Computation in Physics

(1)

Computation in Physics: Challenges in

”Non-observable” Phenomena

L.T. Handoko

Group for Theoretical High Energy Physics, Research Center for Physics, Indonesian Institute of Sciences

and

Department of Physics, University of Indonesia

handoko@teori.fisika.lipi.go.id ⇔ http://teori.fisika.lipi.go.id handoko@fisika.ui.ac.id ⇔ http://staff.fisika.ui.ac.id/handoko/

SMIC - 8 August 2020

(2)

Outline

1 Intermesso

2 Introduction Physical scale

An example on material dynamics Micromolecul dynamics : protein Micromolecul dynamics : DNA

3 Davydov-Scott protein dynamics Problem setup

The model Partition function

Thermodynamic properties Results

4 Summary

(3)

Intermesso: LIPI

Indonesian Institute of Sciences

Focus programme 2020-2024

(4)

Intermesso: LIPI

Indonesian Institute of Sciences

Focus programme 2020-2024

(5)

Intermesso: LIPI

LIPI’s supporting programme

Recuitment of highly skilled candidate researchers: PhD, diaspora, etc.

Encouraging researchers to compete for external (local or global) grants.

LIPI’s budget is focused on research infrastucture.

Open research infrastructure for academics and industries incl.

SME’s, startups.

Incentive for local and global collaborations through multi schemes:

by-research program, research assistantship, visiting fellow, postdoctoral fellow.

(6)

Problem setup

Physical scale

Physical scale Theory

macro : Newtonian mechanics

∼10⁻¹⁸mt : elementary particle

∼10⁻⁹ mt : mesoscale ?

Basic idea !

Treating the intermediate scale phenomena as a result of ’macroscopic ensemble’ of many point-particle interactions !

(7)

Problem setup

Physical scale

Basic idea !

(8)

Problem setup

Physical scale

Basic idea !

(9)

Problem setup

Physical scale

Basic idea !

(10)

Problem setup

Physical scale

Basic idea !

(11)

Problem setup

Physical scale

Basic idea !

(12)

Problem setup

Physical scale

physical vs chemical

(13)

Problem setup

Physical scale

NTI-US

(14)

An example: material dynamics

Nanopowder dynamics in a mechanical nanomaterial process

Comminution processes in ball mills as a canonical ensemble

(15)

An example: material dynamics

H_m=H₀+V_m−m+V_m−v+V_m−m⁰+V_ext with,m,m⁰ : vial, powder, ball

G. K. Sunnardianto, Muhandis, F. N. Diana, LTH,J. Compt. Theor. Nanoscience8 (2011) 194-200

H0 = 1 2mm

nm

X

i=1

|(~pm)_i|²

V_m−m^imp 0(~r,t) = −

nm

X

i=1 n_m0

X

j=1

Z (ξ_mm0)_ij 0

d(ξ_mm⁰)_ij n~· F~_mm^imp0

ij

V_m−m^Coul0(~r) = QmQm⁰ n_m

X

i=1 n_m0

X

j=1

1

(~r_m)_i−(~r_m⁰)_j

(16)

An example: material dynamics

Thermodynamics quantities in the system

Thetemperature-dependentpressure is obtained, P_P⁰ = 1−βF ln⁻¹

2m_Pπ β

F ≡ 2 Z ⁿ^P

Y

i=1

d~ri



Q_Pφ− 2 15n_P

nP

X

i(6=j)=1 nP

X

j=1

ΥPP

1−v_PP² q

R_PP^eff (ξ_PP)^5/2_ij

− 4 15n_P

X

m:b,v nP

X

i=1 nm

X

j=1

Υ_Pm 1−v_Pm²

q

R_P^eff_m(ξPm)^5/2_ij





Some points :

The geometry of vial motions are less important than considered.

The external magnetic force affects nothing.

The dissipative term contributes nothing

(17)

An example: material dynamics

F vs ratio of powder and ball radius (nickel)

(18)

An example: material dynamics

The fundamental problem

Is the system at (thermal) equilibrium?

E =

1 2 + n

~ ν

(19)

Problem setup

Micromolecul dynamics : protein

Davydov-Scott model of protein / monomer / α−helix / single strand

(20)

Problem setup

Micromolecul dynamics : DNA

Peyrard-Bishop model of DNA / β−helix / double strand

(21)

Problem setup

Interaction based model of DNA

(22)

Problem setup

Interaction based model of DNA

(23)

Problem setup

Micromolecul dynamics : DNA, protein

Based on previous works :

A. Sulaiman, F.P. Zen, H. Alatas, LTH, Phys. Rev. E81 (2010) 061907

A. Sulaiman, LTH, J. Compt. Theor. Nanoscience 8 (2011) 124-132 A. Sulaiman, F.P. Zen, H. Alatas, LTH, Physica D 86 (2012)

1640-1647

A. Sulaiman, F.P. Zen, H. Alatas, LTH, Physica Scripta241 (2012) 015802

M. Januar, A. Sulaiman, LTH, J. Compt. Theor. Nanoscience 10 (2013) 2106-2112

(24)

Problem setup

Micromolecul dynamics : points

Both are considered as a complicated life system triggered by a kind of ’vibration’

to keep themself alive in an ’open system’

⇒interacting continously with surrounding environment

⇒its quantum effects→open quantum system?

PROTEIN CellE.g. energy transfer of 10 kcal/mol (0.42 eV) in hidrolisis reaction of ATP to ADP,

DNA Genetic bequeathing⇒transcription, replication, denaturation, etc

Some effects of pH, heating, electromagnetic force, surrounding fluid, dissipation (damping), etc

Most models consider the system is solitonic, realizing the non-linear world

⇒Non-linear Klein-Gordon eq. (DNA), non-linear Schrodinger eq. (protein), etc

∂_x²0Φ + ˜m²_ΦΦ + ˜λΦ³= 0→Φ(x⁰) =|m˜_Φ| r2

˜λsech(|m˜_Φ|x⁰)

(25)

Davydov-Scott protein dynamics

Problem setup

Facts of biomolecule :

Formed by atomic / molecular chain in a regular way → life crystal.

Not a rigid body, but elastic.

Each life matter has different chain content.

Davydov-Scott model

The transfer energy is described by the excitation energy of amide-I which is stabilized by phonon, but...

⇒ no interaction with thermal bath. Exp. shows temperature dependences on crystal (CH3CONHC6H5)_x

⇒ too short life-time (∼1 ps)

⇒ external forces due to laser, fluid, etc

(26)

Davydov-Scott protein dynamics

Problem setup

Davydov-Scott model

(27)

Davydov-Scott protein dynamics

Problem setup

Davydov-Scott model

(28)

Davydov-Scott protein dynamics

Problem setup

Davydov-Scott model

(29)

Davydov-Scott protein dynamics

Problem setup

Davydov-Scott model

(30)

Davydov-Scott protein dynamics

Problem setup

Davydov-Scott model

(31)

Davydov-Scott model

Optical tweezer / trapping

(32)

Davydov-Scott model

The model

Hamiltonian of coupled double oscilator harmonics

Using the operators of amide-I (x,p) and amide-site (Q,P) H= p²

2m +1

2mω²x²+ P² 2M +1

2κQ²+χxQ

χ=χ⁰p

~/(2MΩ) : coupling constant; Ω, ω: frequency protein model

Environment effect ?

⇒irreversible →dissipation →open quantum system

Linblad master equation←preserve the density operator properties (hermiticity, trace 1, positivity)

∂ρ

∂t =−i

~

[H, ρ]+X

j

(L_jρL^†_j −1

2L^†_jL_jρ−1 2ρL^†_jL_j)

H: system interior;L: external effects, linear in (x,p) and (Q,P)

(33)

Davydov-Scott model

The model

Hamiltonian of coupled double oscilator harmonics

Using the operators of amide-I (x,p) and amide-site (Q,P) H= p²

2m +1

2mω²x²+ P² 2M +1

2κQ²+χxQ

χ=χ⁰p

~/(2MΩ) : coupling constant; Ω, ω: frequency protein model

Environment effect ?

⇒irreversible →dissipation →open quantum system

Linblad master equation←preserve the density operator properties (hermiticity, trace 1, positivity)

∂ρ

∂t =−i

~

[H, ρ]+X

j

(L_jρL^†_j −1

2L^†_jL_jρ−1 2ρL^†_jL_j)

(34)

Davydov-Scott model

The model

Linblad operator

L₁ = p

γ(1 +ν)

rMΩ 2~ Q+i

r 1

2M~ΩP + χ⁰

~Ω rmω

2~x

!

L₂ = √ γν

rMΩ 2~ Q−i

r 1

2M~Ω+ χ⁰

~Ω rmω

2~x

!

γ: damping constant; Ω =p

κ/M;ν= (e^~Ω/^{kB T}−1)⁻¹: Bose-Einstein distribution

Another alternative using Caldirola-Kanai Hamiltonian : dt →exph Rt

0γ(t)dti

i~

∂Ψ

∂t =−~²

2me^−γt∇²Ψ +e^γtV(x)Ψ and,4x4p≥ ^~₂e^−γt.

(35)

Davydov-Scott model

Partition function

∂ρ

∂t = −i

~[H0, ρ]− i

2~ζ1[Q, ρP +Pρ]− i

2~ζ2[x, ρP +Pρ]

+ζ₃

2~[Q,[Q, ρ]] + ζ₄

2~[x,[Q, ρ]] + ζ₅

2~[x,[x, ρ]] + ζ₆

2~[P,[P, ρ]]

ζ1= ₂^γ

~(1 + 2ν);ζ2=γ₄^χ⁰

~Ω

pω

Ω(1 + 2ν) ζ₃=γ^MΩ₂

~(1 + 2ν);ζ₄=γ₂^χ⁰

~Ω

√MmΩω(1 + 2ν);

ζ5=γ^χ₄⁰²^mω

~Ω² (1 + 2ν);ζ6= ₂ ^γ

~MΩ(1 + 2ν)

⇒ diffusion terms frictional damping rate

(36)

Davydov-Scott model

Partition function

∂ρ

∂t = i~ 2m

∂²

∂x² − ∂²

∂x⁰²

ρ−

imω² 2~ + δ₃

2~²

x²−x⁰² ρ +

i~

2M + δ1

2~

∂²

∂Q² − ∂²

∂Q⁰²

ρ−

iMΩ² 2~ + δ2

2~

(Q²−Q⁰²)ρ

−i χ

~ + δ4

2~

xQ−x⁰Q⁰ ρ

δ₁=γ((1 + 2ν)/(2M~Ω)),δ₂=γ(MΩ/2)(1 + 2ν), δ3=γ(1 + 2ν)(χ²mω/(~²Ω²)) danδ4= (χ⁰/(~Ω))√

mωMΩ(1 + 2ν)

The solution is just the partition function of the related hamiltonian under consideration !

Z= Z

D[x(t)]e⁻~¹S(x(t))

(37)

Davydov-Scott model

Partition function

Further assumptions :

1 Amide-site is more rigid than amide-I→quantum fluctuation is dominated by amide-I

2 Q=Q+ ˘Q, only the classicalQ is considered.

⇒ Z =Z_xZ_Q Sx =

Z τ 0

dt 1

2mx˙²+1

2kx²−1

2δx⁴+ ˜χx Q

SQ = Z τ

0

dt 1

2MQ˙²+1

2˜κQ²−1 4λQ⁴

(38)

Davydov-Scott model

Thermodynamic properties

Specific heat

Concerning the quantities of heat added to the system in order to increase the temperature :

C =k_Bβ²∂²ln(Z)

∂β² and,

C =Camide−site+Camide−I+C_mixing

(39)

Davydov-Scott model

Thermal equilibrium

1 amide-site :

E = ~Ω

2 + ~Ωe^~^Ωβ 1−e⁻^~^Ωβ n_Q = 1

e^~^Ωβ−1

2 amide-I :

E= ~ω

2 +~ωe^−~^ωβ

1−e⁻^~^ωβ +mω²hhx Qii nx = 1

e^~^ωβ−1+mω

~ hhx Qii

(40)

Davydov-Scott model

Thermal equilibrium

1 amide-site :

E = ~Ω

2 + ~Ωe^~^Ωβ 1−e⁻^~^Ωβ n_Q = 1

e^~^Ωβ−1

2 amide-I :

E= ~ω

2 +~ωe^−~^ωβ

1−e⁻^~^ωβ +mω²hhx Qii nx = 1

e^~^ωβ−1+mω

~ hhx Qii

(41)

Davydov-Scott model

Specific heat

Temperature dependence of normalized C_V for various damping

(42)

Davydov-Scott model

Specific heat

Temperature dependence of normalized C_V for under-damped

(43)

Davydov-Scott model

Specific heat

Temperature dependence of normalized C_V for critical-damped

(44)

Davydov-Scott model

Specific heat

Temperature dependence of normalized C_V for over-damped

(45)

Davydov-Scott model

Specific heat

Temperature dependence of normalized C_V vs absolute temperature

(46)

Summary

1 Some examples on how to model the nanomaterial dynamics, and the life matter at nano scale have been described.

2 The interaction of Davydov-Scott monomer with thermal bath is investigated using the Lindblad formulation of master equation including the anharmonic oscillation term of amide-site. The thermodynamic partition function and in particular specific heat are calculated using the path integral methods.

The environment contributes to the kinetic term, the harmonic potential of amide-site vibration and the anharmonic term of amide-I.

3 The anomaly of specific heat that becomes negative for certain parameter sets at high temperature region is observed (as already pointed out in some previous works)..

(47)

Summary

(48)

Summary

(49)

Computation in Physics