Chap 3. Statistical thermodynamics

3-1. Irreversibility and Equilibrium

□□ of isolated system in equilibrium

□ depends on "constraint" which are specified by some macroscopic parameters (external parameters and any other conditions in 2-2.),

{y1, y2, ... ,yn}	(eg. yi=V, H, E, ...)

[eqn: therefore Omega = Omega (y_1 , y_2 , ... , y_n )]

= # of accessible states with [eqn: y_i ↔ y_i + delta y_i] (i = 1, ... , n)

□removal of some constraint

# of accessible states must increase or be unchanged.

[eqn: therefore Omega _f >= Omega _i]

In the final equilibrium situation, systems in ensemble are distributed over all possible □f states. If [eqn: Omega _f >> Omega _i], it is very unlikely that the initial distribution of systems is restored after removal of constraints.

[its probability [eqn: P _i = {Omega _i } over Omega_f]]

Consider only one parameter y.

With constraint n y, initially y=yi.

After removal of constraint, y can take any value.

Usually, □(y) has very pronounced maximum at some value .

∀ In final equilibrium situation, practically all systems will have values of y very close to .

If some constraints of an isolated system are removed, the parameters tend to readjust themselves to minimize [eqn: Omega (y_1 , ... , y_n )]

□irreversibility

□[eqn: Omega _f > Omega _i] ∀ redistribution of systems in ensemble until □f states are equally likely. By putting back constraints on the system, the systems cannot return to initial distributions over □i states only.

(eg. free exp.) ∀ irreversible process and, during the process, the system is in non-equilibrium

□[eqn: Omega _ f = Omega _i ~]∀ no redistributions of systems and all states are always equally likely.

∀ during the process, the system is always in equilibrium and reversible process.

3-2 Thermal interactions

(1)Entropy and temperature

Consider two macroscopic system A and A' which are initially separated in equilibrium and isolated. Suppose their respective mean energies are  = Ei and ' = Ei'. Then, make thermal contacts between them so that energies can be exchanged.

ⓐ equilibrium property

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

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

(*weak interaction assumptions : [eqn: H _ { i n t }]negligible )

What value does E assume in equilibrium ?

Let [eqn: Omega ^(0) (E )]= # of accessible states of [eqn: A^(0)] when A has an energy between [eqn: E ~ and ~ E + delta E``].

∀ "Fundamental postulate " tells us that

[eqn: bold P(E)] = probability that A has an energy between [eqn: bold {E ↔ E + delta E ```}][eqn: bold{ PROPTO Omega ^(0) (E)````}]

[eqn: Omega ^(0) (E) ~=~under Omega(E) ~cdot ~ under {Omega'(E^(0) - E )}~~]

[eqn: Omega(E)]       =  # of accessible states of system A when A has an energy between [eqn: E ↔ E + delta E``].

[eqn: Omega'(E^(0) - E )]   =  # of accessible states of system A' when A'  has an energy  between [eqn: E ^(0) - E] ↔ [eqn: E^(0) - E + delta E``]

[eqn: therefore P(E ) = C Omega(E) Omega' (E^(0) - E )``] where C is a normalization constant.

□(E) is the extremely rapidly increasing function of E.

< remember [eqn: Omega (E ) ∼ E^f ``]>

As E increases, □ □(E) → fast increasing

□ [eqn: Omega ' (E^(0) - E) ~→ ~~fast~~ decresing]

∀  assumes  and ' assumes [eqn: E^(0) - tilde E```]in equilibrium

[Mean value]

□ How to locate [eqn: tilde E ~~]?

[eqn: {∂ l n P } over ∂E right|_ {E= tilde E} = 1 over P ∂P over ∂E right | _ {E = tilde E } = phi]

[eqn: ln P = ln C + ln Omega (E) + ln Omega' &(under{E ^(0) - E} ) #&E' = E^(0) - E~]

[eqn: therefore ~~0 = {∂ln Omega(E)} over ∂E - {∂ ln Omega' (E') } over ∂E' right| _ {lpile {E={tilde E } ~,~ E' = tilde E ' = E^(0) - tilde E } }]

Define [eqn: beta (E) == {∂ ln Omega(E) } over ∂E``]

∀  where [eqn: tilde E ' = E^(0) - tilde E~]

[eqn: beta (tilde E ) = beta ' ( tilde E' )]

□Temperature T

□ → unit of inverse energy.

Define [eqn: k_B T == 1 over □] ([eqn: k_B]: Boltzmann constant)

[eqn: therefore equilibrium~is ~ ob tained ~ when~ T = T'. ( or ~ □ = □' )]

□entropy S

Define [eqn: S == k_B ln Omega]

[eqn: 1 over T = ∂S over ∂E~]

∀

[eqn: ln P = ln C + 1 over k_B [ S + S' ]] ; maximized in equilibrium

∀ S + S' is maximized in equilibrium

∀

total entropy is maximized in equilibrium.

derive T = T'

ⓑ approach to equilibrium

In final equilibrium,

[eqn: bar E = tilde E ( == bar E _F ) ~ and ~ bar E ' = E^(0) - tilde E (== bar E _f ' )]

[eqn: S (bar E _f ) + S ' (bar E ' _f ) >= S( bar E _i ) + S' (bar E '_i )~]

[eqn: left { lpile { △ S = S (bar E _f ) - S (bar E _i ) ~&:&~ e n t ropy ~ ch a n g e ~ of ~ A # △ S' = S' (bar E ' _f ) - S' (bar E ' _i ) ~&:&~ e n t ropy ~ ch a n g e ~ of ~ A' }]

[eqn: therefore ~ △ S + △ S' ge phi] ( total entropy increases !! but each may decrease or increase)

And [eqn: bar E _f + bar E ' _f = bar E _i + bar E ' _i]

[eqn: left { lpile { Q & = & bar E _f - bar E _i & : & hear ~ absorbed ~ by ~ system~ A # Q' & = & bar E ' _f - bar E ' _i & : & hear ~ absorbed ~ by ~ system~ A' }]

[eqn: ∴ Q + Q' = phi]

If Q > 0 , we call A colder and A' hotter.

(Q'<0 )         system which absorbs hear.

∀ hear flows from hot to cold.

ⓒ temperature

[eqn: □ = 1 over {k_B T } = { ∂ ln Omega } over ∂E ``]

(i) □ : increasing function of E (ordinarily)

∀ □ > 0 and T > 0

(ii) □ is roughly ∼ Ef

∀ [eqn: k_B T ∼ E over f ~~] : rough measure of the mean energy per degree of freedom.

(iii) cold and hot?

Q : infinitesimal > 0   [Q' = -Q ]

[eqn: lpile {△ S + △ S' &= S( bar E _ f ) - S ( bar E _ i ) + S ' ( bar E ' _f ) - S' ( bar E ' _i )`` # &= S( bar E _i +Q ) - S ( bar E_i ) + + S' ( bar E ' _i - Q ) - S' ( bar E ' _i ) # &= Q ∂S over ∂E right |_{E= bar E_i }-Q ∂S' over ∂E' right |_ {E'=bar E' _i} # &= Q over T_i - Q over T'_i ge phi # & ∴~~~T_i < T'_i ~~∀~~left { lpile { c o l d & ; & l o w ~T# h o t & ; & h i g h ~T } right } }]

(2) Heat reservoirs

heat reservoir (heat bath)

[eqn: A' >> A]

(many more degrees of freedom)

Temperature change of heat reservoir (A') is almost negligible even if A and A' change heat because A' is much bigger than A.

Change in [eqn: ln Omega '] by absorbing heat Q'

[eqn: = ln Omega ' (E' +Q') - ln Omega'(E')]

[eqn: = {∂ ln Omega ' } over ∂E' Q' + 1 over 2 { ∂^2 ln Omega' } over ∂E'^2 Q'^2 + ...]

[eqn: = □' Q' + 1 over 2 under{∂□' over ∂E'} Q '^2 + ...]

temperature change of A' is negligible.

∴ Entropy change of heat reservoir :

[eqn: △ S ' = k_B [ ln Omega ' (E' +Q') - ln Omega '(E') ]~]

[eqn: △ S' = Q' over T']

Any system at T which absorb an infinitesimal amount of heat [eqn: □ Q~] :

the entropy change of this system after absorbing heat

[eqn: lpile { dS & = K_B [ln Omega (E + □Q ) - ln Omega(E)] # & = k_B left [ {∂ ln Omega } over ∂E □Q + O (□Q)^2 right] #& = k_B □ □q = □Q over T # & ~~∴ dS = □Q over T }]

(3) Sharpness of the prob. distribution P(E)

Skip     (각자공부)

3-3. General interactions

(1) Generalized force and entropy

Consider a single "external: parameter x only

[eqn: Omega = Omega (E, x)~]

Now, [eqn: d~ ln Omega = under { {∂ ln Omega } over ∂E } dE + under {{∂ ln Omega } over ∂x} dx~]

=□            ?? generalized force

Consider [eqn: △ Omega = left( ∂Omega over ∂x right ) dx ``]	["Not" increase of # of accessible states by change [eqn: x -> x +dx``]because total energy E also changes as [eqn: x -> x +dx``]. So E must be fixed.]

[eqn: △ Omega = Omega (E, x+dx ) - Omega (E, x)~]

△□ = ( # of microscopic states entering the range [E ↔ E + □E] - # of microscopic states leaving this range ) when [eqn: x -> x +dx``]

Consider a particular state r with energy Er(x).

when [eqn: x -> x +dx``], [eqn: d E_r = E_r (x + dx ) - E_r (x) = ∂E_r over ∂x dx ~~]

[eqn: = under {- X _r dx }] : energy shift.

equilibrium ensemble average : [eqn: dE = - under { bar X } dx ~]

mean generalized force.

# of microstates entering the range at E :

[eqn: Omega (E) cdot dE over deltaE = - bar X Omega dx over deltaE == sigma (E)~]

# of microstates leaving the range at E + □E

[eqn: sigma (E + delta E)~]

[eqn: ∴ ~~△ Omega = sigma (E) - sigma (E + delta E ) = - ∂sigma over ∂E delta E ~~]

[eqn: = ∂ over ∂E ( bar X Omega ) dx ~]

[eqn: ∀ ∂Omega over ∂x = ∂ over ∂E ( bar X Omega ) = ∂barX over ∂E Omega + bar X ∂Omega over ∂E~]

[eqn: ∀ {∂ ln Omega } over ∂x = under { {∂ bar X } over ∂E} + under {bar X {∂ ln Omega } over ∂E }~]

[eqn: O left ( barX over E right)]   [eqn: O left( f barX over E right )]     [eqn: ( BECAUSE Omega ∼ E ^f )]

∴  ( or [eqn: ∂S over ∂x = barX over T~])

[eqn: {∂ ln Omega } over ∂x = □ bar X ~]

many external parameters { X1, X2, ..., Xn}

([eqn: bar X_alpha] : mean generalized force conjugate to x□)

[eqn: {∂ ln Omega } over ∂x_alpha = □ bar X_alpha ~]

Infinitesimal quasistatic process

[eqn: lpile{ d ln Omega &={∂ln Omega}over ∂E dE+ sum from alpha=1 to n {∂ln Omega}over ∂x_□ d x_□# & = □ [ dE + sum from alpha=1 to n bar X_alpha d x_□ ] # & = □ [ d E + □W ] = □□Q }]

∀

[eqn: □ Q = T dS = dE + □W~]

(2)Equilibrium conditions

□thermal interaciton

∀ entropy maximized ∀ □ = □'

□general interaction

[eqn: A^(0) &= &A + A ' & : & i s olated~]

[eqn: E^(0) & = & E + E ' & : & constant~]

{X□} and {X'□} are not free to each other  (eg. V + V ' = V(0) : constant)

[eqn: Omega ^(0) ( E , \{ x _ alpha \} )] = # of accessible states of A(0) when A has an energy between E↔E+□E and external parameters {x□}

[eqn: Omega ^(0)~] ∀ product of □ and □' ∀ sharp maximum at [eqn: E = tilde E ~ and ~ x_□ = tilde x_ □ ~~]

∀ [eqn: left ( {∂ ln Omega } over ∂E -> right ) □ = □' ~]and

[eqn: left ( {∂ ln Omega } over ∂x_□ -> right ) under { bar X _ □ = bar X _ alpha '}~]

(example) single parameter x = V

[eqn: lpile { E^(0) = E + E' # V^(0) = V + V' }]

[eqn: Omega ^(0) (E, V ) = Omega (E, V ) cdot Omega ' (E', V' )]

[eqn: ln Omega^(0) = ln Omega + ln Omega ' ~~~~[ S^(0) = S + S ' -> m a x i m ized ~]~]

[eqn: d ln Omega ^(0) = phi~]

∀ [eqn: lpile { ~#~# +#~ }] [eqn: lpile {d ln Omega & = & □ [dE + PdV ] # d ln Omega ' & = & d ln Omega ' ( E^(0) - E , V^(0) - V ) # & = & - □ [ d E + P' d V ] #&# d ln Omega ^(0) & = & ( □ - □' ) dE + (P-P')dV =phi```` }]

□ = □'  and P = P'

∴

(3) Entropy

ⓐ exact differential

□Q : not exact differential (like □W)

[eqn: int from i to f □Q~~~,~~int from i to f □W ~~:~~path-dependent~~~]

But dS : exact differential

[eqn: int from i to f dS = S_f - S_i] : path-independent

[eqn: dS = □Q over T~] (quasi-static process)

∀   [eqn: S_f - S_i = int from i to f dS = int from i to f □Q over T~] : path-independent

(eq.) "quasi-statically" [[eqn: 1 over T]: integrating factor]

ⓑ uniqueness

[eqn: S = k_B ln Omega]

In classical mechanics, cell size [eqn: h_0~]is arbitrary

∀ NOT unique

In Quantum mechanics, cell size [eqn: h_0 = hbar~]

∀ unique

ⓒ limiting behavior

T → 0 ; E approaches its ground-state value [eqn: E_0]

# of states at [eqn: E_0] : [eqn: Omega_0]= 1

or  = O(1) : degenerated.

Usually [eqn: Omega ∼ E^f ~~∀ ~~S ∼f~]

But, even in degenerated case,  [eqn: S ∼ O(1) << f~]

So, the entropy is vanishingly small [S/f → 0]

∴ As T → 0 ,  S → 0.

However, for very small T, the entropy of the system can be finite due to degrees of freedom of nuclear spins which can be aligned by very very small interaction energy. So, even for very small T, nuclear spins are still random.

∀ As T → O+, S → S0

finite, independent of all parameters for atoms.

3-4 Thermodynamic laws

□ 0th law :

[eqn: lpile{ A&↔&B&: e q u i l. &(□_A& = &□_B ) # B&↔&C&: equil. &(□_B& = &□_C ) } right]]

∀ [eqn: A↔C ~:~ equil.~ (□_A = □_A )]

□ 1th law : (Energy conservation)

Q = △E + W

[eqn: left { lpile { Q&:& heat~absorbed~ under{by} ~the ~system # W&:& work~done~under by ~ the~system# △E& : & chante~of~i nternal~energy }]

quasi-static process :

□Q = dE + □W

□ 2nd law : (Entropy)

isolated system : △ S ≥ 0	(entropy tends to increase)

(equal sign for reversible process)

not isolated and quasi-static process:

[eqn: dS = □Q over T]

□3rd law : As T →0+, S → S0 (constant)

* microscopic relation

[eqn: S = k_B ln Omega~]

∀ In equilibrium, the probability of finding an isolated system specified by {y1, ..., yn} is

[eqn: P PROPTO Omega ( y_1 , ... , y_n ) propto e^{S/k_B~}~]

* Combining 1st & 2nd law

∀	[eqn: T dS = dE + □W = dE + sum from □ bar X _ □ d x_□~]

[eqn: S = S (E; x_1 , ... , x_n )~]

[eqn: dS = left ( ∂S over ∂E right ) dE + sum from alpha=1 to n left ( ∂S over ∂x_□ right ) dx_□ ~]

[eqn: 1 over T = ∂S over ∂E ~~ and ~~ bar X_□ over T = ∂S over ∂x_□~]

∴                                        (can be derived)

(example) Ideal gas

[eqn: Omega PROPTO V^N chi (E)~~∴ S = k_B [ N ln V + ln chi (E) + ln C ] ~]

[eqn: left { lpile { P over T = ∂S over ∂V = Nk_B over V&∀& under{PV = N k_B T} # && equation~of ~state# 1 over T = k_B {∂ ln chi(E) } over ∂E &∀ & E = E(T)~~(n o t~func. ~ of ~V )# }]

(H. W.) 1, 2, 3, 4, 5