^{beta mu H s_2 } cdot ... sum from {s_N = +-1} e^{beta mu H s_N } # & =left ( sum from {s = +-1} e^{beta mu H s } right )^N = [2 cosh(beta mu H) ] ^ N }]

· magnetization

[eqn: lpile{ M & = < sum from i=1 to N mu s_i > = 1 over Z sum from {\{ s \} } left [ mu cdot left ( sum from i=1 to N s_i right ) e^{ beta mu H sum from i=1 to N s_i } right ] # & = 1 over Z 1 over beta ∂ over ∂H Z = 1 over beta ∂ over ∂H ( ln Z) = - ∂ over ∂H F }]

( [eqn: H]: external parameter [eqn: M]: conjugate force)

[eqn: dF = - sdT - M dH]

[eqn: F = - 1 over beta ln Z = - N over beta ln [ 2 cosh ( beta mu H ) ]]

[eqn: M = - ∂F over ∂H = N mu tanh ( beta mu H )]

· magnetic susceptibility

[eqn: chi = left ( ∂M over ∂H right )_T = N mu ^2 beta sech ^2 ( beta mu H)]

zero-field susceptibility

[eqn: chi _0 = chi | _{ H-> 0^+ } = M mu ^2 beta] ( finite for [eqn: beta ne inf])

< linear slope>

∀ like in the ideal gas, non-interacting atoms.

[eqn: Z_a = sum from {s = +- 1 } e^ { beta mu H s }]	and   [eqn: Z = ( Z_a ) ^N]

magnetization for an atom

[eqn: lpile {<mu s > &= 1 over beta ∂ over ∂H (ln Z_a ) = - ∂F_a over ∂H # & = mu tanh(beta mu H)}]

∴ [eqn: M = N <mu s> = N mu tanh ( beta mu H )]

6-3 Grandcanonical ensemble

similar to canonical ensemble except particle exchange

[eqn: E, N]: not fixed.

Question :

In equilibrium, what is [eqn: P_r]for a given [eqn: E_r , N_r]?

[eqn: lpile { E^(0) & = & E_r & + & E'# N^(0) & = & N_r & + &N' }]

[eqn: P_r PROPTO Omega ' ( E^(0) - E_r , N^(0) - N_r )]

[eqn: lpile { ln Omega ' ( E^(0) - E_r , N^(0) - N _r )& □ ln Omega ' (E^(0) , N^(0) ) #&- { ∂ ln Omega' (E' , N' ) } over ∂E' right | _ { lpile { E' = E^(0) # N' - N^(0) } } E_r #&- { ∂ ln Omega ' } over ∂N' right | _ { lpile { E'=E^(0) # N' = N^(0) }} N_r }]

[eqn: ∴ P_r propto e^{- beta E_r - 2 N_r } ~~left [ alpha ={ ∂ ln Omega' } over ∂N' right | _(0) right ]]

H. W. 1, 4, 7, 11