1

Pair Contact Process withDiffusion

Uwe Tauber, Olivier Deloubriere and Frederic van Wijland

Department of Physics, Virginia Polytechnic Institute and State University,Blacksburg, Virginia, USA.

ITF, Utrecht ; LPT, Orsay and Pole Matiere et Systemes Complexes, ParisVII.

fvw@th.u-psud.fr

1Non-Equilibrium Statistical Physics in Low Dimensions and Reaction Diffusion Sys-tems, Dresden, 22/09/2003 – 10/10/2003

Outline

•Motivations

•PCPD (definition, known facts)

•State of the art

•Systematic study of fluctuations (technical)

•Consequences for the critical behavior

•Other processes

Phase transitions in NonEquilibrium Steady-States

•Understanding NESS

•Long-range order without macroscopic currents

•Universality

•Absence of rationalized classification

Absorbing State Transitions

•A particular subclass

•Transition from an active state to a frozen state (possibly degenerate)

•Specific analytical and numerical difficulties

•Experimental issue

Pair Contact Process with Diffusion

•Contact ProcessA → 2A, A → ∅

with exclusion, without diffusion;

•Pair Contact Process

2A → 3A, 2A → ∅

with exclusion, without diffusion;

•�

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�Pair Contact Process with Diffusion

2Aλ→ 3A, 2A

k→ ∅

with exclusion, with diffusion;

•Higher order processes

•Coupling to conserved modes

Known analytical facts about the PCPD

•Mean-field equation of the local density field ρ(x, t) = 〈nA(x, t)〉:

∂tρ = D∆xρ − 2kρ2 + λρ2(1 − ρ)︸ ︷︷ ︸

λρ2−λ′ρ3

•Predicted phase diagram

•For k ≥ kc the system falls into an absorbing, particle-free, state.

•For k < kc it reaches a stationary nonzero density

ρ(∞) =λ − 2k

λ′∼ ∆β , ∆ = kc − k

•At k = kc,

ρ(t) =1√2λ′t

∼ t−δ

•Relaxation to the absorbing state is as 1/t

•Relaxation to the active state is exponentially fast, with typical timeτ ∼ ∆−2.

•Correlation length in the active phase is ξ ∼ τ 1/2.

•β = 1, δ = 1/2, ν = 1 and z = 2.

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�Keep in mind the λ′ dependence.

Relevant questions

•Upper critical dimension above which mean-field is valid

•Critical properties below dc

– .New universality class?

– .or known one, like the contact process or the parity conservingone.

– .Understanding why simulations are difficult.

Answers from the simulation experts in d = 2

•G. Odor, M.C. Marques and M.A. Santos (2002): Effective exponentδ

0.48 ≤ δ ≤ 0.55

depending on the diffusion constant.

•H. Chate (2003): Again a slightly faster-than-mean-field decay atcriticality.

•Similar behavior for P. Grassberger (19??).

• Probably mean-field with strong corrections. Hints at dc = 2.

Answers from the simulation experts in d = 1

•Odor, Chate, Kockelkoren: There is an independent universality classwith δ = 0.20.

•Carlon, Barkema: This is the Directed Percolation (or Contact Pro-cess) class.

•All: z ' 1.7 � 2.

•All: Absorbing phase follows A + A → ∅ kinetics. There is an entirecritical line.

•Barkema, Chate: The density of p-uplets scales with time as theparticle density.

•Dornic, Chate: Anticorrelations, which are present in the absorbingphase do not survive up to the critical point.

Analytic strategies

•Howard, Tauber: without mutual exclusion of particles there is nei-ther a critical point (see poster by Paessens and Schutz) nor a steady-state and there are an infinite number of generated vertices;

•Hinrichsen: Phenomenological Langevin equations for ρ(x, t). Doesthis exist? How to choose the noise?

∂tρ = ∆ρ + (λ − 2k)ρ2 − λ′ρ3 + η

〈η(x, t)η(x′, t′)〉 = ...ρ2δ(t − t′)δ(d)(x − x′)

Master equation

•P (n = {ni}, t) is the probability to observe the microscopic config-uration of occupation numbers n.

•|Ψ〉 ≡ ∑

n P (n, t)|n〉.

•Master equation

∂tP (n) =∑

n′

W (n′ → n)P (n′) −∑

n′

W (n → n′)P (n)

is equivalent tod|Ψ〉dt

= −H|Ψ〉

with appropriately chosen H.

• Examples:

Hannihilation = k∑

x

(a†2

x − 1)a2

x

Bosonic field theory with built-in exclusion

• H may evolve only |0〉 or |1〉 states.

• Spin chain formulation not convenient when nonintegrable (all phasetransitions in d ≥ 1, all in d ≥ 2).

• Incorporate exclusion at least for the branching process:

i

Ai+1

Ai+2

∅ →i

Ai+1

Ai+2

A

H = λ[

1 − a†i+2δni+2,0 − δni+2,0ai+2

]

δni,1δni+1,1δni+2,0

• Exact mapping to a field theory: normal order, replace creation a†x

and annihilation ax operators by fields a(x, t) and a(x, t).

Interaction vertices (bare action)

branching and exclusion

branching

annihilation and branching

Bubbles everywhere

=

a2a5

•Problem: all branching processes

2A → (n + 2)A, n ≥ 1

are wildly generated;

•Big problem: all of them are equally relevant;

•Renormalizable theory?

+

+ +...

= + +

u3

2

u1u2u3

u2

2u2

u2

u3u1

Simple functional RG

• Write the action in the form

Sinteraction =

∫

∑

n≥1

unana2 +∑

n≥1

vnana3

• Identify dc = 2 as the upper critical dimension, and set ε = 2 − d.

• Microscopic couplings are

u1 = 2k − λ, u2 = k − 2λ

• DefineU(x) =

∑

n≥1

unxn

V (x) =∑

n≥1

vnxn (irrelevant)

• Sit at space scale e` and define scale-dependent couplings:

U(x, `) =∑

n≥1

un(`)xn

• Sit at space scale e` and define scale-dependent couplings:

U(x, `) =∑

n≥1

un(`)xn

•�

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�Beautiful miracle

∂`U = εU − 1

2U∂2

xU

with ε = 2 − d.

•Neither the field nor the diffusion constant pick up any anomalousdimension.

Analysis of the flow

• Fixed function is

G∗(x) = ∆x + εx2, ∆ = a constant

• Effective action at the fixed point is

S =

∫[∆aa2 + εa2a2

]

• Problem: such an effective action does not yield the expected phasediagram.

•�

�

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�Do not throw all irrelevant couplings

• Keep those that are responsible for producing a nontrivial phasediagram.

A dangerously irrelevant coupling

•RG irrelevance is not synonymous for physical irrelevance.

•To one-loop,

∂`V = (−2 + 2ε)V − 1

2∂2

xV U − 3

2V ∂2

xU

•Among the infinite number of couplings flowing to 0 only one reallymatters.

•The v1(0) = λ′ eventually flows to zero as

v1(`) ∝ exp(y`), y = −2 − ε + 3 ln4

3ε2 + O(ε2)

•Find the effective equation of state at scale e` with v1 and solve it.

Critical behavior of the PCPD near dc = 2

• Critical exponents:

δ =1

2− 3

4ε2 +

3

4ln

4

3ε2 + O(ε3)

β = 1 − 2ε + O(ε2)

ν = 1 +1

2ε + O(ε2)

• But there are two results valid to all orders in ε:

z = 2, η = 0

and a hyperscaling relation

β =2δ

d − 2δ

• Relaxation to the absorbing state follows the annihilation kinetics(as t−d/2).

PCPD at the upper dc

• Logarithmic corrections in every corner.

• Active phase:

ρ(t → ∞) ∼ ∆ ln2 ∆, ξ ∼ 1

∆ ln1/2 ∆

• Critical point:

ρ(t) ∼ ln3/2 t

t1/2

• Relaxation to the absorbing state:

ρ(t) ∼ ln t

t

Critical comments

• Do the initial conditions lie in the basin of attraction of our fixedpoint?

• Couplings eventually responsible for anticorrelations starts in a pos-itive correlation region.

• Weird things happening at finite ε? Note the bad convergence ofthe ε expansion. Or is it just bad luck?

Comparison with numerical results

•Agreement that dc = 2.

•But the faster-than-mean-field decay in d = 2 is a sign that asymp-totics might not be reached.

•Needless to say both exact results on the exponents disagree withexisting simulations.

•Anticorrelations at variance with the clustering observed in all simu-lations.

Triplet Contact Process with Diffusion

•Triplets involved:

3Aλ→ 4A, 3A

k→ ∅with exclusion, with diffusion;

•dc = 1,

ρ(t) ∼ ln4/3 t

t1/3

•Quite consistent with δ = 0.27 in Kockelkoren-Chate’s simulations.is mean-field in d = 1.

•Similar but higher order processes are mean-field down to d = 1.

Conserved Pair Contact Process with Diffusion

• Idea: introduce an auxiliary species tuned to ensure conservation ofparticles. For instance

2A + Bλ→ 3A, 2A

k→ 2B

• The auxiliary species may be diffusing (faster or slower than theorder parameter) or static.

• Number of accessible absorbing states may become infinite.

• First order transition for DA < DB .

• Discrepancies with simulations expected to be even stronger, but forDA > DB ≥ 0 all exponents are known exactly

d ≤ 2, δ =d

4, z = 2

Final comments

•Approaching a complete classification of absorbing phase transitions.

•Hope to bring some measurable quantities in d = 2 and a couple ofexact results in d = 1 or 2.

•Numerical issue.