# Mô hình Ising trong mô phỏng vật lý

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• 1M hnh Ising

• 2Mt s kin thc v thng k

• 3Mt s kin thc v thng k

Khng gian pha

Xt mt h c in N ht

Trng thi ca h c xc nh bi ta r v xung lng p ca tt c cc ht

Khng gian pha: 6N bin, = (r,p) hoc (q,p)

S thay i trng thi theo thi gian tun theo cc phng trnh c hc c in

qk=H pk

, pk=Hqk

H=KV p

• 4

Chuyn ng ca h theo thi gian m t bi mt qu o trong khng gian pha (t)

Do tnh tt nh ca cc phng trnh Newton, qu o ny khng bao gi ct chnh n!

Poincare: nu i lu th h c th quay tr v trng thi ban u!

Poincare recurrence time > tui v tr i vi h v m

• 5Mt s kin thc v thng k

i lng o c A()

Gi tr o c bng thc nghim l gi tr trung bnh theo thi gian

Gibbs: ly trung bnh theo tp hp vi phn b cn thit!

(): mt xc sut trng thi iu kin v m nht nh: NVE, NVT, NPT...

Aobs=Atime=A t time=1

tobs0

t obs

A t dt

Aobs=Aens=

A

Tp hp thng k

• 6Mt s kin thc v thng k

Tp hp: bao gm cc bn sao ca h nhiu trng thi khc nhau

(,t) mt xc sut

nh l Louville:

s h trong tp hp khng thay i theo thi gian

tp hp chuyn ng theo thi gian trong khng gian pha nh mt cht lng c nn bng 0!

d dt=0

t

=i=1

N

ri ri pi pi

• 7Mt s kin thc v thng k

Khi t v cng ln, ta c tp hp cn bng:

khi , khng ph thuc thi gian!

v ta c

H ergodic: any point in phase space is accessible from any other point

H non-ergodic: some region of phase space is not accessible from outside

t

=0

Atime=Aens

• 8Mt s kin thc v thng k

Trng s & hm phn hoch:

ty thuc vo cch ly trng s ta c cc tp hp khc nhau

M phng Monte Carlo: cho php to ra mt tp hp cc trng thi theo mt xc xut cho trc, khi

=Q1w

Q=

w

A=Q1

A w

A=1

Kk=1

K

Ak

• 9Mt s kin thc v thng k

Tp hp vi chnh tc N,V,E = constants

Phng php ng lc hc phn t (MD): to ra tp vi chnh tc (E=constant), ng thi bo ton xung lng tng cng

QNVE=

H E

QNVE=1

N !

1

h3N dr dpH r , pE

S=k B lnQNVE entropy

• 10

Mt s kin thc v thng k

Tp hp chnh tc N,V,T = constants

w ()=eH ()/k BT

QNVT=eH ()/k BT

F=k BT lnQNVT

QNVT=1

N !

1

h3N dpe

K /k BTdr eV pr /k BT

Nng lng t do Helmholtz

Z NVT=

QNVT=1

N ! h2/2mk BT 3N /2 Z NVE

dr eV pr /k BTTch phn cu hnh

• 11

Mt s kin thc v thng k

Tp hp ng nhit ng p

N,P,T=constants

w =eH PV /k BT

QNPT=V

eHPV /k BT=

V

eP /k BT QNVT

G=k BT lnQNPT

Z NPT=dV ePV /k BTdr eV pr /k BTNng lng t do Gibbs

• 12

Mt s kin thc v thng k

Tp hp chnh tc ln

,V,T=constants

w =eH N /k BT

QVT=N

eH N /k BT=

N

e N /k BT QNVT

PV=k BT lnQVT phng trnh trng thi

• 13

Mt s kin thc v thng k

nh lut ng phn

Mi bc t do ng vi kch thch nng lng kT

S bc t do =

Nc l s rng buc (constraint)

pk H pk =k BT qkHqk =k BT

3NN c

• 14

Mt s kin thc v thng k

Nhit tc th Nhit o c bng thc nghim l nghit

trung bnh theo thi gian

Trong m phng c th tnh nhit t mt trng thi vi m ca h

T nh lut ng phn ta c:

Nhit tc th:

K =i=1

N pi2

2mi =3N2 k BT

T=2K

3NkB

=1

3NkBi=1

N pi2

mi

• 15

Mt s kin thc v thng k

Trong trng hp c Nc rng buc:

Nhit trung bnh:

T=2K

3NN ck B=

1

3NN ck Bi=1

N pi2

mi

T= T

• 16

Mt s kin thc v thng k

p sut tc th T trng thi vi m ca h c th tnh c p sut

tc th

T nh lut ng phn ta c:

suy ra:

Lc tng cng bng ngoi lc + ni lc:

qk pk =k BT pk= f ktot=

qk

V p

1

3 i=1N

rif itot =N kBT

f itot=f i

extf iinternal

• 17

Mt s kin thc v thng k

Ngoi lc cn bng vi p sut ln cc bc tng:

Hm virial

p sut tc th:

1

3 i=1N

rif iext =PV

W1

3i=1

N

rif iinternal=

1

3i=1

N

riri V p

PV=N kBTW

P= k B TW

V= Pideal gas Pex

P= k BTW

V= Pideal gas Pexhoc

• 18

Mt s kin thc v thng k

Tng tc cp

W=1

3i

i j

rif ij=1

3i

i j

ri rij v rij

V p=i j

v rij

W=1

3i

i j

w rij

w r =rdv r dr

hm virial cho tng tc cp

• 19

Mt s kin thc v thng k

Nhit dung ring

N,V,T=constants

N,P,T=constants

E=H

E 2=H 2H 2

Cv=H 2H 2

k BT2

C p=H 2H 2

k BT2

• M hnh Ising

M hnh Ising l g? V sao n quan trng?

M hnh Ising l m hnh ton hc c t theo tn ca nh Vt l Ernst Ising (Ngi c).

M hnh Ising l m hnh dng m t hin tng chuyn pha st t m ch s dng cc spin-up v down

• M hnh Ising Ising gii bi ton 1D nm 1924 trong lun

vn Tin s ca mnh (thun tu Ton). Trng hp mng vung 2D c th gii chnh xc c bng gii tch (Onsager, 1944)

n nay, bi ton v m hnh Ising c p dng trong rt nhiu lnh vc: vt l, sinh hc (lin quan n t) n cc vn x hi (m hnh n gin 2 la chn)

M hnh Ising l m hnh chun th xem mt thut ton trong khun kh p dng ca m hnh c hiu qu khng

• St tCc domain t sp xp thng hng theo mt hngThng thng, cc domain khngsp xp thng hng theo mt hng

Tuy nhin, cc domain c th c p

Ti nhit thp th cu hnh n nh l cu hnh vi tt c spinu hng ln hoc hng xung (2 cu hnh)

Nhit Curie (nhit ti ton b tnh st t bin mt). Vi st l 1043 K

im ti hn: l im xy ra s chuyn pha (loi II)

• Gin pha

Nhit thp Nhit cao

• M hnh

Universality Class l mt lp ca cc h Vt l c chung mt tnh cht ng m khng ph thuc vo cc tnh cht ng lc ca h. V d: h hp kim 2 cht, h 2 cht lng trn ln, hay h siu chy ca Helium trong 3 chiu u thuc vo mt lp

M hnh Ising ch s dng cc vector UP v DOWN nhng li m t c rt nhiu pha khc nhau ca vt cht

- Hp kim 2 cht - Trn 2 cht lng - Cht lng v kh trn ln

- Siu chy ca Helium- Hin tng siu dn trong kim loi

• M hnh IsingGii tch

Ising 1924

Onsager 1944

Gii s, v d pp Monte Carlo

Nhit cao

2-D

3-D

Nhit thp

1-D

• 26

M hnh Ising

E=J ij

si s jHi=1

N

si

cc cp ln cn gn nht

J - nng lng tng tc trao iJ > 0 - st t (ferromagnet)J < 0 - phn st t (anti-ferromagnet)

s=1

M=i=1

N

si

cm t (susceptibility)

m=1

Ni=1

N

si

C H=E2E 2

k BT2

T=M 2M 2

k BT

H spin trn mng Ernst Ising (1924)

t ha (magnetization)

i khi s dng B thay v H

• Gii tch m hnh Ising 1 chiu

The partition function is given by

Z =+1

s1=1

+1s2=1

...+1

sN=1eEI{Si} (3)

One Dimensional Ising Model and Transfer MatricesLet us consider the one-dimensional Ising model where N spins are on a chain. We

will impose periodic boundary conditions so the spins are on a ring. Each spin onlyinteracts with its neighbors on either side and with the external magnetic field B. Thenwe can write

EI{Si} = JNi=1

SiSi+1 BNi=1

Si (4)

The periodic boundary condition means that

SN+1 = S1 (5)

The partition function is

Z =+1

s1=1

+1s2=1

...+1

sN=1exp

[

Ni=1

(JSiSi+1 +BSi)

](6)

Kramers and Wannier (Phys. Rev. 60, 252 (1941)) showed that the partition functioncan be expressed in terms of matrices:

Z =+1

s1=1

+1s2=1

...+1

sN=1exp

[

Ni=1

(JSiSi+1 +

1

2B (Si + Si+1)

)](7)

This is a product of 2 2 matrices. To see this, let the matrix P be defined such thatits matrix elements are given by

S|P |S = exp{[JSS +

1

2B(S + S )

]}(8)

where S and S may independently take on the values 1. Here is a list of all the matrixelements:

+1|P |+ 1 = exp [(J +B)]1|P | 1 = exp [(J B)]+1|P | 1 = +1|P | 1 = exp[J ] (9)

Thus an explicit representation for P is

P =

(e(J+B) eJ

eJ e(JB)

)(10)

2

The partition function is given by

Z =+1

s1=1

+1s2=1

...+1

sN=1eEI{Si} (3)

One Dimensional Ising Model and Transfer MatricesLet us consider the one-dimensional Ising model where N spins are on a chain. We

will impose periodic boundary conditions so the spins are on a ring. Each spin onlyinteracts with its neighbors on either side and with the external magnetic field B. Thenwe can write

EI{Si} = JNi=1

SiSi+1 BNi=1

Si (4)

The periodic boundary condition means that

SN+1 = S1 (5)

The partition function is

Z =+1

s1=1

+1s2=1

...+1

sN=1exp

[

Ni=1

(JSiSi+1 +BSi)

](6)

Kramers and Wannier (Phys. Rev. 60, 252 (1941)) showed that the partition functioncan be expressed in terms of matrices:

Z =+1

s1=1

+1s2=1

...+1

sN=1exp

[

Ni=1

(JSiSi+1 +

1

2B (Si + Si+1)

)](7)

This is a product of 2 2 matrices. To see this, let the matrix P be defined such thatits matrix elements are given by

S|P |S = exp{[JSS +

1

2B(S + S )

]}(8)

where S and S may independently take on the values 1. Here is a list of all the matrixelements:

+1|P |+ 1 = exp [(J +B)]1|P | 1 = exp [(J B)]+1|P | 1 = +1|P | 1 = exp[J ] (9)

Thus an explicit representation for P is

P =

(e(J+