Chap 2. Statistical description of many-body systems

How to understand many-body systems?

① State <microscopic>	[eqn: <2 ~-~ coin~tossing># left {lpile{ & 1&2&state ~\# # & ↑ & ↑ & 1 # & ↑ & ↓ & 2 # & ↓ & ↑ & 3 # & ↓ & ↓ & 4 } right }]

② Statistical ensemble

- a collection of many

experiments(randomness) = { 1, 2, 1, 3, 4, 2, 4, ... }

- relative probabilities ?

[eqn: p(1), ~ p(2),~p(3),~ p(4)]

③ Postulates

about relative probabilities	(eg.)no preference

between two faces

p(up) = p(down)

∀ p(1) = p(2) = p(3) = p(4) = 1/4

equally likely

④ Probability theory prob. that the same faces are up

∀ results !!	= p(1) + p(4) = 1/2

2-1. statistical formulation

(1) Specification of the microscopic state of a system

□quantum state

wave function [eqn: Psi (q_1 , q_2, ... , q_f )]

[eqn: left { lpile { q_i & : & g e n eralized ~ coordinates~i n cluding~spin~variables# f&:& \# ~ of ~ d e grees ~ of ~ freedom }]

f quantum numbers [eqn: pile { ――-> # specify }] a quantum state.

(ex1)	N particles fixed in position with spin 1/2

spin → magnetic quantum number [eqn: M = +- 1over 2]

[eqn: \{ m_1 , m_2 , ... , m_N \} ~pile {――-> # spacify } ~a~ microstate]

(ex2)	N non-interacting 1-dimensional S. H. O.

S. H. O. → [eqn: &E_n = (n + 1 over 2 ) hbar omega # &\{ n ~:~ quantum ~ n umber~ =~ 0,~1,~2,~...~ \}]

[eqn: therefore \{ n_1 , n_2 , ... , n_N \} -> ~a ~ microstate]

□ classical description

Classical approximation sometimes allow us to study a given system more easily.

-simplest case : one-particle system in one dimension

< point particle, no spin, ...>

∀ (p, q) determines the system completely.

∀ a state ↔ (p,q) ↔ a point in phase space.

<phase space>

∀ continuous

∀ uncountable # of states

Divide the phase space in to small cells

such as [eqn: delta p cdot delta q == h_0]

→ Discrete → countable # of states.

[eqn: delta p cdot delta q == h_0] → [eqn: phi~] ∀ specification of a state becomes more precise!

But, the uncertainty principle of QM tell us that

[eqn: h_0 >= hbar = left ( h over 2pi right )~~~h =~ :~ Plank's~ constant]

So, when the system is treated fully in QM, one should take h0 finite.

Division of the phase space is "essential" (not fictitious)

- N particle system in 3 dimensions.

[eqn: left { lpile { q_1 , & q_2, & ...~, & q_f # p_1 , & p_2, & ...~, & p_f }right }~ <-> ~a ~ microstate]

, where f = 3N

cell size = [eqn: delta q_1 ~ delta q_2~ ...~ delta q_f ~ delta p_1 ~ delta p_1 ~...~ delta p_f = h_0 ^f~~]

□ macroscopic state (macrostate)

defined by macroscopic parameters like pressure, volume, temperature, internal energy, ...

(2) Statistical ensemble

: a collection of many experiments in similarly prepared systems subject to whatever conditions are specified as known.

□ Consider a simple system consisting of three spin -1/2 fixed point particles in magnetic field H in z-direction

- possible states? [□ : magnetic moment in z-direction ]

23 state   (each particle has two states [eqn: m_i = + 1over 2 , - 1 over 2])

[eqn: pile { state ~ i ndex & state & total ~magnetization & total ~energy ~E~~ # &&& <Zeeman ~energy>~~ # 1& +~+~+ & 3mu &-3 mu H # 2& +~+~- & mu & -mu H # 3& +~-~+ & mu &- mu H # 4& -~+~+ & mu & -mu H # 5& +~-~- & -mu & mu H # 6& -~+~- & -mu & mu H # 7& -~-~+ & -mu & mu H # 8& -~-~- & -3mu & 3mu H }]

We usually do not know the details of a given system, but sometimes know "partial" information like total energy, volume, total magnetization, ... etc.

(mostly macroscopic parameters)

With fixed values of these parameters, only part of possible states are "accessible" to the system.

eg. E=-□H fixed

∀ only 3 states out of 9 [2, 3, 4] are accessible to the system.

Statistical ensemble of this system

={ 2, 3, 3, 2, 4, 3, 2, ... ... ...}

What are relative probabilities of these accessible states(2, 3, 4)??  p(2), p(3), p(4)

DO NOT KNOW A PRIORI!

CAN BE DETERMINED BY EXPERIMENTS!

(3) Fundamental postulate on relative probabilities.

Consider an isolated system ∀ total energy E is constant in time.

→ still great many microstates accessible to the system.

→ relative probability of these microstates??

□Simple case - "equilibrium"

equilibrium :

The probability of finding the system in any one of its accessible states is independent of time

p(2), p(3), p(4) : constant of time.

∀ all macroscopic parameters are also constants of time.

isolated system in equilibrium :

only information → the system is in any one of its accessible states with constant energy.

∀ 'reasonable' to assume that the system is "equally likely" to be found in any one of its accessible states!! (equal probabilities)

p(2) = p(3) = p(4) = 1/3

Question : equally likely postulate ∃ equilibrium

( independent of time)

Answer : Liouville's theorem tells you that if the accessible states are equally likely at the beginning, then the probabilities of finding the system in any of its accessible states do not change in time.

i.e., equally likely → equilibrium

but we do not know whether equally likely probability distribution is	a unique probability distribution in equilibrium.

<proof of the answer>

Consider an isolated classical system of one particle in one dimension

state ↔ (q, p)

[eqn: rho (q, p ; t) dq dp] = # of systems in the ensemble with  [eqn: q <-> q + dq~] and [eqn: p<-> p + dp~] at time t

[ □ : number density ]

Classical equation of motion.

[eqn: dot { q} \ ∂H over ∂p ~ and dot p = - ∂H over ∂q ~~[H(q, p) : Hamiltonian]]

In time dt, the change in the number of systems within this volume

[eqn: == left [ ∂rho over ∂t dt right ] ~dq ~dp]

[eqn: lpile { Inflow& = & dot q dt dp rho (q,p)& +& dot p dt dq rho(q,p)~ # &&~~~~DOWNARROW& &~~~DOWNARROW # Outflow & = &\{ q → q + dq \} & & \[p.I p+dp \} }]

[eqn: left { lpile { dot q ( q + dq,p) & = & dot q (q,p) & + & dq {∂{dot q }} over ∂q + O(dq)^2 # dot rho ( q + dq,p) & = & dot rho (q,p) & + & dq {∂{dot rho }} over ∂q + O(dq)^2 }]

[eqn: lpile { Outflow & = left(dot q + dq {∂ dot q } over ∂q right) left( rho + dq {∂ rho } over ∂q right)dt` dp + [p] # & = dot q ~rho~ dt dp + left( {∂ dot q } over ∂q rho + ∂rho over ∂q dot q right) dq` dt `dp + [p] }]

[eqn: lpile { therefore~ Inflow ~-~Outflow &= - left ( {∂ dot q } over ∂q rho + ∂rho over ∂q dot q right ) dq` dt `dp - [p] # & = ∂rho over ∂t dt` dq` dp }]

[eqn: pile { therefore ∂rho over ∂t = - left[ &under{ {∂ dot q } over ∂q rho} &+ ∂rho over ∂q dot q + &under { {∂ dot p } over ∂p rho }& + ∂rho over ∂p dot p right ] # & DLINE &&DLINE & # &{∂^2 H } over ∂q∂p rho && - {∂^2 H } over ∂p∂q rho}]

[eqn: ∂rho over ∂t = - ∂rho over ∂q dot q - ∂rho over ∂p dot p~]

If equally likely at time t , □ is constant independent of states specified by (q, p) at time t.

∴ [eqn: ∂rho over ∂q = ∂rho over ∂p = 0`]

[eqn: ∂rho over ∂t = phi]

∀                    ∀ □ does not change in time

So, if equally likely at time t, □ remains unchanged after that ∀ equilibrium

[Q. E. D]

[* simple formulation.

[eqn: 0 = ∂rho over ∂t + ∂rho over ∂q dot q + ∂rho over ∂p dot p = {d rho } over dt# => ~~{d rho } over dt = 0] [total derivative : rate of change of □if the system moves along in phase space according to the classical equations of motion]

Generalization to many particle system in 3D is straight forward.

* Fundamental postulate

Right or wrong?

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

∀ Make theoretical predictions based on this postulate and check them by experiments

∀ Confirmed !!

□ Non-equilibrium

accessible states are not equally likely.

Fundamental postulate → relative problem changes in time.

- where to go ? How fast to go?

∀ equally likely (equilibrium) [H-theorem]

---maximum entropy---

"relaxation time" ; microsecond ∼ century < not easy to calculate >

(4) Probability and Density of states

□Consider an isolated system in equilibrium with[eqn: E <-> E + delta E. ~~[under {E over N } << delta E << E ]]

QM energy gap.

[eqn: Omega (E ) ==]total # of accesible states with E ↔ E + □E

eg. 1 dim'l SHO

[eqn: E = P^2 over 2m + 1 over 2 K q^2]

[eqn: Omega(E) = {Shaded~ area } over {cell ~ size(h_0 ) }]

□ Suppose the parameter y represents a physical quantity other than the energy (e. g. magnetic moment, velocity, ...) and assumes the values of yk (k=0, 1, 2, ...)

[eqn: Omega ( E` ; y_k ) ==]total # of accessible states with	[eqn: E <-> E + delta E] and [eqn: y = y_k .]

[eqn: [ sum from k=0 to inf Omega(E, y_k ) = Omega(E)]]

Probability that y assumes the value of yk :

[eqn: P(y _k ) = {Omega ( E ; y_k ) } over {Omega (E) } ~~~ (← equally ~likely)]

[eqn: therefore < y > = sum from k y_k P(y_k) = { sum from k y_k Omega ( E ~;y_k ) } over { Omega (E) }]

(eg.) three spin -1/2 particles in magnetic field

with E = - □H.

Say y represents the magnetic moment of the 1st particle [ yk = □ , -□]

2	+	+	-	□

3	+	-	+	□

4	-	+	+	-□

[eqn: lpile { ∀ Omega(E) = 3 ~~~ & Omega (E~; y_k = □) = 2 #& Omega (E~; y_k =- □) = 1 # therefore <y> = & { □cdot 2 - mu } over 3 = 1 over 3 □ }]

□ Density of states

For [eqn: E over N << delta E << E , ~~ ∀ ~ Omega(E) ∝ delta E ~]

[eqn: ∀ Omega(E ) == under { omega (E ) }delta E]

# of states per unit energy range

≡ "density of states"

☆ rough calculation of □(e) or □(e) as a function of E :

Consider a system  of [eqn: f~] degrees of freedom.

∀ Needs [eqn: f~] quantum numbers to specify a state.

Let □E = energy measured from the QM ground state energy.

□□(E) = total # of states with energy < E.

□(E) should increase monotonically but how rapidly??

First, consider one typical degree of freedom.

□ : energy contributed by this degree of freedom to the total energy E [eqn: [□ ∼ E over f ]]

[eqn: □ _1 ( epsilon )] : total # of possible values of the quantum number associated with this degree of freedom of the energy < □.

[eqn: pile { SIM # roughly}~~ epsilon over {△ epsilon }] : mean spacing between possible quantized energy.

[eqn: therefore Phi _ 1 (epsilon ) sim epsilon]

[eqn: ∀ Phi (E) sim [ Phi _1 (epsilon) ] ^f``]

[eqn: therefore~Omega(E) = Phi ( E+delta E) - Phi ( E)]

[eqn: CONG {∂Phi} over ∂E delta E~~]

∀ [eqn: omega (E) & = {∂Phi } over ∂E ∼ ∂ over ∂E left( E over f right)^2 = left( E over f right) ^f-1 ~~ #&approx left( E over f right) ^f ~~~[ f approx 10^26 ]]

[eqn: therefore ~~ omega(E) ∼ E^f ∼ Omega(E)] : extremely rapidly increasing function

[eqn: ln Omega ∼ ln omega ∼ f ln E``] : slowly varying function of E.

□ Classical Ideal gas

[eqn: E = sum from i=1 to N { vec p_i ^2 } over 2m + under{ U ( `vec r_1, ... , vec r_N `)``}]

= □ ( for Ideal gas )

: small density → good approximation.

[eqn: Omega(E) = ?``]

[eqn: lpile {Omega (E ) & = 1 over { h_0 ^3N } int _ {E ↔ E + delta E } ... int 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 } del ^N chi (E) ~~~[E ~ is ~ i nd ependent~ of ~ vec r_i ]}]

[eqn: lpile {,where ~~ chi(E) & = int _ {E↔E+delta E } ... int d^3 vec p_1 ... d^3 vec p_N #&[ V ~i ndependent ] # & = int _ { E < sum from i {{vec p_i} ^2 } over 2m < E + delta E } d^3 vec p_1 ... d^3 vec p_N }]

Consider N =1

[eqn: chi & = int ... int dp_x dp_y dp_z # &{ 2mE < p_x ^2 + p_y ^2 + p_z ^2 < 2m(E + delta E )} &]

〓 volume of the spherical shell (shaded volume)

〓 [eqn: 4pi R^2 d R]

N = N ; volume of the spherical shell in 3N dimension

[eqn: chi ∼ R ^3N-1 dR ∼ E ^ { 3N-1 over 2 } cdot { delta E } over sqrt E =E ^{ 3over2 N -1 } delta E ∼ E ^ { 3 over 2 N }~~]

[eqn: therefore ~~Omega(E) ∼ 1 over { h_0 ^3N } del ^N E ^ { 3 over 2 N } ∼ E ^ {f over 2}~~]

2-2. Interactions between macroscopic systems

(1) Macroscopic state

specified bye "external parameters " [x1, x2, ..., xn ] and any other conditions imposed on the system.

□-external parameters ; magnetic field, volume, ...

[can change energy Levels of the system]

-any other conditions ; total energy, temperature, pressure, ...

For a given macroscopic state, there are many possible microscopic states.

∀	Er	=     Er ( X1, X2, ... , Xn)

(energy Level)

□ r : denotes a quantum state of the system.

(2) Interactions

Consider                [eqn: A ^(0) = A + A ' ~~~~: ~ i solated~~~]

two typical interactions :

(i) external parameters are unchanged

(energy Levels are unchanged) ∀ "thermal"

(ii) external parameters are changed (lile volume change)

(energy Levels are changed) but thermally insulated ∀ "mechanical"

@ "thermal" interactions

ensemble :

mean energy transfer via pure thermal interactions ≡ "heat"

mean energy change of system A

:[eqn: △ _t bar E == _def Q ~~] : heat absorbed by system A.

[eqn: A ^ (0)] : isolated → total energy conserved

[eqn: therefore ~ △ _t bar E + △ bar e' = phi ~~~∀ ~~~ Q + Q ' = phi]

ⓑ □mechanical" interactions

thermally insulated, but energy transfer can occur due to mechanical work. ( changes of external parameters

ensemble :

mean energy change of system A

[eqn: : ~ △_m bar E == _ def. -W ~ :]macroscopic work done by system A.

A(0) : isolated  ∀  [eqn: △ _m bar E + △ _m bar E ' = phi ~~~ ∀ ~~ W + W' = phi]

system A

<energy levels are changed>

ⓒ general interactions

[eqn: △ bar E = △_t bar E + △ _m bar E = -W + Q]

[eqn: bold { therefore ~~Q~~~=~~ △ E ~~+~~ W }~~]

thermal    +       mechanical    →    thermodynamics

□infinitesimal changes

[eqn: bold {□ Q ~~~= ~~~d bar E ~~~ +~~~ □ W}~~]

,where □ : not difference but infinitesimal quantity

(3) Quasi-static processes

so slow that the system remains arbitrarily close to "equilibrium" at all stages of the process.

(time scale in the process is much longer than the relaxation time □)

@ generalized force

Er = Er (X1, ... , Xn)

external parameters are infinitesimally changed.

(mechanical interactions)

[eqn: X _ alpha → X_ alpha + d X _alpha]

[eqn: pile { ∀ # chain~ role ~~~}]  [eqn: dE_r = sum from alpha=1 to n left ( {∂E_r } over {∂ X_alpha } right) d X_alpha]

If the system remains in this microstate r, the work done by the system is

[eqn: □ W_r =~~~~ -d E_r ~~~~~== sum from alpha X_alpha,r d X_alpha]

<energy level shift>

,where [eqn: X_alpha,r == - left ( {∂ E_r } over {∂ X_alpha } right )]: "generalized force" conjugate to X□ in state r.

[ X□ : distance ∀ X□ : ordinary force]

Consider ensemble.

If the change of external parameters is quasi-static, X□ have "well-defined" mean values at any time and distributions of ensemble are subject to equilibrium conditons.

[eqn: therefore]Macroscopic work

[eqn: □W = sum from alpha=1 to n bar X_alpha~ dX_alpha ~,where~~bar X_alpha = - bar { left ( {∂E_r } over {∂ X_alpha } right)}]

(mean generalized force)

ⓑ quasi-static work done by pressure

external parameter V → V +dV

Let the pressure exerted on the piston Pr if the system is in state r.

∀ force = [eqn: P_r cdot A]

∀ [eqn: □ W_r = (P_r A ) ds = P_r dV]

[eqn: therefore ~~~P_r = - {∂ E_r} over ∂V]

(∴ pressure is the generalized force conjugate the volume)

Mean in quasistatic process

[eqn: □ W = bar P dV]

Suppose [eqn: V_i → V_f] (quasi-static) and [eqn: bar P = bar P (V)]

[eqn: ∀~~ W_if ~~ =~~ int from V_i to V_f ~ bar P ~dV]

(HW) 2, 4, 7, 10