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Introduction Directed Percolation
Non-equilibrium phase transitionsAn Introduction
Lecture I
Haye Hinrichsen
University of Würzburg, Germany
March 2006
Introduction Directed Percolation
Outline
1 IntroductionStochastic Many-Particle SystemsEquilibrium Statistical MechanicsEquilibrium and Nonequilibrium Dynamics
2 Directed PercolationFrom Isotropic to Directed PercolationThe DP Phase TransitionDP Lattice Models
Introduction Directed Percolation
Outline
1 IntroductionStochastic Many-Particle SystemsEquilibrium Statistical MechanicsEquilibrium and Nonequilibrium Dynamics
2 Directed PercolationFrom Isotropic to Directed PercolationThe DP Phase TransitionDP Lattice Models
Introduction Directed Percolation
Stochastic Many-Particle Systems– Basic Assumptions
Classical Statistical Physics
At a given instance of time the system is in a certainconfiguration c.
All possible configurations c form a set of configurations C.
Transitions c → c′ from one configuration to a different oneoccur spontaneously at a certain rate wc→c′ .
Introduction Directed Percolation
Transition rates
...trivial, but often a source of mistakes...
Rates are probabilities per unit time.
Rates carry the same dimension as 1/t .
The numerical value of a rate can be larger than 1.
In a computer simulation rates have to be multiplied by atime interval ∆t to get a probability.
if (w*dt < random(0,1)) ...
Introduction Directed Percolation
Direct Monte-Carlo Simulations
1. Choose an initial configuration and set t = 0.
2. Compute total outgoing rate Wc =∑
c′∈C wc→c′ .
3. Select new configuration c′ with probability wc→c′/Wc .
4. Increase t by 1/Wc*.
5. Monitor quantities of interest and continue while t < tmax .
* Approximation, good if t � 1/Wc
Introduction Directed Percolation
Direct Monte-Carlo Simulations
Approximation:
Poisson statistics (like radioactive decay) → regular intervals
Introduction Directed Percolation
Time-dependent Probability Distribution
The individual trajectory of aclassical stochastic system is unpredictable.
Probability Distribution
Pt(c) is the probability to find the system at time tin the configuration c.∑
c∈CPt(c) = 1
The evolution of Pt(c) is predictable.
Introduction Directed Percolation
Master Equation
The master equation is a linear deterministic evolutionequation that describes the flow of probability:
∂
∂tPt(c) =
∑c′
wc′→cPt(c′)︸ ︷︷ ︸gain
−∑c′
wc→c′Pt(c)︸ ︷︷ ︸loss
.
In the continuum also called Fokker-Planck equation.(do not mix up with Langevin equations, see below.)
Introduction Directed Percolation
Compact Notation – Time Evolution Operator
List all the probabilities in a vector:
|Pt〉 =
Pt(c1)Pt(c2)Pt(c3). . .
∈ [0,1]⊗|C|
Write Master equation in a compact form:
∂t |Pt〉 = −L|Pt〉
L is called Liouville operator.
Introduction Directed Percolation
Compact Notation – Formal Solution
The Lioville operator L is defined by the matrix elements
〈c′|L|c〉 = −wc→c′ + δc,c′∑c′′
wc→c′′ .
Formal solution:|Pt〉 = e−Lt |P0〉 ,
where |P0〉 is the initial probability distribution at t = 0
...looks like quantum mechanics...
Introduction Directed Percolation
Compact Notation – Conservation of Probability
Probability conservation ∑c∈C
Pt(c) = 1
can be expressed as〈1|Pt〉 = 1 ,
where〈1| :=
∑c∈C
〈c| = (1,1,1,1, . . . ,1)
⇒〈1|L = 0
Introduction Directed Percolation
Compact Notation – Structure of L
The matrix of the time evolution operator in configuration basis
has positive entries on the diagonal,has negative non-diagonal entries,and the sum over columns vanishes.
Example: L =
2 −3 −5−1 5 −1−1 −2 6
Such matrices are called intensity matrices.
Introduction Directed Percolation
Solving the Master Equation by Diagonalization
The master equation can be solved by diaogalization of L:1. Solve eigenvalue problem:
L|φi〉 = λi |φi〉
2. Express initial state |P0〉 in terms of eigenvectors:
|P0〉 =∑
i
ai |φi〉
3. Write down the solution:
|Pt〉 = e−Lt |P0〉 =∑
i
ai e−λi t |φi〉
solution = sum over exponentially decaying eigenmodes
Introduction Directed Percolation
Spectrum of the Time Evolution Operator
The time evolution operatoris non-Hermitean, meaning that left and right eigenvectorsto the same eigenvalue may have different representations.
Because of 〈1|L = 0 there is at least one eigenvalueλ0 = 0, representing the stationary state.
All eigenvalues have a non-negative real part.
Eigenvalues may occur in complex-conjugate pairs.
complex? damped oscillations?
Introduction Directed Percolation
Complex Eigenvalues of L – Chemical Oscillations
Belusov-Zhabotinsky-Reaction
http://online.redwoods.cc.ca.us/instruct/darnold/deproj/
Introduction Directed Percolation
Complex Eigenvalues - Chemical Oscillations
Chemical oscillations occur in activator-inhibitorreactions:
Introduction Directed Percolation
Complex Conjugate Eigenvalues - ChemicalOscillations
Even spatio-temporal oscillations can be observed...
Introduction Directed Percolation
Complex Conjugate Eigenvalues - ChemicalOscillations
...and may be simulated by partial differential equations:
Introduction Directed Percolation
Classical Stochastic Dynamics compared toQuantum Mechanics
Quantum Mechanics∫ψ∗(x)ψ(x)dx = 1 〈ψ|ψ〉 = 1 H = H† U−1 = U†
Classical Stochastic Dynamics∑c Pt(c) = 1 〈1|Pt〉 = 1 〈1|L = 0 〈1|U = U
...look similar, but differ significantly.
Introduction Directed Percolation
Spectrum of the Time Evolution Operator
Relaxation times
The relaxation time of an eigenmode |φi〉 is (Reλi)−1
Systems with finite configuration space relax to theirstationary state exponentially within finite timedetermined by the eigenvalue with the smallest real part.
Conversely: Systems exhibiting infinite relaxation times orpower laws must have an infinite configuration space.
Introduction Directed Percolation
Spectrum of the Time Evolution Operator
Relaxation times
The relaxation time of an eigenmode |φi〉 is (Reλi)−1
Systems with finite configuration space relax to theirstationary state exponentially within finite timedetermined by the eigenvalue with the smallest real part.
Conversely: Systems exhibiting infinite relaxation times orpower laws must have an infinite configuration space.
Introduction Directed Percolation
Spectrum of the Time Evolution Operator
Relaxation times
The relaxation time of an eigenmode |φi〉 is (Reλi)−1
Systems with finite configuration space relax to theirstationary state exponentially within finite timedetermined by the eigenvalue with the smallest real part.
Conversely: Systems exhibiting infinite relaxation times orpower laws must have an infinite configuration space.
Introduction Directed Percolation
Outline
1 IntroductionStochastic Many-Particle SystemsEquilibrium Statistical MechanicsEquilibrium and Nonequilibrium Dynamics
2 Directed PercolationFrom Isotropic to Directed PercolationThe DP Phase TransitionDP Lattice Models
Introduction Directed Percolation
Temperature
In equilibrium Nature maximizes theentropy S of the entire system.
The total entropy is extensive: S = S1 + S2
The total energy is conserved: E = E1 + E2
⇒ ∂S1
∂E1= −∂S1
∂E1=∂S2
∂E2
⇒ The quantity 1T := ∂S
∂E is the same in both subsystems.
Temperature ⇔ equilibrium & energy conservation.
Introduction Directed Percolation
Canonical Ensemble – Boltzmann Distribution
Compute the probability to findsystem 1 in a certain configuration c.
P(c) ∝ # of possible configurations of the bath.
P(c) ∝ exp(
1kB
S2(E2)
)=; exp
(1kB
S2(E − E1)
)
≈ exp(
1kB
(S2(E)− ∂S2(E)
∂EE1 + . . .)
)
∝ exp(− E1
kBT
).
Introduction Directed Percolation
Equilibrium Statistical Mechanics
In equilibrium statistical mechanicsthere is no notion of time.
Equilibrium models are defined by anenergy functional c → E(c).
In equilibrium statistical mechanics we getthe probability distribution for free!
——————————————————–
Nonequilibrium models are defined by aset of transition rates.
In nonequilibrium statistical mechanicsthe probability distribution is obtainedby solving the master equation.
Introduction Directed Percolation
Outline
1 IntroductionStochastic Many-Particle SystemsEquilibrium Statistical MechanicsEquilibrium and Nonequilibrium Dynamics
2 Directed PercolationFrom Isotropic to Directed PercolationThe DP Phase TransitionDP Lattice Models
Introduction Directed Percolation
Equilibrium and Nonequilibrium Dynamics
There are two types of stochastic dynamical processes:
1 Equilibrium dynamics:Dynamical processes which relax towards a stationarystate described by an equilibrium ensemble.
2 Nonequilibrium dynamics:All other processes.
Introduction Directed Percolation
Equilibrium Dynamics
Example:The following dynamical processes
1 Ising Heat bath dynamics2 Glauber-Ising model3 Ising Metropolis dynamics4 Ising Kawasaki dynamics∗5 Swendsen-Wang cluster algorithm6 Wolf cluster algorithm7 · · ·
relax towards a stationary state which is precisely theequilibrium ensemble of the Ising model.
Proof needed!
Introduction Directed Percolation
Detailed Balance
A system is said to obey Detailed Balance if the probabilitycurrents between pairs of configurations compensate eachother:
Peq(c) wc→c′ = Peq(c′) wc′→c .
Introduction Directed Percolation
Detailed Balance implies Stationarity ...
...but not vice versa:
P(A) = P(B) = P(C) = 13
Introduction Directed Percolation
Non-equilibrium Dynamics
Broken detailed balance in the stationary state impliesthat the stationary state of the system is not in equilibrium
In particular, irreversible one-way flow...
...is always out of equilibrium.
Introduction Directed Percolation
Physical Conditions for Nonequilibrium
Flow of energy through the system:
Introduction Directed Percolation
Physical Conditions for Nonequilibrium
Flow of particles through the system:
Introduction Directed Percolation
Continuous Phase Transitions
Continuous phase transitions at equilibrium:
can be categorized into universality classes .
are particularly well understood in two dimensions,where conformal invariance leads to a classificationscheme.
Nonequilibrium phase transitions ?
Introduction Directed Percolation
Phase transitions into absorbing states
An absorbing state is a configuration from wherethe system cannot escape:
borrowed from stud.fh-wedel.de
Systems with reachable absorbing states are out of equilibrium.
Standard example: Directed Percolation (DP).
Introduction Directed Percolation
Outline
1 IntroductionStochastic Many-Particle SystemsEquilibrium Statistical MechanicsEquilibrium and Nonequilibrium Dynamics
2 Directed PercolationFrom Isotropic to Directed PercolationThe DP Phase TransitionDP Lattice Models
Introduction Directed Percolation
From Isotropic to Directed Percolation
isotropic percolation directed percolation
Introduction Directed Percolation
Directed Percolation as a Dynamical Process
The preferred direction can be interpreted as time t .
Introduction Directed Percolation
DP as a Reaction-Diffusion Process
a) b) c) d) e)
(a) death process A → 0(b-c) diffusion
(d) offspring production A → 2A(e) coagulation 2A → A
Introduction Directed Percolation
Outline
1 IntroductionStochastic Many-Particle SystemsEquilibrium Statistical MechanicsEquilibrium and Nonequilibrium Dynamics
2 Directed PercolationFrom Isotropic to Directed PercolationThe DP Phase TransitionDP Lattice Models
Introduction Directed Percolation
DP exhibits a Phase Transition
...starting with a fully active lattice:
Introduction Directed Percolation
DP exhibits a Phase Transition
...starting with a single active site:
Introduction Directed Percolation
Order Parameter: Number of Active Sites N(t)
Let us measure 〈N(t)〉 for various q.
Introduction Directed Percolation
This little program does it:
#include <fstream.h>using namespace std;const int T=1000; // number of updatesconst int R=10000; // number of independent runsconst double p=0.6447; // percolation probability
double rnd(void) { return (double)rand()/0x7FFFFFFF; }
int main (void){int s[T][T],N[t]i,t,r;for (t=0; t<T; t++) N[t]=0; // clear cumulative particle numberfor (r=0; r<R; r++) { // loop over R runs
s[0][0]=1; // place initial seedfor (t=0; t<T-1; t++) { // temporal loop
for (i=0; i<=t+1; ++i) s[t+1][i]=0; // clear new configfor (i=0; i<=t; ++i) if (s[t][i]==1) { // loop over active sites
N[t]++; // count active sitesif (rnd()<p) s[t+1][i]=1; // random activation leftif (rnd()<p) s[t+1][i+1]=1; // random activation right
} } }ofstream os ("N.dat"); // write average N(t) to filefor (t=0; t<T-1; t++) os << t << ' ' << (double)N[t]/R << endl;}
Introduction Directed Percolation
This is the Output (via xmgrace):
1 10 100 1000t
0,01
0,1
1
10
100
<N
(t)>
p=0.7p=0.65p=0.6447 (critical)p=0.64p=0.6
0 100 300 400t10
-2
10-1
100
<N
(t)>
Introduction Directed Percolation
Result for 〈N(t)〉:
For q < qc 〈N(t)〉 crosses over to an exponential decay.
For q > qc 〈N(t)〉 tends to a linear increase.
At q = qc 〈N(t)〉 increases algebraically as tθ
with an exponent θ ≈ 0.302.
Introduction Directed Percolation
UNIVERSALITY
The exponent
θ = 0.313686(8)
is the same in all realizations of DP.It only depends on the dimension d .
DP stands for a universality classof non-equilibrium phase transitions.
DP is the ’Ising model’ of non-equilibrium.
Introduction Directed Percolation
UNIVERSALITY
All long-range properties, especially critical exponentsand scaling functions, are universal, i.e., they are thesame for all models in the DP universality class.
Short-range properties and critical threshold (pc) arenon-universal.
→ Examples of different DP models?→ Can we characterize the range of DP models?
Introduction Directed Percolation
Outline
1 IntroductionStochastic Many-Particle SystemsEquilibrium Statistical MechanicsEquilibrium and Nonequilibrium Dynamics
2 Directed PercolationFrom Isotropic to Directed PercolationThe DP Phase TransitionDP Lattice Models
Introduction Directed Percolation
Domany-Kinzel Cellular Automaton
Transition probabilities:
P[1|0,0] = 0P[1|0,1] = P[1|1,0] = p1
P[1|1,1] = p2
Introduction Directed Percolation
Domany-Kinzel Cellular Automaton
p 2p1s p 1i
t+1
t
i-1s i+1s
Algorithm:
For all i generate a random number zi ∈ [0,1] and set:
si(t + 1) =
1 if si−1(t) 6= si+1(t) and zi(t) < p1 ,1 if si−1(t) = si+1(t) = 1 and zi(t) < p2 ,0 otherwise .
Introduction Directed Percolation
Domany-Kinzel Cellular Automaton
Special cases:
case restriction p1,c p2,c
Wolfram 18 p2 = 0 0.801(2) 0
Site DP p2 = p1(2− p1) 0.70548515(20) 0.70548515(20)
Bond DP p1 = p2 0.644700185(5) 0.873762040(3)
Compact DP p2 = 1 1/2 1
Introduction Directed Percolation
Domany-Kinzel Model: Phase Diagram
0 1p
1
0
1
p2
inactive phase
activephase
compact DP
bond DP
site DP
rule 18Wolfram
Introduction Directed Percolation
Domany-Kinzel Model: Phase Diagram
site DP bond DP compact DPWolfram 18
Compact DP is not DPbut it is the same as Glauber-Ising at T = 0.
All the rest of the transition line belongs to DP.