Effective long-range interactions in driven systems David Mukamel.
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Transcript of Effective long-range interactions in driven systems David Mukamel.
Effective long-range interactions in driven systems
David Mukamel
Systems with long range interactions
in d dimensions
two-body interaction
dVVRE /1
dr
rv1
)(
for σ<0 the energy is not extensive
-strong long-range interactions
self gravitating systems (1/r) σ=-2
ferromagnets σ=0
2d vortices log(r) σ=-2
)/1( 3r
These systems are non-additive BA FFF
A B
As a result, many of the common properties of typicalsystems with short range interactions are not sharedby these systems.
Driven systems
T1 T2
T1>T2
E
heat current
charge current
Local and stochastic dynamics
No detailed balance (non-vanishing current)
What is the nature of the steady state?
drive in conserving systems result in many cases in long range correlations
What can be learned from systems with long-range interactions
on steady state properties of driven systems?
TSEF Free Energy:
VSVE d , /1 since
S E the entropy may be neglected in thethermodynamic limit.
In finite systems, although E>>S, if T is high enoughE may be comparable to TS, and the full free energyneed to be considered. (Self gravitating systems, e.g.globular clusters)
Systems with long range interactions
Globular clusters are gravitationally bound concentrationsof approximately ten thousand to one million stars, spreadover a volume of several tens to about 200 light years in diameter.
KT 62102
3
1MvTkB km/sec 30v
Thus although andE may be comparable to TS
3/5VE VS
TSEF
For the M2 clusterN=150,000 starsR= 175 light years
3010 2M Kg
1~1
~
~ ~22
)O(R
GNM
TTS
E
NMSR
MGNE
One may implement the large T limit by rescalingthe Hamiltonian
VEHVH d /
1S )(2 i
2
1
N
iiSN
JH
Although the canonical thermodynamic functions (free energy,entropy etc) are extensive, the system is non-additive
+ _
4/JNEE 0E
21 EEE
For example, consider the Ising model:
Features which result from non-additivity
Negative specific heat in microcanonical ensemble
Inequivalence of microcanonical (MCE) andcanonical (CE) ensembles
Breaking of ergodicity in microcanonical ensemble
Slow dynamics, diverging relaxation time
Thermodynamics
Dynamics
Temperature discontinuity in MCE
Some general considerations
Negative specific heat in microcanonical ensembleof non-additive systems.Antonov (1962); Lynden-Bell & Wood (1968); Thirring (1970), Thirring & Posch
coexistence region in systems with short range interactions
E0 = xE1 +(1-x)E2 S0 = xS1 +(1-x)S2
hence S is concave and the microcanonicalspecific heat is non-negative
S
1E 2E E0E
On the other hand in systems with long range interactions(non-additive), in the region E1<E<E2
S
1E 2E E
E0 = xE1 +(1-x)E2
S0 xS1 +(1-x)S2
The entropy may thus follow the homogeneoussystem curve, the entropy is not concave. andthe microcanonical specific heat becomesnegative.
0 222 EECT V
compared with canonical ensemble where
0E
Ising model with long and short range interactions.
)1(2
)(2 1
1
2
1
i
N
ii
N
ii SS
KS
N
JH
d=1 dimensional geometry, ferromagnetic long range interaction J>0
The model has been analyzed within the canonical ensemble Nagel (1970), Kardar (1983)
Canonical (T,K) phase diagram
0M
0M
Microcanonical analysis
)1(2
)(2 1
1
2
1
i
N
ii
N
ii SS
KS
N
JH
KUMN
JE 2
2
NNN
NNMU = number of broken bonds in a configuration
Number of microstates:
2/2/),,(
U
N
U
NUNN
U/2 (+) segments U/2 (-) segments
Mukamel, Ruffo, Schreiber (2005); Barre, Mukamel, Ruffo (2001)
s=S/N , =E/N , m=M/N , u=U/N
)1ln()1(2
1)1ln()1(
2
1
ln)1ln()1(2
1)1ln()1(
2
1
ln1
),(
umumumum
uummmm
Nmus
but
Maximize to get ),( ms )(m
))(,()( mss
KumJ
2
2
//1 sT
),(),( msmus
canonical microcanonical
359.0/
317.032
3ln/
JT
JT
MTP
CTP
The two phase diagrams differ in the 1st order region of the canonical diagram
Ruffo, Schreiber, Mukamel (2005)
s
m
discontinuous transition:
In a 1st order transition there is a discontinuity in T, and thus thereis a T region which is not accessible.
//1 sT
S
E
In general it is expected that whenever the canonical transitionis first order the microcanonical and canonical ensemblesdiffer from each other.
S
1E 2E E
Dynamics
Systems with long range interactions exhibit slow relaxation processes.
This may result in quasi-stationary states (long livednon-equilibrium states whose relaxation time to the equilibrium state diverges with the system size).
Non-additivity may facilitate breaking of ergodicity which could lead to trapping of systems in non-Equilibrium states.
Long range correlations in driven systems Conserved variables tend to produce long range correlations.
Thermal equilibrium states are independent of the dynamics(e.g. Glauber and Kawasaki dynamics result in the same Boltzmann distribution)
Non-equilibrium steady states depend on the dynamics(e.g. conserving or non-conserving)
Conserving dynamics in driven, non-equilibrium systemsmay result in steady states with long range correlationseven when the dynamics is local
Can these correlations be viewed as resulting from effectivelong-range interactions?
Driven systems
ABC model
One dimensional driven model with stochastic local dynamicswhich results in phase separation (long range order) where thesteady state can be expressed as a Boltzmann distribution of aneffective energy with long-range interactions.
A B C
ABC Model
AB BA1
q
BC CB1
q
CA AC1
q
dynamics
Evans,Kafri, Koduvely, Mukamel PRL 80, 425 (1998)A model with similar features was discussed by Lahiri, Ramaswamy PRL 79, 1150 (1997)
Simple argument:
AB BA1
q
BC CB1
q
CA AC1
q
ACCCC CCCCA
CBBBB BBBBC
BAAAA AAAAB
…AACBBBCCAAACBBBCCC…
…AABBBCCCAAABBBCCCC…
…AAAAABBBBBCCCCCCAA…
fast rearrangement
slow coarsening
The model reaches a phase separated steady state
logarithmically slow coarsening
…AAAAABBBBBCCCCCCAA…
tlqt l ln
needs n>2 species to have phase separation
strong phase separation: no fluctuation in the bulk;only at the boundaries.
…AAAAAAAAAABBBBBBBBBBBBCCCCCCCCCCC…
Phase separation takes place for any q (except q=1)
Phase separation takes place for any density N , N , N A B C
N N NA B C Special case
The argument presented before is general, independent of densities.
For the equal densities case the model has detailed balance for arbitrary q.
We will demonstrate that for any microscopic configuration {X}
One can define “energy” E({X}) such that the steady state
Distribution is
})({})({ XEqXP
AAAAAABBBBBBCCCCC E=0
……AB….. ……BA….. E E+1
……BC….. ……CB….. E E+1
……CA….. ……AC….. E E+1
...)(...)(...)(...)( BAPABBAWABPBAABW
With this weight one has:
=q =1
qABPBAP ...)(.../...)(...
AAAAABBBBBCCCCC AAAABBBBBCCCCCA
E E+NB-NC
NB = NC
Thus such “energy” can be defined only for NA=NB=NC
This definition of “energy” is possible only for N N NA B C
AABBBBCCCAAAAABBBCCCC
The rates with which an A particle makes a full circle clockwiseAnd counterclockwise are equal
CB NN qq
Hence no currents for any N.
For the current of A particles satisfies CB NN CB NNA qqJ
The current is non-vanishing for finite N. It vanishes only in thelimit . Thus no detailed balance in this case.N
…AAAAAAAABBBABBBBBBCCCCCCCCCAA…
The model exhibits strong phase separation
The probability of a particle to be at a distance on the wrong side of the boundary is lql
The width of the boundary layer is -1/lnq
N N NA B C The “energy” E may be written as
summation over modulo )( ki N• long range
Local dynamics
xEqxP
1
2
1 1
( / 3)N N i
i i k i i k i i ki k
E x C B AC B A N
N
i
N
kkiikiikii ABCABC
N
kxE
1
1
1
1
Partition sum
nqnpqZ )()(1
Excitations near a single interface: AAAAAAABBBBBB
P(n)= degeneracy of the excitation with energy n
P(0)=1P(1)=1P(2)=2 (2, 1+1)P(3)=3 (3, 2+1, 1+1+1)P(4)=5 (4, 3+1, 2+2, 2+1+1, 1+1+1+1)
P(n)= no. of partitions of an integer n
nn
nnp
34
)3/2exp(~)(
Weakly asymmetric ABC modelNeq /
The model exhibits a phase transition atfor the case of equal densities
32 c
c
c
homogeneous
inhomogeneous
9/1 1
1 1
NABCABCN
xEN
i
N
kkiikiikii
effective rescaled “energy”
Clincy, Derrida, Evans, PRE 67, 066115 (2003)
Generalized ABC model
add vacancies: LNNNNN CBA , 3/
A , B, C, 0
AB BA1
q
BC CB1
q
CA AC1
q
0X 0X1
1X=A,B,C
Dynamics
A. Lederhendler, D. Mukamel
grand-canonical dynamics
AB BA1
q
BC CB1
q
CA AC1
q
0X 0X1
1
ABC 0001
q
(6 1) 3N L
X=A,B,C
…A00ACBABCCA00AACBBB00000CCC…
ABC000
1
2
1 1
L L i
i i k i i k i i ki k
E x C B AC B A N NL
3)16()},({)3},({ NNXENXE
The dynamics is local for 6/1
For NA=NB=NC: there is detailed balance with respect to
})({})({ XEqXP
/1 , , )),(( / TkeqTSETxF BN
1 , 0CBA
continuum version of the model
21
0
1
0
1
0
)()()()()()()(
xdxzxxzxxzxxdzdxE BCCAAB
)(ln)()(ln)()(ln)()(ln)( 00
1
0
xxxxxxxxdxS CCBBAA
canonical
grand canonical
)),(( TxF
)),((),( TxFTG
6
1 ,T=0.02, Grand canonical
6
1 ,T=0.02, Canonical – 2nd order transition at 2177.0c
6
1 ,T=0.02, Grand canonical – 1st order transition
049.0Correlations for both solutions with
Summary
Local stochastic dynamics may result in effective long-range interactions in driven systems.
This is manifested in the existence of phase transitionsin one dimensional driven models.
Existence of effective long range interactions can be explicitlydemonstrated in the ABC model.
The model exhibits phase separation for any drive
Phase separation is a result of effective long-rangeinteractions generated by the local dynamics.
Inequivalence of ensembles in the driven model.
1q