Chap 1. Introduction to statistical methods.

▣ Many particle systems

▣ Microscopic laws of physics

-→ describe behavior of many particle systems(1026)

(a) Is it possible?

hopeless!! insufficiency of memory & speed,  chaos

(b) Are we interested in microscopic details?

Usually not!

Only interested in macroscopic parameters like volume,  pressure, conductivity, ...

(kind of averaged quantity over very many particles)

(c) Then, is that possible to understand macroscopic behavior?

[eqn: left { lpile{ & Yes !& i n~~ many ~~useful~~ examples~~ # & No! & lpile{ i n ~~ many ~~ move~~i n teresting~~ examples# like~~biological~~ growth. }}]

▣ Equilibrium & nonequilibrium

*equilibrium; macroscopic parameters do not change in times easier to handle than non-equilibrium situation.

[eqn: left{lpile {& thermodynamics`` ;& concerns~ r e l a t i o n ships# && between~ macroscopic~ parameters # &statistical~ mechanics``;& based ~on ~microscopic~ properties#& &→derive~ all ~thermodynamic ~results ~and ~more~ }]

*non-equilibrium ; much more difficult

[eqn: left {lpile{& irreversible~~ thermo dyna mics # & or ~~ nonequilibrium ~~ statistical~~ mechanics# &kinetic ~~ theory}]

▣statistical average

so many particles(1026) → great to apply statistical arguments!

Elementary probability theory

→	take a example of "random walks"

(most important and simple problem in probability theory)

1-1 Random walks

* Statistical treatment → probability theory.

* need "an assembly of very large # N of similarly prepared systems" <ensemble>

(eg. dice throw, public opinion survey, . . .)

* Outcome depends crucially on the characteristics of the ensemble chosen.

* Simple example : "random-walk problem"

drunken walker

(Brownian motion of a suspended particle

in a fluid < diffusion>)

- How far would he reach in time [eqn: t`]?

- What is the probability of his being at [eqn: vec r`] after time [eqn: t`] elapsed? [eqn: P_t ( vec r` )`]

[eqn: P_t ( vec r` )`]: probability distribution function

<large number os similar trials>

∀then average over trials with equal weight.

1-2. One-dimensional random walks

Why? easy and applications

( magnetism: each atom either spins up or down

total magnetization of N atoms)

[eqn: p ~+~ q~ =~ 1] (always walk)

[eqn: x~ =~ m~l``]

Question : What is the probability of his being at [eqn: x~ =~ m~l``] after N steps? [eqn: P_N (m)]

(1)  Probability distribution

Fundamental assumptions:

Steps are "statistically independent"

- no history dependency -

[eqn: left{ lpile{ n_1 ~ = ~ \# ~ of ~ steps~ t o ~ the ~& r ight # n_2 ~ =~ ~~~~~〃~~~~〃 & l e ft }]

[eqn: P _ N (m) = left( underline{ lpile { N # n_1 } right ) }p ^{n_1} q ^{n_2 } ~~~ : under{B i n ominal}~distribution]

# of different possible ways.

[eqn: lpile{ N = n_1 + n_2 # m = n_1 - n_2 } right }~ ∀ ~left { lpile { n_1 = { N+m} over 2 # n_2 = {N-m} over 2 }~ bmatrix{ -N le m le N~ # 0 le n_1 le N}]

[eqn: ~※~~ lpile { N ~even ~ ->~ m~even # N~ odd~-> ~ m~odd } => triangle m = 2]

[eqn: bmatrix { P_N (m) = pmatrix { N # {N+m}over 2 } p ^ { {N+m}over 2 } q ^ { {N-m}over 2 } # == W_N ( n_1 ) = pmatrix { N~ # n_1 } p ^n_1 q ^ N-n_1 }}]

normalization : [eqn: sum from { m=-N } to N P_N (m) = sum from {n_1 =0 } to N W _N (n_1 ) = (p+q) ^ N =1]

(2) Mean and dispersion

* Mean :

[eqn: lpile {< n_1 > & = sum from {n_1 =0 } to N n_1 W _N (n_1 ) = p ∂ over ∂p left[ sum from {n_1 =0} to N pmatrix { N~ # n_1 } p^n_1 q ^N-n_1 right ] # & = p ∂ over ∂p ( p+q)^N = Np(p+q)^N-1 = Np}]

[eqn: lpile { <m> & =< 2n _1 - N > = 2Np-N=(2p-1)N # & = (p-q)N }]

[eqn: p=q ~∀~ <m> =0~~~∀ ~~<x> = <ml> = phi]

* Dispersion

[eqn: lpile { <( triangle n_1 )^2 > & == < (n_1 - < n_1 > )^2 > = <n_1 ^2 - 2 n_1 < n_1 > + < n_1 > ^2 > #& = <n_1 ^2 > - < n_1 > ^2 }]

[eqn: lpile { <n _1 ^2 > & = sum from {n_1 =0 } to N n_1 ^2 W_N ( n_1 ) = ( p ∂ over ∂p ) ^2 ( p+q) ^N #&= p ∂ over ∂p [ N p ( p+q ) ^N-1 ] #&= np(p+q ) ^N-1 + N ( N -1 ) p ^2 (p+q ) ^N-2 # & = Np [ 1 + (N-1 ) p ] = N^2 p^2 + N pq }]

[eqn: therefore < ( triangle n_1 ) ^2 > = N pq]

[eqn: lpile { < ( triangle m ) ^2 > & = < ( m- <m> ) ^2 > = < ( 2n_1 - N - 2 <n _1 > + N )^2 > #& = 4 < ( triangle n_1 ) ^2 > }]

[eqn: therefore < ( triangle x ) ^2 > = < ( triangle m )^2 > l^2 = 4 l^2 Npq]

□ RMS deviation                   [eqn: triangle ^ * n_1 == sqrt { < ( triangle n_1 ) ^2 > }]

□ Relative width[eqn: == { triangle ^* n_1 } over { < n_1 >} = sqrt { q over p } 1 over sqrt { N } ~ _ { N -> inf } ^ --->~~~ phi]

normalize by <n1>

sharpens pronounced maximum !

(3) large N limit

Consider a continuous function W(n1) for large N.

[eqn: lpile { W ( n_1 ) ~ & : ~ m axi m um ~~ at ~~ n_1 = tilde {n_1} # & => ~ dW over dn_1 right| _{n_1 =tilde n_1 }=phi~~left[or{d ln W}over dn_1 right|_{ n_1 = tilde n_1 } = phi right ] ``}]

W(n1) : rapidly varying function of n1

So consider [eqn: ln W``] (much more slow varying function) [(e. g.) en → n]

-Expand [eqn: ln W~] near [eqn: n _1 = tilde n_1]

[eqn: ln W(n_1 ) = ln w ( tilde { n_1 } ) + B_1 eta + 1over2 B_2 eta ^2 + . . .]

, where [eqn: eta = n_1 - tilde n_1 ~ and ~ B_k = {∂^k ln W } over { ∂ n_1 ^k } right | _{n_1 = tilde n_1 }]

In order that [eqn: ln W~] is maximum at [eqn: n _1 = tilde n_1],

B1 = □ and B2 < 0

-How to calculate B1 & B2 ?

[eqn: lpile {ln W(n_1 )& = ln left [ N! over {n_1 ! (N-n_1 ) ! } p^n_1 q^{N-n_1 } right ] # & = ln N! - ln n_1 ! - ln ( N - n_1 ) ! # & + n_1 ln p + (N-n_1 ) lnq }]

[eqn: {∂ [ln n_1 ! ]} over ∂n_1 pile{ approx # { n_1 ->inf}} {ln ( n_1 +1 )! - ln n_1 ! } over {(n_1 + 1 ) - n_1 } = ln(n_1 +1 ) approx ln n_1~]

[eqn: {∂ [ln(N- n_1 )! ] } over ∂n_1 & pile { approx # N-n_1 -> inf } {ln(N-n_1 -1 )! - ln(N-n_1 )! } over { (n_1 +1) - n_1 }#&~~~~~ = - ln ( N-n_1 ) }]

[eqn: left [ lpile { * ~S t e r l i n g ~~ f o r m u l a # ln n! pile { approx # n-> inf } n ln n - n + 1 over 2 ln n + 1 over 2 ln(2 pi ) + . . . } right ]]

[eqn: therefore {∂ ln W(n_1 ) } over ∂n_1 cong - ln n_1 + ln(N-n_1 ) + ln p - lnq]

[eqn: ∀ { N - tilde n_1 } over {tilde n_1 } = q over p = 1-p over p]

∴=[eqn: <n_1 >]

[eqn: tilde n_1 = Np]

[eqn: lpile { B_2 & = {∂^2 ln W (n_1 ) } over {∂ n_1 ^2 } right | _{n_1 = tilde n_1 } = - 1 over { tilde n_1 } - 1 over { N- tilde n_1 } # &= - 1 over N left ( 1 over p + 1over q right ) = - 1 over Npq = - 1 over { < ( triangle n_1 ) ^2 > } < ~0 }]

[eqn: left [ B_2 = O ( 1 over N ) , B_3 = O ( 1 over N^2 ) , ... right ]]

[eqn: therefore ln W(n_1 ) cong ln W ( tilde n_1 ) - 1 over 2 |B_2 | eta ^2]

[eqn: ∀ ~ W(n_1 ) cong W ( tilde n_1 ) exp left[ - 1 over 2 |B_2 | eta ^2 right]]

- Normalization

[eqn: lpile { 1 = sum from { n_1 =0 } to N W (n_1 ) & pile { approx # n->inf } int from 0 to inf W (n_1 ) dn_1#& pile { cong # n_1 = tilde n_1 + eta } int from {- tilde n_1 } to inf W ( tilde n_1 + eta ) d eta`` # & pile { cong # (tilde n_1 ∼ N ) } int from -inf to inf W ( tilde n_1 ) exp [ - 1over2 | B_2 | eta ^2 ] d eta~ }]

[eqn: [ Gaussian ~~ i n t e g r al ~ : ~ int from -inf to inf e^ { -a x^2 } dx = sqrt { pi over a } ]]

[eqn: therefore 1 = W ( tilde n_1 ) sqrt { 2pi over { |B_2 | }}]

[eqn: lpile {=>~ W(n _1 ) & = sqrt { {|B_2 | } over 2pi } exp left[ - 1 over 2 |B_2 | (n_1 - tilde n_1 )^2 right ]# & = 1 over { sqrt 2pi triangle ^ * n_1 } exp left [ - {(n_1 - <n_1 > ) ^2 } over { {2 (triangle ^ * n_1 )^2 }} right ] # & [ where ~ <n_1 > = Np ~, ~ (triangle ^* n_1 ) = sqrt Npq ]}]

(4) Gaussian distribution

[eqn: P_N ( m) = 1 over { sqrt { 2 pi N pq }} exp left [ - 1 over 8 {(m-N ( p-q))^2 } over Npq right ]]

[eqn: << n_1 = N+M over 2 >>]

Use continuous variable X  instead of m.

[eqn: X == ml~~~(Say~~ l ~<< 1 )]

How many discrete m's between x and x + dx?

remember △m=2!

[eqn: lpile { P(x ) & = 1 over { sqrt 2pi sigma } exp left [ - {(x-mu)^2 } over 2sigma^2 right ] #&, where ~~ sigma = 2l sqrt Npq ~~and ~~mu = N(p-q)l ~~}]

[eqn: =>~~ dx over 2l]

Define

[eqn: P(x)dx] = probability that the walker locates in the range between x and [eqn: x + dx] after very many steps ([eqn: N -> inf])

[eqn: => P(x) dx = P_N (m) cdot dx over 2l]

[eqn: =>] Probability density(function)

Gaussian probability distribution

normalization   [eqn: int from -inf to inf P(x) dx =1]

[eqn: lpile { <x> = int from -inf to inf x P(x) dx = mu # <(△x)^2 > = <x^2 > - <x> ^2 = sigma ^2 }]

[eqn: lpile { left [ f(a) == & int from -inf to inf e^{-a x^2 } dx = sqrt {pi over a} # & int from -inf to inf x e^{-a x^2 } dx = - 1 over 2 df(a) over da , ... right] }]

1-3 Probability distributions involving several variables

So far, we used the single variable n1 or m

(n1 = 0, 1, 2, ... , N )

Consider two-variable problem. ( u and v )

{ui ; i = 1, 2, ... , M }

{vi ; i = 1, 2, ... , N }

[ (eg.) calculating walker ; v=n1

take a card out of 10 cards (1, 2, ... , 10 ) and add up ∀ ui ]

[eqn: P(u_i , v_j )] : probability that [eqn: u = u_i ~\& ~ v = v_j]

normalization : [eqn: sum from i=1 to M sum from j=1 to N P(u_i , v_j ) = 1]

define[eqn: left { lpile { P_u ( u_i ) == probabilities ~that ~ u=u_i ~\& ~ v ~ : ~ arbitrary~~ # P_v ( v_j ) == ~~~〃~~~〃~~~〃 u~ :~ arbitrary~\& ~ v=v_j }]

[eqn: P_u ( u_i ) = sum from j=1 to N P ( u_i , v_j )~~~, ~~ P_v ( v_j) = sum from i=1 to M P ( u_i , v_j )]

(1) Statistically independent variables

u & v are "statistically independent" (un-correlated) if the probability that u assumes a certain value does not depend on the value assumed by v.

[eqn: => ~ P( u_i , v_j ) = P_u ( u_i ) P_v ( v_j )]

[eqn: [ (eg.)~~ v_0 = n_1 =1 ~,~ u=2 ~ => P(1,2) = p cdot 1 over 10 ]]

Important result:

[eqn: lpile { < f(u) g(v) > & = sum from i,j P ( u_i , v_j ) f(u_i ) g(v_j ) # & = sum from i P_u (u_i ) f(u_i ) sum from j P_v (v_j ) g(v_j ) # & = <f(u) > <g(v)> #& [ BECAUSE sum from i,j' P(u_i , v_j' ) f(u_i ) ]}]

[eqn: therefore]separable if stat. independent.

(2) Generalized random walks

[eqn: S_i] ; displacement in the ith step.

[eqn: W(S_i ) dS_i ~ = ~]prob. that the ith displacement lies between [eqn: S_i~] and [eqn: S_i +dS_i`]

[eqn: left [ lpile { ORW ~=>& W(s_i ) ~ has ~ two~ delta-functions# & at ~ S=+1 ~and~ -l .} right ]]

Assume that [eqn: W(S_i )] is i-independent . [eqn: ( == W(S))]

(ex)

[eqn: left [ lpile { ORW ~=>& W(s_i ) ~ has ~ two~ delta-functions# & at ~ S=+1 ~and~ -l .} right ]]

Question : After N steps, <x>, <(△x)2>, P(x) ?

total displacement [eqn: x = sum from i=1 to N S_i]

[eqn: lpile { therefore <x> & = sum from i=1 to N <S_i > = sum from i=1 to N int from -inf to inf dS_i S_i W(S_i ) # & = sum from i+1 to N int from -inf to inf dS` S` W(S) = N<S> }]

[eqn: <(△x)^2 >& = <x^2 > - < x>^2]

[eqn: lpile { <x^2 > &= sum from i sum from j < S_i S_j > =sum from i=1 to N < S_i ^2 > + sum from {i ne j } <S_i S_j ># & = N ( S^2 > + sum from { i ne j } <S _ i > <S _j > # & = N < S^2 > + N(N-1)<S>^2}]

[eqn: lpile { (S_i , ~S_j ~ for ~ i ne j # because statistically ~~ i n dependent) }]

[eqn: therefore <(△x)^2 > = N \{ <S^2 > - <S> ^2 \} = N < (△S)^2 >]

□RMS deviation	[eqn: lpile { △^* x & = sqrt { <(△x)^2 > } = sqrt N sqrt {<(△S)^2 > } # & = sqrt N △^* S }]

□relative width	[eqn: = {△^* x } over <x> = {△^* S } over <S> 1 over sqrt N~~ pile { ――-> # N->inf }~~ phi]

* ordinary random walk

[eqn: W(S) = p delta (S-l) + q delta ( S+l)]

[eqn: left [ int from -inf to inf W(S) dS = p+q = 1 ~ : ~properly~normalized right ]]

[eqn: lpile { <S> &= int S(W(S)dS = (p-q)l # <S^2 >& = int S^2 W (S)dS = (p+q)l^2 = l^2 }]

[eqn: # <(△S)^2 >& = l^2 - (p-q)^2 l^2 #& = l^2 [1-p+q]~[1+p-q] # & = 4pql^2]

[eqn: lpile { therefore& <x> = N(p-q)l &[=mu`]# &<(△x)^2 > = 4Npql^2 &[ = sigma ^2 `]``````````````````under O.K. }]

* probability density [eqn: P(x)]?  [eqn: x = sum from i=1 to M S_i`]

[eqn: lpile { &&probability # 1st~ step& S_1 <-> S_1 + dS_1&~~~~=> & W(S_1 ) dS_1 # 2nd~step & S_2 <-> S_2 + dS_2&~~~~=> & W(S_2 ) dS_2 # & cdot& cdot& cdot# & cdot& cdot& cdot }]

[eqn: lpile { P(x) dx & = int from -inf to inf ... int W(S_1 )W(S_2 ) ... W(S_N ) dS_1 dS_2 ... dS_N~ # & \{ x < sum from i=1 to N S_i < x +dx \} ~~~under {restriction} # &= int from -inf to inf ... int W(S_1 ) ... W(S_N )#&~~~~~~~~~ times delta ( x-sum from i=1 to N S_i ) dx dS_1 ... dS_N # & left [ delta (x) = 1 over 2pi int from -inf to inf dk ~ e ^{+- ikx} right ] }]

[eqn: lpile { therefore ~P(x)& = 1 over 2pi int from -inf to inf dk~e^{-ikx} int from -inf toinf dS_1 W(S_1 ) e^{ikS_1 ~} #&~~times int from -inf to inf dS_2 W(S_2 ) e^{ikS_2 } ... # & = 1 over 2pi int from -inf to inf dk ~e^{-ikx } ~[Q(k)] ^N# &,where ~~Q(k) = int from -inf to inf dS ~e^{ikS } W(S) #& ~~(Fourier~transform) }]

Assume that W(S) varies slowly and falls off rapidly enough as [eqn: |S| -> inf].

Also consider N is very large.

Q(k) becomes smaller as k becomes large (rapid oscillation)

[eqn: [Q(k)] ^N``]for large N should decrease very rapidly as k becomes large [exponential decay]. So expand [eqn: e^iks ~] for small k.

[eqn: lpile { Q(k) & = int from -inf to inf dS W(S) [ 1 + ikS - 1 over 2 k^2 S^2 + ...]# &=1 + ik<S> - 1 over 2 k^2 <S^2 > + ... }](for finite moments, need that W(S) falls off rapidly).

[eqn: lpile{ therefore Q^N (k) & = N ln [1 +ik<S> - 1 over 2 k^2 <S^2 > + ... ] #& = N~[ik<S> - 1 over 2 k^2 <S^2 > + 1 over 2 k^2 <S>^2 + ...] #& = N~ [ ik<S> - 1 over 2 k^2 <(△S)^2 > + ... ] # therefore Q^N (k)& CONG e^{iNk(S> - 1 over 2 N k^2 <(△S)^2 >} }]

[eqn: lpile { => P(x) & = 1 over 2pi int from -inf to inf dk ~e ^{ i(N<S> -X ) k - 1 over 2 N <(△s)^2 > k^2 } # & = 1 over {sqrt 2pi sigma } e^{ 1 { (x-mu)^2 } over 2sigma^2 }~~~~~left [ pile { u &=& N<S> # sigma &=&N <(△S)^2 > } right ] }]

Gaussian distribution again!

General statement :

□W(S) : arbitrary but falls off rapidly enough as [eqn: |S|-> inf]

[exponential decay]

□each step : statistically independent

∀  Gaussian distribution as N→∞

[Central limit theorem]

→ accounts for the fact that so many phenomena in nature

obey approximately a Gaussian distribution [e. g. stat.

errors in experiments]

[But some take power-law distribution like stock market price, ...]