Efficient Simulations Efficient Simulations of Gas-Grain Chemistry of Gas-Grain Chemistry Using Moment EquationsUsing Moment Equations
M.Sc. Thesisby
Baruch Barzelpreformed under the supervision of
Prof. Ofer Biham
The Interstellar Clouds
•Molecular and atomic H
•Density: ~10 -1000 (atoms cm-3)
•Gas Temperature: 50 -150 K
7
The H2 PuzzleH2 Production in the gas phase:
H + H → H2
Gas-Phase Reactions Cannot Account for the Observed Production Rates
Observed Production Rates in ISC:
RH ~ 10-15 (mol cm-3s-1)2
9
The Interstellar Dust Grains•Composition:
Carbons, Silicates, Olivine, H2O, SiC
•Temperature: ~5-20 K
•Size Range:
10-6-10-3 (cm) → 100-108 sites
•Activation Energies: (meV) MaterialE0 (diffus)E1 (disorp)
Carbon44.056.7
Olivine24.732.1
10
kBT
-E0
AH = (1/S) e
= FH - WH‹NH› - 2AH‹NH›2d‹NH› dt
The Rate Equation
Incoming fluxDesorption
Recombination
WH = e kBT
-E1
The Production Rate of H2 Molecules:
RH = AH‹NH›2 (mol s-1)2
11
Mean-field approximation
= FH - WH‹NH› - 2AH‹NH›2d‹NH› dt
When the Rate Equation Fails
•Neglects fluctuations•Ignores discretization
Not valid for small grains and low flux
12
Probabilistic ApproachP(0)
P(1)
P(NH-1)
P(NH)
P(NH+1)
P(NH+2)
P(Nmax)
Flux term:
FH[PH(NH-1) - PH(NH)]
Desorption term:
WH[(NH+1)PH(NH+1) - NHPH(NH)]
Reaction term:
AH[(NH+2)(NH+1)PH(NH+2) - NH(NH-1)PH(NH)]
FH
WHAH
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The Master Equation
= FH[PH(NH-1) - PH(NH)]
+ WH[(NH+1)PH(NH+1) - NHP(NH)]
+ AH[(NH+2)(NH+1)PH(NH+2) - NH(NH-1)PH(NH)]
dPH(NH)
dt
‹NH›= NHPH(NH)NH= 0
S
RH = AH (‹NH2› - ‹NH›)2
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The Rate Equations
= F1 - W1‹N1› - 2A1‹N1›2 - (A1+A2)‹N1›‹N2›
- (A1+A3)‹N1›‹N3›
d‹N1›
dt
= F2 – W2‹N2› - 2A2‹N2›2 - (A1+A2)‹N1›‹N2›d‹N2›
dt
= F3 - W3‹N3› - (A1+A3)‹N1›‹N3›+(A1+A2)‹N1›‹N2›d‹N3›
dt
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The Master Equation
P(N1,N2,N3) = Fi[P(…,Ni-1,…)-P(N1,N2,N3)]
+ Wi[(Ni+1)P(..,Ni+1,..)-NiP(N1,N2,N3)]
+ Ai[(Ni+2)(Ni+1)P(..,Ni+2,..)-Ni(Ni-1)P(N1,N2,N3)]
+ (A1+A2)[(N1+1)(N2+1)P(N1+1,N2+1,N3-1)-N1N2P(N1,N2,N3)
+ (A1+A3)[(N1+1)(N3+1)P(N1+1,N2,N3+1)-N1N3P(N1,N2,N3)
3
i=1
3
i=1
2
i=1
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P(N1,N2,N3) = Fi[P(…,Ni-1,…)-P(N1,N2,N3)]
+ Wi[(Ni+1)P(..,Ni+1,..)-NiP(N1,N2,N3)]
+ Ai[(Ni+2)(Ni+1)P(..,Ni+2,..)-Ni(Ni-1)P(N1,N2,N3)]
+ (A1+A2)[(N1+1)(N2+1)P(N1+1,N2+1,N3-1)-N1N2P(N1,N2,N3)
+ (A1+A3)[(N1+1)(N3+1)P(N1+1,N2,N3+1)-N1N3P(N1,N2,N3)
3
i=1
3
i=1
2
i=1
Rij = (Ai + Aj) ‹NiNj›
Rii = Ai (‹Ni2› - ‹Ni›)
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The Rate vs. The MasterRate equations:
•Mean field approximation
•High efficiency
•Not reliable for surface reactions (at low coverage)
Master equation:•Microscopic probability distribution
•Accurate model of grain surface reactions
•Low efficiency (exponential growth)
•Hard work
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The Moment Equations
‹NHk› = NH
kPH(NH)NH=0
8
After applying the summation:
‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›
‹NH2› = FH + (2FH + WH - 4AH)‹NH›
+ (8AH - WH)‹NH2› - 4AH‹NH
3›
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Truncating the Equations 1. Set the cutoff
2. Express the (k+1)th moment by the first k moments
‹NH1› = PH(1) + 2PH(2) + +kPH(k)
‹NH2› = PH(1) + 22PH(2) + +k2PH(k)
‹NHk› = PH(1) + 2kPH(2) + +kkPH(k)
PH(NH > k) = 0
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Truncating the Equations 1. Set the cutoff
2. Express the (k+1)th moment by the first k moments
3. Plug into the first k moment equations
‹NH1› = PH(1) + 2PH(2) + + kPH(k)
‹NH2› = PH(1) + 22PH(2) + +k2PH(k)
‹NHk› = PH(1) + 2kPH(2) + +kkPH(k)
PH(NH > k) = 0
‹NHk+1› = Ci‹NH
i›i=0
k
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Moment Equations for H2 Production
‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›
‹NH2› = FH + (2FH + WH - 4AH)‹NH›
+ (8AH - WH)‹NH2› - 4AH‹NH
3›
1. Set the cutoff → k=2
‹NH3› = 3‹NH
2› - 2‹NH›2. Reduce excessive moments →
3. Plug into the equations…
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‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›
‹NH2› = FH + (2FH + WH - 4AH)‹NH›
+ (8AH - WH)‹NH2› - 4AH‹NH
3›
‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›
‹NH2› = FH + (2FH + WH + 4AH)‹NH›
- (4AH + 2WH)‹NH2›
Moment Equations for H2 Production
‹NH3› = 3‹NH
2› - 2‹NH›
1. Set the cutoff → k=2
2. Reduce excessive moments →
3. Plug into the equations…
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Moments for Complex Networks
OH O2
H2
O
H
H2O OH
The probability: P(N1,N2,N3)
The moments: ‹N1aN2
bN3c›
The cutoff: Ni < ki
The challenge: Reduction of the excessive moments
‹N1aN2
bN3c› = Clnm‹N1
lN2nN3
m› lmn=0
k-1
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Reduction of Excessive MomentsThe probability: P(N1,N2)
V(a,b) M(N1,N2,a,b) P(N1,N2)
v = M p
‹N1aN2
b› = Cnm‹N1nN2
m› mn=0
k-1
‹N1aN2
b› = N1aN2
b P(N1,N2)N1N2=0
k-1
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‹N1›, ‹N3›‹N2›,
Setting the Cutoffs
OH O2
H2
O
H
H2O OH‹N1N2›
‹N1N3›
‹N22›
‹N12›
3 vertices + 2 edges + 2 loops = 7 equations
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Multi-Specie NetworkH2CO H3CO
OH
HCO H
O CO
CO2 + H
O2
H2
HCO
H2CO
OH
CO2
H3CO CH3CO
H2O
7 vertices
8 edges
2 loops
17 equations
+
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Summary• The advantages of the moment equations:
Reliable even for low coverage Efficient Linear Easy to incorporate into rate equation models Directly generate the required moments
• Further applications should be tested.