Efficient Simulations of Gas-Grain Chemistry Using Moment Equations M.Sc. Thesis by Baruch Barzel preformed under the supervision of Prof. Ofer Biham

2 Complexity in the Universe

3 Horse-Head Nebula

The Interstellar Clouds (ISC)

The Interstellar Clouds Molecular and atomic H Density: ~ (atoms cm -3 ) Gas Temperature: K

6 The Role of H 2 H2H2 Complexity Complex Molecules Star Formation

7 The H 2 Puzzle H 2 Production in the gas phase: H + H → H 2 Gas-Phase Reactions Cannot Account for the Observed Production Rates Observed Production Rates in ISC: R H ~ (mol cm -3 s -1 ) 2

8 The Solution

9 The Interstellar Dust Grains Composition: Carbons, Silicates, Olivine, H 2 O, SiC Temperature: ~5-20 K Size Range: (cm) → sites Activation Energies: (meV) E 1 (disorp)E 0 (diffus)Material Carbon Olivine

10 kBTkBT -E0-E0 A H = (1/S) e = F H - W H ‹ N H › - 2A H ‹ N H › 2 d ‹ N H › dt The Rate Equation Incoming flux Desorption Recombination W H = e kBTkBT -E1-E1 The Production Rate of H 2 Molecules: R H = A H ‹ N H › 2 (mol s -1 ) 2

11 Mean-field approximation = F H - W H ‹ N H › - 2A H ‹ N H › 2 d ‹ N H › dt When the Rate Equation Fails Neglects fluctuations Ignores discretization Not valid for small grains and low flux

12 Probabilistic Approach P(0) P(1) P(N H -1) P(N H ) P(N H +1) P(N H +2) P(N max ) Flux term: F H [P H (N H -1) - P H (N H )] Desorption term: W H [(N H +1)P H (N H +1) - N H P H (N H )] Reaction term: A H [(N H +2)(N H +1)P H (N H +2) - N H (N H -1)P H (N H )] FHFH WHWH AHAH

13 The Master Equation = F H [P H (N H -1) - P H (N H )] + W H [(N H +1)P H (N H +1) - N H P(N H )] + A H [(N H +2)(N H +1)P H (N H +2) - N H (N H -1)P H (N H )] dP H (N H ) dt ‹ N H › =  N H P H (N H ) N H = 0 S R H = A H ( ‹ N H 2 › - ‹ N H › ) 2

14 R H vs. Grain Size 2 F H = S (atoms s -1 ) E 0 = 22 E 1 =32 (meV) T surface = 10 K

15 Complex Reactions OHO2O2 H2H2 O H H2OH2O The parameters: F i ; W i ; A i (i=1,2,3) 13 2

16 The Rate Equations = F 1 - W 1 ‹ N 1 › - 2A 1 ‹ N 1 › 2 - (A 1 +A 2 ) ‹ N 1 ›‹ N 2 › - (A 1 +A 3 ) ‹ N 1 ›‹ N 3 › d ‹ N 1 › dt = F 2 – W 2 ‹ N 2 › - 2A 2 ‹ N 2 › 2 - (A 1 +A 2 ) ‹ N 1 ›‹ N 2 › d ‹ N 2 › dt = F 3 - W 3 ‹ N 3 › - (A 1 +A 3 ) ‹ N 1 ›‹ N 3 › +(A 1 +A 2 ) ‹ N 1 ›‹ N 2 › d ‹ N 3 › dt

17 The Master Equation P(N 1,N 2,N 3 ) =  F i [P(…,N i -1,…)-P(N 1,N 2,N 3 )] +  W i [(N i +1)P(..,N i +1,..)-N i P(N 1,N 2,N 3 )] +  A i [(N i +2)(N i +1)P(..,N i +2,..)-N i (N i -1)P(N 1,N 2,N 3 )]  + (A 1 +A 2 )[(N 1 +1)(N 2 +1)P(N 1 +1,N 2 +1,N 3 -1)-N 1 N 2 P(N 1,N 2,N 3 )  + (A 1 +A 3 )[(N 1 +1)(N 3 +1)P(N 1 +1,N 2,N 3 +1)-N 1 N 3 P(N 1,N 2,N 3 ) 3 i=1 3 i=1 2 i=1

18 P(N 1,N 2,N 3 ) =  F i [P(…,N i -1,…)-P(N 1,N 2,N 3 )] +  W i [(N i +1)P(..,N i +1,..)-N i P(N 1,N 2,N 3 )] +  A i [(N i +2)(N i +1)P(..,N i +2,..)-N i (N i -1)P(N 1,N 2,N 3 )]  + (A 1 +A 2 )[(N 1 +1)(N 2 +1)P(N 1 +1,N 2 +1,N 3 -1)-N 1 N 2 P(N 1,N 2,N 3 )  + (A 1 +A 3 )[(N 1 +1)(N 3 +1)P(N 1 +1,N 2,N 3 +1)-N 1 N 3 P(N 1,N 2,N 3 ) 3 i=1 3 i=1 2 i=1 R ij = (A i + A j ) ‹ N i N j › R ii = A i ( ‹ N i 2 › - ‹ N i › )

19 The Rate vs. The Master Rate 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

20 The Moment Equations ‹ N H k › =  N H k P H (N H ) NH=0NH=0 8 After applying the summation: ‹ N H › = F H + (2A H - W H ) ‹ N H › - 2A H ‹ N H 2 › ‹ N H 2 › = F H + (2F H + W H - 4A H ) ‹ N H › + (8A H - W H ) ‹ N H 2 › - 4A H ‹ N H 3 ›

21 Truncating the Equations 1. Set the cutoff 2. Express the (k+1)th moment by the first k moments ‹ N H 1 › = P H (1) + 2P H (2) + +kP H (k) ‹ N H 2 › = P H (1) P H (2) + +k 2 P H (k) ‹ N H k › = P H (1) + 2 k P H (2) + +k k P H (k) P H (N H > k) = 0

22 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 ‹ N H 1 › = P H (1) + 2P H (2) + + kP H (k) ‹ N H 2 › = P H (1) P H (2) + +k 2 P H (k) ‹ N H k › = P H (1) + 2 k P H (2) + +k k P H (k) P H (N H > k) = 0 ‹ N H k+1 › =  C i ‹ N H i › i=0 k

23 Moment Equations for H 2 Production ‹ N H › = F H + (2A H - W H ) ‹ N H › - 2A H ‹ N H 2 › ‹ N H 2 › = F H + (2F H + W H - 4A H ) ‹ N H › + (8A H - W H ) ‹ N H 2 › - 4A H ‹ N H 3 › 1. Set the cutoff → k=2 ‹ N H 3 › = 3 ‹ N H 2 › - 2 ‹ N H › 2. Reduce excessive moments → 3. Plug into the equations…

24 ‹ N H › = F H + (2A H - W H ) ‹ N H › - 2A H ‹ N H 2 › ‹ N H 2 › = F H + (2F H + W H - 4A H ) ‹ N H › + (8A H - W H ) ‹ N H 2 › - 4A H ‹ N H 3 › ‹ N H › = F H + (2A H - W H ) ‹ N H › - 2A H ‹ N H 2 › ‹ N H 2 › = F H + (2F H + W H + 4A H ) ‹ N H › - (4A H + 2W H ) ‹ N H 2 › Moment Equations for H 2 Production ‹ N H 3 › = 3 ‹ N H 2 › - 2 ‹ N H › 1. Set the cutoff → k=2 2. Reduce excessive moments → 3. Plug into the equations…

25 R H vs. Grain Size 2

26 Moments for Complex Networks OH O2O2 H2H2 O H H2OH2O The probability: P(N 1,N 2,N 3 ) The moments: ‹ N 1 a N 2 b N 3 c › The cutoff: N i < k i The challenge: Reduction of the excessive moments ‹ N 1 a N 2 b N 3 c › =  C lnm ‹ N 1 l N 2 n N 3 m › lmn=0 k-1

27 Reduction of Excessive Moments The probability: P(N 1,N 2 ) V(a,b) M (N 1,N 2,a,b) P(N 1,N 2 ) v = M p ‹ N 1 a N 2 b › =  C nm ‹ N 1 n N 2 m › mn=0 k-1 ‹ N 1 a N 2 b › =  N 1 a N 2 b P(N 1,N 2 ) N 1 N 2 =0 k-1

28 ‹N1›,‹N1›, ‹N3›‹N3›‹N2›,‹N2›, Setting the Cutoffs OHO2O2 H2H2 O H H2OH2O ‹N1N2›‹N1N2› ‹N1N3›‹N1N3› ‹N22›‹N22› ‹N12›‹N12› 3 vertices + 2 edges + 2 loops = 7 equations

29 Production Rates vs. Grain Size

30 Multi-Specie Network H 2 COH 3 CO OH HCOH OCO CO 2 + H O2O2 H2H2 HCO H 2 CO OH CO 2 H 3 COCH 3 CO H2OH2O 7 vertices 8 edges 2 loops 17 equations +

31 Production Rates vs. Grain Size

32 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.

33 Revealing the Trick The moment equations validity - For small grainsFor large grainsCutoff justifiedP H (N H ) is Poisson Second order: (  << 1) The equations are valid First order: (  ≈ 1) Production rate is accurate but population size maydeviate Moment equations valid under all circumstances