Monte Carlo in different ensembles Daan Frenkel Different Ensembles - PowerPoint PPT Presentation

Monte Carlo in different ensembles Daan Frenkel

Different Ensembles Ensemble Name Constant Fluctuating (Imposed) (Measured) NVT Canonical N,V,T P NPT Isobaric-isothermal N,P,T V µ VT Grand-canonical µ ,V,T N 2

Statistical Thermodynamics Partition function 1 $ & ( ) dr N exp − β U r N ∫ Q NVT = ' . % Λ 3 N N ! I will come back to this Ensemble average 1 1 $ & ( ) exp − β U r N ( ) dr N A r N ∫ A NVT = ' . % Λ 3 N N ! Q NVT Probability to find a particular configuration 1 1 ( ) N N ( N N ) ( ) N ( ) N N r dr' r' r exp U r' exp U r # $ # $ = δ − − β ∝ − β ∫ 3 N ' ( ' ( Q N ! Λ NVT Free energy ( ) F ln Q β = − 3 NVT

Detailed balance o n K o ( n ) K n ( o ) → = → K o ( n ) N o ( ) ( o n ) acc( o n ) → = × α → × → K n ( o ) N n ( ) ( n o ) acc( n o ) → = × α → × → acc( o n ) N n ( ) ( n o ) N n ( ) → × α → = = acc( n o ) N o ( ) ( o n ) N o ( ) → × α → 4

NVT -ensemble N n ( ) exp U n ( ) ∝ ⎡ − β ⎤ ⎣ ⎦ acc( o n ) N n ( ) → = acc( n o ) N o ( ) → acc( o n ) → exp U n U o ( ) ( ) ⎡ ⎤ ⎡ ⎤ = − β − ⎣ ⎦ ⎣ ⎦ acc( n o ) → 5

NPT ensemble We control the • Temperature (T) • Pressure (P) • Number of particles (N) 6

Scaled coordinates Partition function 1 $ & ( ) dr N exp − β U r N ∫ Q NVT = ' . % Λ 3 N N ! Scaled coordinates s r i L / = The energy depends on i the real coordinates This gives for the partition function 3 N L ( ) N N Q ds exp U s ; L " # = ∫ − β NVT 3 N % & N ! Λ N V ( ) N N ds exp U s ; L " # = ∫ − β 3 N % & N ! Λ 7

N in volume V M in volume V 0 -V 8

N V ( ) N N Q ds exp U s ; L " # = ∫ − β NVT % & 3 N N ! Λ M − N ( ) V 0 − V V N $ & ( ) ds M − N exp − β U 0 s M − N ; L Q MV 0 , NV , T = ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! $ & ( ) ds N exp − β U s N ; L ∫ × % ' V 0 − V ( ) M − N Q MV 0 , NV , T = V N ∫ ds N exp − β U s N ; L $ ( ) & Λ 3( M − N ) M − N ( ) ! Λ 3 N N ! % ' 9

M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , NV , T = ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! To get the Partition Function of this system, we have to integrate over all possible volumes : M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , N , T = d V ∫ ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! Now let us take the following limits: M M → ∞$ constant ρ = → % V V → ∞ & 0 As the particles in the reservoir are an ideal gas, we have: P ρ = β 10

M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , N , T = d V ∫ ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! We have M N M N − − M N M N V V V − 1 V V V − exp M N V V ( ) ( ) ( ) " # − = − ≈ − − % & 0 0 0 0 0 M N − M N M N V V V − exp [ V ] V − exp [ PV ] ( ) − ≈ − ρ = − β 0 0 0 This gives: P β N N ( N ) Q d V exp PV V ds exp U s ; L [ ] " # = − β − β Λ ∫ ∫ NPT 3 N % & N ! 11

NPT Ensemble Partition function: P β N N ( N ) Q d V exp PV V ds exp U s ; L [ ] ⎡ ⎤ = Λ ∫ − β ∫ − β NPT 3 N ⎣ ⎦ N ! Probability to find a particular configuration: ( N ) N ( N ) N V , s V exp PV exp U s ; L [ ] " # ∝ − β − β NPT & ' Sample a particular configuration: • change of volume • change of reduced coordinates Acceptance rules ?? 12

Detailed balance o n K o ( n ) K n ( o ) → = → K o ( n ) N o ( ) ( o n ) acc( o n ) → = × α → × → K n ( o ) N n ( ) ( n o ) acc( n o ) → = × α → × → acc( o n ) N n ( ) ( n o ) N n ( ) → × α → = = acc( n o ) N o ( ) ( o n ) N o ( ) → × α → 13

NPT -ensemble ( ) ( ) N N N N V , s V exp PV exp U s ; L [ ] " # ∝ − β − β NPT & ' acc( o n ) N n ( ) → = acc( n o ) N o ( ) → Suppose we change the position of a randomly selected particle N ( N ) V exp PV exp U s ; L [ ] " # − β − β acc( o n ) → n & ' = acc( n o ) N ( N ) → V exp [ PV ] exp U s ; L " # − β − β o & ' ( N ) exp U s ; L " # − β n % & { } exp U n U o ( ) ( ) = = − β " − # % & ( N ) exp U s ; L " # − β o % & 14

NPT -ensemble ( ) ( ) N N N N V , s V exp PV exp U s ; L [ ] " # ∝ − β − β NPT & ' acc( o n ) N n ( ) → = acc( n o ) N o ( ) → Suppose we change the volume of the system N ( N ) V exp PV exp U s ; L [ ] " # − β − β acc( o n ) → n n n & ' = acc( n o ) N ( N ) → V exp [ PV ] exp U s ; L " # − β − β o o o & ' N V " # { } n exp P V V exp U n U 0 ( ) ( ) ( ) = $ − β − % − β $ − % ' ( ) * ) * n o V + , o 15

Algorithm: NPT • Randomly change the position of a particle • Randomly change the volume 16

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Measured and Imposed Pressure • Imposed pressure P • Measured pressure <P> from virial # & ∂ F ds N e − β U ( s N ) ( ) dVV N e − β PV ∫ ∫ − % ( # & $ ∂ V ' P = − ∂ F N , T = % ( ds N e − β U ( s N ) dVV N e − β PV $ ∂ V ' ∫ ∫ N , T * , ( ) p ( V ) = exp − β F ( V ) + PV + - Q NPT * , ∫ ( ) Q NPT = β P dV exp − β F ( V ) + PV + - 20

# & β P dV ∂ F ) + ( ) ∫ ( ) + PV P = − exp − β F V % ( * , Q ( NPT ) ∂ V $ ' N , T ) + ( ) ∂ exp − β F V [ ] dV exp − β PV β P * , ∫ P = Q ( NPT ) ∂ V β 21

Measured and Imposed Pressure b b b − • Partial integration ∫ ∫ [ ] a f dg = fg gdf a a [ ] 0 ( ( ) ) exp F V PV • For V=0 and V= ∞ − β + = • Therefore, [ ] [ ] ( ) P exp PV exp F V β − β ∂ − β P dV = ∫ Q ( NPT ) V β ∂ P β [ ] ( ( ) ) P dVP exp F V PV P = = − β + = ∫ Q ( NPT ) 22

Grand-canonical ensemble What are the equilibrium conditions? 23

Grand-canonical ensemble We impose: – Temperature (T) – Chemical potential ( µ ) – Volume (V) – But NOT pressure 24

System in reservoir Here they don’t Here particles interact What is the statistical thermodynamics of this ensemble? 25

N V ( ) N N Q ds exp U s ; L " # = ∫ − β NVT % & 3 N N ! Λ M − N ( ) V 0 − V V N $ & ( ) ds M − V exp − β U 0 s M − N ; L Q MV 0 , NV , T = ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! $ & ( ) ds N exp − β U s N ; L ∫ × % ' V 0 − V ( ) M − N Q MV 0 , NV , T = V N ∫ ds N exp − β U s N ; L $ ( ) & Λ 3( M − N ) M − N ( ) ! Λ 3 N N ! % ' 26

M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , NV , T = ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! To get the Partition Function of this system, we have to sum over all possible number of particles M − N ( ) V 0 − V N = M V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , N , T = ∑ ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! N = 0 Now let us take the following limits: M M → ∞$ constant ρ = → % V V → ∞ & 0 As the particles are an ideal gas in the big reservoir we have: ( ) 3 k T ln µ = Λ ρ B N exp N V ( ) N = ∞ β µ ( ) N N Q ds exp U s ; L ∑ # $ = ∫ − β VT 27 3 N & ' µ N ! Λ N 0 =

Q tot = Q R ( M − N ) Q sys ( N ) = e − β F R ( M − N ) Q sys ( N ) Expand F R ✓ ∂ F R ◆ F R ( M − N ) = F R ( M ) − N + · · · ∂ M ✓ ∂ F R ◆ But: = µ And hence : ∂ M Q tot = Q R ( M − N ) Q sys ( N ) = e − β F R ( M ) e β µN Q sys ( N ) Sum over all N: N exp N V ( ) N = ∞ β µ ( ) N N Q ds exp U s ; L ∑ # $ = ∫ − β VT 28 3 N & ' µ N ! Λ N 0 =

µ VT Ensemble Partition function: N exp N V ( ) N = ∞ β µ N ( N ) Q ds exp U s ; L # $ ∑ = ∫ − β VT & ' 3 N µ N ! Λ N 0 = Probability to find a particular configuration: N exp N V ( ) β µ ( ) ( ) N N N V , s exp U s ; L " # ∝ − β VT & ' µ 3 N N ! Λ Sample a particular configuration: • Change of the number of particles • Change of reduced coordinates Acceptance rules ?? 29