1. by Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations

2. 2 Molecular Formation in the ISM

3. 3 Horse-Head Nebula Molecular Formation in the ISM

4. 4 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 ~ 10-15 (mol cm -3 s -1 ) 2 The H 2 Puzzle

5. 5 The Solution

6. 6 kBTkBT -E0-E0 A H = (1/S) e = F H - W H ‹ N H › - 2A H ‹ N H › 2 d ‹ N H › dt 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 The Rate Equation

7. 7 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

8. 8 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 Probabilistic Approach

9. 9 = 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 The Master Equation

10. 10

11. 11 OHO2O2 H2H2 O H H2OH2O The parameters: F i ; W i ; A i (i=1,2,3) 13 2 Complex Reactions

12. 12 OHO2O2 H2H2 O H H2OH2O 13 2 The Master Disaster: P(N 1,N 2,N 3 ) Exponential Growth Complex Reactions

13. 13 ‹ 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 › The Moment Equations

14. 14 We need more knowledge… Imposing a cutoff on P(N) The Daring Imposition: P(N>2) = 0 Truncating the Equations

15. 15 ‹ 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 › And after imposing the cutoff… Moment Equations for H 2 Production

16. 16 ‹ 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

17. 17 R H vs. Grain Size 2

18. 18 ‹N1›,‹N1›, ‹N3›‹N3›‹N2›,‹N2›, OHO2O2 H2H2 O H H2OH2O ‹N1N2›‹N1N2› ‹N1N3›‹N1N3› ‹N22›‹N22› ‹N12›‹N12› 3 vertices + 2 edges + 2 loops = 7 equations A View to Complex Networks

19. 19 Production Rates vs. Grain Size

20. 20 15 vertices 30 edges + 3 loops 48 equations Multi-Specie Network

21. 21 Summary The advantages of the moment equations: Reliable even for low coverage EfficientLinear Easy to incorporate into rate equation models Directly generate the required moments Further applications should be tested.