Chi-Jen Wang�Da-Jiang Liu, James W. Evans
NSTC, 26 Nov 2025
Triplet approximation in Schloegl’s second model for non-equilibrium phase transitions in lattice bistable systems
Schloegl第二模型於晶格雙穩態系統中非平衡相變的三階近似
p
1/3
1
NSTC Grand 111-2628-M-194-001-MY3 �Schloegl第二模型於空間流行病模型中的高階近似
…共15組方程式
TODAY�…共6組方程式
二維網格的三階近似
Bethe 網格的三階近似
二維網格的近似滴液解
DISCONTINUOUS PHASE-TRANSITIONS IN
NON-EQUILIBRUM SYSTEMS
Background: Discontinuous Transitions in Equilibrium Systems
P
pressure
n
non-ideal gas
Liquid = L
Gas = G
density
Van der Waals equation of state
(with van der waals loop)
P
pressure
n
non-ideal gas
Liquid = L
Gas = G
density
Metastable Gas = MG
ML
Maxwell
construction
Actual
behavior
Pe
Discontinuous phase transition
at “equistability” pressure Pe
gas
liquid
Coexistence at p=pe
Stationary planar interface
P
L
G
MG
ML
Pe
n
CONDENSATION P>Pe
L
L
MG
Super-
critical
liquid
droplet
grows
Sub-critical
shrinks
Bethe lattice
Bethe lattice is an infinite connected cycle-free graph where
each node is connected to z neighbors
z=3
z=4
finite graph�z=3
infinite graph
Related Works – finite graph
Kouvaris, Sebek, Mikhailov, Kiss 2016
ui is the chemical concentration, f(u) = −u(u − h)(u − a) with 0<h<a,
K is the strength of diffusive coupling,�Aij=1 if there is a edge connects node i and node j, Aij=0 otherwise.
Related Works
Kouvaris, Sebek, Mikhailov, Kiss 2016
Related Works – 1st order phase trans
Chatterjeea and Durrett, 2013
Consider the discrete time threshold-θ contact process on a random r-regular graph and on trees in which all vertices have degree r.
Sites having at least θ many occupied neighbors at time t become occupied at time t + 1 with probability p.
Schloegl’s second model (quadratic contact process) on a lattice involves �1. spontaneous particle annihilation at rate p, and
2. autocatalytic particle creation at empty
sites with two or more occupied neighbors.
Models of the work
p
1/3
1
p
1/3
1
Models of the work on Bethe lattice(z=3)
Labeling the nodes j on Bethe lattice
Labeling the rings k
�
Quadratic contact process
i i i
Radius symmetry�set Ck = P∙k for the “concentration” of particles in ring k.
1. annihilation rate p at filled site i
2. particle creation rate rn at empty site i
P∙k + Pok = 1
Liu Wang Evans �Phys. Rev. E. 104, 014135 (2021)
Creation rate: rn=nC2/zC2 =n(n-1)/z(z-1) , �for n occupied neighbors out of z neighbors
p
1/3
1/3
Mean Field (MF) approximation
Visualizaton for z=3
Radius symmetry assumption
Master Equation:
P∙k-1–ok–∙k+1 ≈ P∙k-1 Pok P∙k+1 = Ck-1(1-Ck)Ck+1
∙k+1 � Pok–∙k+1 ≈ Pok P∙k+1 P∙k+1 = (1-Ck)Ck+1Ck+1
Ck = P∙k for the “concentration” of particles in ring k.
Mean Field
Approximation
d/dt Ck = -p Ck + (1-Ck) Ck+1 [2 Ck-1 + (z-2) Ck+1]/z, for ring k ≥1,
MF version lattice differential equations (discrete reaction-diffusion equations) (dRDE)
p
Mean Field (MF) approximation
d/dt C0 = -p C0 + (1-C0)(C1)2 for ring k = 0.
C±(MF) = ½ ± ½ (1-4p)1/2 for 0 ≤ p ≤ ps(MF)= ¼, and Cvac = 0.
Here C+ and Cvac =0 are stable, and C- is an unstable steady state. �C+ and C- disappear beyond the spinodal point ps, a sn-bifurcation.
P∙k-1–ok–∙k+1 ≈ P∙k-1 Pok P∙k+1 = Ck-1(1-Ck)Ck+1
∙k+1 � Pok–∙k+1 ≈ Pok P∙k+1 P∙k+1 = (1-Ck)Ck+1Ck+1
Radius symmetry assumption
Ck = P∙k for the “concentration” of particles in ring k.
1/3
dC/dt = - pC + (1-C)C2� loss gain
Homogeneous assumption Ck = C for all sites.
KINETICS AND STEADY-STATES
BISTABILITY
populated stable
vacuum stable
PS =1/4
sn bifurcation
or “spinodal”
C
unstable
steady state
active stable =C+(MF)
= [1+√(1-4p)] /2
Steady-States
Spatial homogeneous case - MF
annihilation rate
ubstable = C-(MF) � =[1-√(1-4p)] /2
Radius Symmetric (Non-homogeneous)
- MF estimate of the propagation velocity
(Left) the populated steady state embedded in the vacuum state;
(Right) the vacuum state embedded in the populated steady state.
V>0
V<0
V>0
V<0
MF estimate of the propagation velocity
V>0
V<0
Pair approximation
Pair approximation : P∙k-1–ok–∙k+1
≈ (P∙k-1–ok) (Pok–∙k+1 ) / Pok
p
p
with neighboring correlations
k k
k-1 k-1
k k
k-2 k-1 k-1 k+2
k
k+2
not ODE yet
Spatially homogeneous – pair approximation
pair-approximation for homogeneous states
d/dt P∙ = -pP∙ + (P∙o)2/ Po
d/dt Poo = 2 P∙o [p – (z-2)P∙oPoo / z(Po)2 ]
Assumption P∙k = Ck = C for all sites.
Pair approximation : P∙–o–∙ ≈ (P∙–o) (Po–∙) / Po
Simplification
Set K = P∙k-1–ok / Pok = Pok–∙k+1 / Pok = P∙o/Po
� (a) d/dt P∙ = -pP∙ + (P∙o)2/ Po becomes � d/dt C = -pC + (1-C)K2
� (b) d/dt Poo = 2 P∙o [p – (z-2)P∙oPoo / z(Po)2 ]
d/dt (C-K(1-C)) = 2 K(1-C) [p – (z-2)K(1-K)/z]
homogeneous pair-approximation for z=3
(a) d/dt C = -pC + (1-C)K2
(b) d/dt (C-K(1-C)) = 2 K(1-C) [p – K(1-K)/3]
pair-approximation for homogeneous states
(a) d/dt P∙ = -pP∙ + (P∙o)2/ Po
(b) d/dt Poo = 2 P∙o [p – (z-2)P∙oPoo / z(Po)2 ]
Spatially homogeneous case - pair
stable populated steady state with
K±(pair) = ½ ± ½ [1 - 12p]1/2
C±(pair) = (K±)2/[p + (K±)2] = 1/{1+1/[K±/p - 3]}
= 1 - ½ {-1 + 4p ± [1 - 12p]1/2}/[1+4p],
for 0 ≤ p ≤ ps(pair) = 1/12≈0.08333, the spinodal point. ��C+ and C- correspond to� stable and unstable steady states.
Vacuum steady state
Cvac(pair) = Kvac(pair) = 0.
homogeneous pair-approximation for z=3
(a) d/dt C = -pC + (1-C)K2
(b) d/dt (C-K(1-C)) = 2 K(1-C) [p – K(1-K)/3]
populated stable
unstable
steady state
vacuum stable
PS =1/12
sn bifurcation
or “spinodal”
BISTABILITY
Steady-States
C
annihilation rate p
Pair estimate of the propagation velocity
p
1/3
1/3
Triplet approximation
for z=3
Radius symmetry assumption
Master Equation:
p
p
k k
k-1 k-1
k
k-1 k+2
k+2
Conti – Master Equation
for z=3
k+1
k-1 k+1
k
k+2
k
k+3
k+1
k+2
k+3
change rate of
P(ok-1–ok–ok+1)
change rate of
ok+1
P(ok–ok+1)
Conti – Master Equation
for z=3
P(ok-1–∙k–ok+1) can’t be expressed as a linear combination of 1, P(oi), P(oi oj), P(oi oj ol).
Unable to enclose (a),(b),(c),(d).
Need the changing rate description of the probability density P(ok-1–∙k–ok+1).
6 ODEs.
Kirkwood approximation
Kirkwood approximation [Kirkwood 1935 JCP]
Triplet approximation
Apply the triplet approximation to the master equations
(a)-(f).
Simplifications
Triplet approximation in Bethe lattice
…3 more eqs
Triplet approximation in Bethe lattice
Homogeneous (site independent)
Spatial homogeneous case - Triplet
3 more eqs for dF/dt, dG/dt, dH/dt.
Steady Staes - Homegeneous
Difficult to find algebraic form of steady states
Numerical� Results
Compare with KMC simulation
global steady-state concentration C versus p for Schloegl’s second model on a finite Bethe lattice with z = 3 and BC at R17. Each data point is obtained as an average over 104 Monte Carlo Steps (MCS).
Boundary conditions
How to characterize behavior on an infinite Bethe lattice?
annihilations and creations on the first k* rings 1≤k≤k*.
boundary conditions (BCs) imposed on ring k*+1
conventional �simulation �ensemble
period BCs in infinite Bethe lattice?
period BCs�
choices (K1, K2, K3) of BCs
KMC simulation results
(K1) sets K = K1 = <C>/(3-2<C>),
(K2) sets K = K2 = [p<C>/(1-<C>)]1/2
motivated by the pair approximation
(K3) sets K = K3, exactly from simulated configurations.
K = P∙o/Po
SUMMARY