Chaos in Many-Particle Systems

Thiago de Freitas Viscondi

Núcleo de Dinâmica e FluidosDepartamento de Engenharia Mecânica

Escola PolitécnicaUniversidade de São Paulo

13/06/18

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 1 / 18

Molecular Dynamics Introduction

Molecular Dynamics: Definition

“...molecular dynamics simulation involves solving the classical many-bodyproblem in contexts relevant to the study of matter at the atomistic level...”1

“Computer simulation generates information at the microscopic level ... andthe conversion of this very detailed information into macroscopic terms ... is

the province of statistical mechanics.”2

1D. C. Rapaport, The Art of Molecular Dynamics Simulation.2M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 2 / 18

Molecular Dynamics Introduction

Molecular Dynamics: Definition

“...molecular dynamics simulation involves solving the classical many-bodyproblem in contexts relevant to the study of matter at the atomistic level...”1

“Computer simulation generates information at the microscopic level ... andthe conversion of this very detailed information into macroscopic terms ... is

the province of statistical mechanics.”2

1D. C. Rapaport, The Art of Molecular Dynamics Simulation.2M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 2 / 18

Molecular Dynamics Introduction

Interaction PotentialsThe Lennard-Jones potential is defined by

u(rj,k ) =

4ε

[(σ

rj,k

)12

−(σ

rj,k

)6], for rj,k < rc ,

0, for rj,k ≥ rc ,

(1)

where rj,k = |~rj −~rk |, ε is the interaction magnitude, and σ defines alength scale.

By removing the attractive part of the Lennard-Jones potential, we obtainthe soft-sphere potential:

u(rj,k ) =

4ε

[(σ

rj,k

)12

−(σ

rj,k

)6]

+ ε, for rj,k < rc ,

0, for rj,k ≥ rc ,

(2)

for rc = 216σ.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 3 / 18

Molecular Dynamics Introduction

Interaction PotentialsThe Lennard-Jones potential is defined by

u(rj,k ) =

4ε

[(σ

rj,k

)12

−(σ

rj,k

)6], for rj,k < rc ,

0, for rj,k ≥ rc ,

(1)

where rj,k = |~rj −~rk |, ε is the interaction magnitude, and σ defines alength scale.

By removing the attractive part of the Lennard-Jones potential, we obtainthe soft-sphere potential:

u(rj,k ) =

4ε

[(σ

rj,k

)12

−(σ

rj,k

)6]

+ ε, for rj,k < rc ,

0, for rj,k ≥ rc ,

(2)

for rc = 216σ.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 3 / 18

Molecular Dynamics Introduction

Potential Sketch

Figure 1: Lennard-Jones (lower curve) and soft-sphere (uppercurve) potentials in dimensionless molecular dynamics units.1

1D. C. Rapaport, The Art of Molecular Dynamics Simulation.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 4 / 18

Molecular Dynamics Introduction

Perturbation

Consider two identical simulations of N two-dimensional particles. At achosen time tp, the following perturbation is applied on every particle inone of the systems:

~v2,j (tp) = ~v1,j (tp) + ε~wj , (3)

where ~v1,j is the velocity of the j-th particle in the original system, ~v2,j isthe corresponding velocity in the perturbed system, ~wj is a random unitvector, and ε is the perturbative parameter.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 5 / 18

Molecular Dynamics Introduction

Criteria for Chaos

Mean-square position deviation:

〈(~r1 −~r2)2〉 =1N

N∑j=1

(~r1,j −~r2,j )2, (4)

where ~r1,j is the position of the j-th particle in the original system and ~r2,jis the corresponding position in the perturbed system.

Position correlation:

corr(~r1,~r2) =〈(~r1 − 〈~r1〉)(~r2 − 〈~r2〉)〉

σ(~r1)σ(~r2), (5)

where σ(~rj ) =√〈(~rj − 〈~rj〉)2〉.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 6 / 18

Molecular Dynamics Introduction

Criteria for Chaos

Mean-square position deviation:

〈(~r1 −~r2)2〉 =1N

N∑j=1

(~r1,j −~r2,j )2, (4)

where ~r1,j is the position of the j-th particle in the original system and ~r2,jis the corresponding position in the perturbed system.

Position correlation:

corr(~r1,~r2) =〈(~r1 − 〈~r1〉)(~r2 − 〈~r2〉)〉

σ(~r1)σ(~r2), (5)

where σ(~rj ) =√〈(~rj − 〈~rj〉)2〉.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 6 / 18

Molecular Dynamics Comparative Simulations

Comparative Simulation

x

-8 -6 -4 -2 0 2 4 6 8

y

-8

-6

-4

-2

0

2

4

6

8

t =0.000

x

-8 -6 -4 -2 0 2 4 6 8

y-8

-6

-4

-2

0

2

4

6

8

t =0.000

Figure 2: Comparative simulation of two-dimensional soft-sphere particles forρ = 0.4, T = 1, N = 121, and ε = 10−6.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 7 / 18

Molecular Dynamics Comparative Simulations

Root-Mean-Square Position Deviation and Correlation

t5 10 15 20 25 30 35 40

log[√

〈(~r1−~r2)2〉]

-14

-12

-10

-8

-6

-4

-2

0

t0 5 10 15 20 25 30 35 40

corr(~r

1,~r

2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3: Root-mean-square deviation (left panel) and correlation (right panel)between particle positions and their perturbed versions. Simulation of soft-sphere particles for d = 2, ρ = 0.4, T = 1, and N = 121.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 8 / 18

Molecular Dynamics Comparative Simulations

Saturation

t

50 100 150 200 250

log[√

〈(~r1−~r2)2〉]

-12

-10

-8

-6

-4

-2

0

2

Figure 4: Root-mean-square deviation between particle positions and their per-turbed versions. Simulation of soft-sphere particles for d = 2, ρ = 0.4, T = 1,and N = 121.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 9 / 18

Molecular Dynamics Comparative Simulations

Saturation Value

According to figure 4, the saturation value of the root-mean-squareposition deviation is

log[√〈(~r1 −~r2)2〉

]≈ 1.947, (6)

for t = 250.

As a consequence of the derivation presented in the appendix, thepredicted value of saturation is

log[√〈(~r1 −~r2)2〉

]= log

(√d12

L

)≈ 1.960,

(7)

for d = 2 and L ≈ 17.39.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 10 / 18

Molecular Dynamics Comparative Simulations

Saturation Value

According to figure 4, the saturation value of the root-mean-squareposition deviation is

log[√〈(~r1 −~r2)2〉

]≈ 1.947, (6)

for t = 250.

As a consequence of the derivation presented in the appendix, thepredicted value of saturation is

log[√〈(~r1 −~r2)2〉

]= log

(√d12

L

)≈ 1.960,

(7)

for d = 2 and L ≈ 17.39.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 10 / 18

Molecular Dynamics Comparative Simulations

Saturation Value

According to figure 4, the saturation value of the root-mean-squareposition deviation is

log[√〈(~r1 −~r2)2〉

]≈ 1.947, (6)

for t = 250.

As a consequence of the derivation presented in the appendix, thepredicted value of saturation is

log[√〈(~r1 −~r2)2〉

]= log

(√d12

L

)≈ 1.960,

(7)

for d = 2 and L ≈ 17.39.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 10 / 18

Molecular Dynamics Comparative Simulations

Maxwell-Boltzmann DistributionVelocity distribution of two-dimensional classical particles in thermalequilibrium:

f (v) =vT

exp(− v2

2T

). (8)

The Boltzmann H-function is defined as

H =

∫f̃ (~v) log

[f̃ (~v)

]ddv

∝∫

f (v) log[

f (v)

vd−1

]dv .

(9)

The H-function satisfies the following relation:⟨dHdt

⟩≤ 0, (10)

with equality only applying when f (v) is the Maxwell-Boltzmanndistribution.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 11 / 18

Molecular Dynamics Comparative Simulations

Maxwell-Boltzmann DistributionVelocity distribution of two-dimensional classical particles in thermalequilibrium:

f (v) =vT

exp(− v2

2T

). (8)

The Boltzmann H-function is defined as

H =

∫f̃ (~v) log

[f̃ (~v)

]ddv

∝∫

f (v) log[

f (v)

vd−1

]dv .

(9)

The H-function satisfies the following relation:⟨dHdt

⟩≤ 0, (10)

with equality only applying when f (v) is the Maxwell-Boltzmanndistribution.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 11 / 18

Molecular Dynamics Comparative Simulations

Maxwell-Boltzmann DistributionVelocity distribution of two-dimensional classical particles in thermalequilibrium:

f (v) =vT

exp(− v2

2T

). (8)

The Boltzmann H-function is defined as

H =

∫f̃ (~v) log

[f̃ (~v)

]ddv

∝∫

f (v) log[

f (v)

vd−1

]dv .

(9)

The H-function satisfies the following relation:⟨dHdt

⟩≤ 0, (10)

with equality only applying when f (v) is the Maxwell-Boltzmanndistribution.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 11 / 18

Molecular Dynamics Comparative Simulations

H-Function

t

0 5 10 15 20 25 30 35 40

H-function

-75

-70

-65

-60

-55

-50

-45

-40

-35

t

0 5 10 15 20 25 30 35 40

RMSDeviation

0.2

0.3

0.4

0.5

0.6

0.7

Figure 5: (Left panel) Boltzmann H-function. (Right panel) Root-mean-squaredeviation of the velocity histogram with respect to the Maxwell-Boltzmann dis-tribution. Simulation of soft-sphere particles for d = 2, ρ = 0.4, T = 1, andN = 121.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 12 / 18

Molecular Dynamics Comparative Simulations

Thermalized Comparative Simulation

x

-8 -6 -4 -2 0 2 4 6 8

y

-8

-6

-4

-2

0

2

4

6

8

t =10.000

x

-8 -6 -4 -2 0 2 4 6 8

y-8

-6

-4

-2

0

2

4

6

8

t =10.000

Figure 6: Comparative simulation of thermalized two-dimensional soft-sphereparticles for ρ = 0.4, T = 1, N = 121, and ε = 10−6.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 13 / 18

Molecular Dynamics Comparative Simulations

Position Root-Mean-Square Deviation and Correlation

t15 20 25 30 35 40 45 50

log[√

〈(~r1−~r2)2〉]

-14

-12

-10

-8

-6

-4

-2

0

t10 15 20 25 30 35 40 45 50

corr(~r

1,~r

2)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 7: Root-mean-square deviation (left panel) and correlation (right panel)between particle positions and their perturbed versions. Simulation of thermal-ized soft-sphere particles for d = 2, ρ = 0.4, T = 1, and N = 121.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 14 / 18

Molecular Dynamics Comparative Simulations

Behaviour Invariance

t

5 10 15 20 25

log[√

〈(~r1−~r2)2〉]

-15

-10

-5

0

ǫ = 10−1

ǫ = 10−3

ǫ = 10−5

ǫ = 10−7

Figure 8: Root-mean-square deviation between particle positions and their per-turbed versions. Simulation of soft-sphere particles for ρ = 0.4, T = 1, andN = 121.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 15 / 18

References

References

D. C. Rapaport. The Art of Molecular Dynamics Simulation. CambridgeUniversity Press, 2004.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 16 / 18

Appendix Saturation

Saturation Value of Mean-Square Position DeviationAs a first step, equation (4) is rewritten in terms of Cartesian coordinates:

〈(~r1 −~r2)2〉 =d∑

j=1

〈(r1,j − r2,j )2〉

=d∑

j=1

〈(∆rj )2〉,

(11)

where r1,j and r1,j are the j-th Cartesian coordinates of a particle in theoriginal and perturbed systems, respectively. Notice that d dimensionsare considered.

Assuming that ∆rj is a uniformly distributed random variable over theinterval [0,Lj/2], the following result is obtained:

〈(∆rj )2〉 =

112

L2j , (12)

where Lj is the length of a d-dimensional box in the j-th direction,considering periodic boundary conditions.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 17 / 18

Appendix Saturation

Saturation Value of Mean-Square Position DeviationAs a first step, equation (4) is rewritten in terms of Cartesian coordinates:

〈(~r1 −~r2)2〉 =d∑

j=1

〈(r1,j − r2,j )2〉

=d∑

j=1

〈(∆rj )2〉,

(11)

where r1,j and r1,j are the j-th Cartesian coordinates of a particle in theoriginal and perturbed systems, respectively. Notice that d dimensionsare considered.

Assuming that ∆rj is a uniformly distributed random variable over theinterval [0,Lj/2], the following result is obtained:

〈(∆rj )2〉 =

112

L2j , (12)

where Lj is the length of a d-dimensional box in the j-th direction,considering periodic boundary conditions.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 17 / 18

Appendix Saturation

Saturation Value of Mean-Square Position DeviationAs a first step, equation (4) is rewritten in terms of Cartesian coordinates:

〈(~r1 −~r2)2〉 =d∑

j=1

〈(r1,j − r2,j )2〉

=d∑

j=1

〈(∆rj )2〉,

(11)

where r1,j and r1,j are the j-th Cartesian coordinates of a particle in theoriginal and perturbed systems, respectively. Notice that d dimensionsare considered.

Assuming that ∆rj is a uniformly distributed random variable over theinterval [0,Lj/2], the following result is obtained:

〈(∆rj )2〉 =

112

L2j , (12)

where Lj is the length of a d-dimensional box in the j-th direction,considering periodic boundary conditions.

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 17 / 18

Appendix Saturation

Result

Upon substitution of the assumption (12) into equation (11):

〈(~r1 −~r2)2〉 =112

d∑j=1

L2j . (13)

In the case of Lj = L, for all j , equation (13) is simplified:

〈(~r1 −~r2)2〉 =d12

L2. (14)

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 18 / 18

Appendix Saturation

Result

Upon substitution of the assumption (12) into equation (11):

〈(~r1 −~r2)2〉 =112

d∑j=1

L2j . (13)

In the case of Lj = L, for all j , equation (13) is simplified:

〈(~r1 −~r2)2〉 =d12

L2. (14)

Viscondi, T. F. (NDF/PME/POLI/USP) Chaos 13/06/18 18 / 18