Точные решения в неравновесной статистической...
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Точные решения в неравновесной статистической механике. В.Б. Приезжев ЛТФ ОИЯИ. Totally Asymmetric Exclusion Process. Totally Asymmetric Exclusion Process. Applications to: Hopping conductivity Queuing problems Directed polymers in random medium Traffic problems. Master Equation. - PowerPoint PPT PresentationTRANSCRIPT
Точные решения в неравновесной статистической механике
В.Б. ПриезжевЛТФ ОИЯИ
Totally Asymmetric Exclusion Process
Totally Asymmetric Exclusion Process
Applications to:
1. Hopping conductivity
2. Queuing problems
3. Directed polymers in random medium
4. Traffic problems
Master Equation
One-particle master equation (Poisson process)
).,(),1(),(
txPtxPt
txP
.1||,)( zzxP x
Substitution
).()1()( xPxPxP
“ Fourier ansatz ”
gives )()exp(),( xPttxP
We put
zzzz xxx 1
1;1
]2,0[, pez pi
)(),(2
0
pfedptxP xipt
From the initial conditions
tyx
yxiptipe
ypiyx
eyx
t
eedpP
epfxP
)!(
2
1t)(x,
Then,
)2()(havewe)0,(
)(2
0
)1(
1,
Poisson distribution
then (2) has the form (1).
Therefore, Eq.(1) + condition P(x,x)=P(x,x+1)
gives the Asymmetric Exclusion Process
Two-particle exclusion process
;t)(x,xP(x,x;t)-P
xxxx
txxPtxxPt
txxP
txxPtxxPtxxPt
txxP
1termstwoaddweif
)2(
havewe,1,If
)1(
);1,();1,1();1,(
21
);2,1(2);12,1();2,11();2,1(
As in the one-particle case, we have
),(2)1,(),1(),(
),();,(
21212121
2121
xxPxxPxxPxxP
xxPetxxP t
Bethe Ansatz
1221
2121211221 ),( xxxx zzAzzAxxP
2
1
21
1212
121
221
21
1
1S:matrix
;1
1;2
11
z
z
A
AS
Az
zA
zz
From condition P(x,x)=P(x,x+1), we have
)(
)();,(
2112221121
2,12
2
01
2
021
xipxpixipxpi
t
eSe
ppfedpdptxxP
Integrating, we obtain
2211221 )2(),( yipyipeppf
From initial conditions
),1(),(),(
);(),(;!
),(
2,1;2,1,|);(|det)0;,|;,(
001
0010
2121
tkFtkFtkF
tjkFtkFk
tetkF
jityxFyytxxP
j
kt
jiji
ASEP as a combinatorial problem
)0;,...,|;,...,( 11 pp yytxxP
Free fermions TASEP
Discrete formulation
of all free paths for time t. M.E. Fisher (1984):
"functionpartition");,( 00
000
xxtxx yzxx
ttxxF
|);(|det)0,|,(
);();();();(
,0
1,202,102,201,10
tyxFytxP
tyxFtyxFtyxFtyxF
ji
Cancellation for the TASEP (step 1)
Reference coordinates for A,B,C,D
)()|)(|1()()|1)(|(
)()|1)(|1()()|1)(|1(
2121
2121
DxxxxBxxxx
CxxxxAxxxx
Shift operators
1
2
1
1:matrix
a
aSS
Cancellation for the TASEP (step 2)
|);(|det)0,|,(
);();();();(
,
1,212,112,201,10
tyxFytxP
tyxFtyxFtyxFtyxF
jiji
Solution for two particles
),1(),(),(
);(),(;!
),(
2,1;2,1,|);(|det)0;,|;,(
001
0
010
2121
tkFtkFtkF
tjkFtkFk
tetkF
jityxFyytxxP
j
kt
jiji
General solution for infinite lattice
);,();,(
);,(1
1);,(
)!();,(
|);,(|det)0;,...,|;,...,(
00
00
0
11
tkyxFk
mtyxF
tkyxFm
mktyxF
yx
tetyxF
tyxFyytxxP
ji
m
kjim
jik
jim
ij
yxt
ji
jijipp
ij