Dynamical equations of periodic systems under constant external stress
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Dynamical Equations Of Periodic Systems Under Constant External Stress. Gang Liu [email protected] HPCVL, Queen's University, Kingston, ON 2014 CAP Congress, Laurentian University , Sudbury June 2014. PURPOSE.

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Dynamical Equations Of Periodic Systems Under Constant External Stress

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Dynamical equations of periodic systems under constant external stress

Dynamical Equations Of Periodic SystemsUnder Constant External Stress

Gang [email protected], Queen's University, Kingston, ON

2014 CAP Congress, LaurentianUniversity, SudburyJune 2014


Purpose

PURPOSE

To derive the concrete mathematical form of dynamical equations for the periodvectors of a periodic system under constant external stress, from Newton's Second Law.

http://arxiv.org/pdf/cond-mat/0209372.pdf


Classical molecular dynamics md

Classical Molecular Dynamics (MD)

widely used in many fields.

Periodic boundary conditions are often employed, then the system becomes a dynamical crystal filled with repeating cells.


In this work the whole system is modeled

In this work, the whole system is modeled

as a limitedmacroscopicbulk, composed of unlimited number of repeated microscopiccells

in three dimensions, with surface effect ignored.


As usual the cell in the center

As usual, the cell in the center

is called MD cell. Particles in it called MD particles

with position vectors , .


For each cell the three edge vectors

For each cell, the three edge vectors

(forming a right-handed triad)

are the period vectors of the system.


Degrees of freedom of the system

Degrees of freedom of the system.

Then the MD particle position vectors and the period vectors are the full degrees of freedom of the system.


What about the periods

What about the periods?

The dynamics of the particles is just Newton's Second Law on them.


Another respect external forces

Another respect: external forces

crystal


Another respect external forces1

Another respect: external forces

External forces will definitely cause its internal structure to change.

external forces

crystal


Another respect external forces2

Another respect: external forces

External forces will definitely cause its internal structure to change.

external forces

crystal


In 1980 parrinello and rahman

In 1980, Parrinello and Rahman

proposed their theory of the period dynamics (PRMD), where they introduced a Lagrangian and brought it into the Lagrangian Dynamical Equation to produce dynamics for both the MD particles and the periods.

and


Equations of prmd constant external pressure

Equations of PRMD (constant external pressure)

with


Dynamical equations of periodic systems under constant external stress

PRMD

combined with the well-known Car-Parrinello MD later, has been used extensively in many kinds of simulations.


In 1983 nose and klein pointed out

In 1983, Nose and Klein pointed out

in the paper

This implies that the generated dynamical equations for the MD particles in PRMD are not that of Newton’s Second Law.


Another drawback in prmd

Another drawback in PRMD

The generated dynamical equation for the periods under constant external stress

in their paper .

It is not in a form where the periods are driven by the imbalance between the internal and external stresses. Then when the systemreaches an equilibrium state, the internal and external stresses may not balance each other.


In any case prmd

In any case, PRMD

can find the true equilibrium states under constant external pressure and zero temperature, when all velocities and accelerations are zero.


Not using lagrangian dynamics we will

Not using Lagrangian Dynamics, we will

Keep Newton’s Second Law for the MD particle as its original

Apply Newton’s Second Law on halves of the system and statistics over system translation and particle moving directions to derive dynamical equations of the periods.


As a result our

As a result, our

dynamical equations are in the form where the periods are driven by the imbalance between internal and external stresses.

The internal stress has both a full interaction term and a kinetic-energy term.


Some notations

Some notations

By using the periods , any cell can be represented with

where are any integers.

For the MD cell .

Cell volume

Cell surface vectors:


More notations

More notations

Only pair potential is considered

Force acting on particle in cell by particle in cell is denoted with

External stress

with external pressure as a special case

where is a unit matrix.


Net force on the md cell

Net force on the MD cell

Since for any action from the green cell on the red MD cell, the reaction is equal to the action from the blue on the red, we have


Dynamical equations of periodic systems under constant external stress

Now let us first cut the system into two parts

with plane , so that for a given period , the right part contains

cells of , the rest in the left part.


A better illustration

A better illustration

The red is the MD cell.


Dynamical equations of periodic systems under constant external stress

The net external force on

is the cross section vector in plane


Dynamical equations of periodic systems under constant external stress

Newton’s Second Law on

is the net force on by


Dynamical equations of periodic systems under constant external stress

Newton’s Second Law on


Dynamical equations of periodic systems under constant external stress

Newton’s Second Law on

is evenly distributed cell by cell in


Dynamical equations of periodic systems under constant external stress

Newton’s Second Law on

“half cell bar”


I s the net force of blues on red and greens

is the net force of blues on red and greens

The red is the MD cell.


T he net force of blues on red

The net force of blues on red

is the negative of that of red on blues.

The red is the MD cell.


T he net force of blues on green

The net force of blues on green

is the negative of that of red on blues.

The red is the MD cell.


T he net force of blues on green1

The net force of blues on green

is the negative of that of red on blues.

The red is the MD cell.


T he net force of blues on green2

The net force of blues on green

is the negative of that of red on blues.

The red is the MD cell.


Dynamical equations of periodic systems under constant external stress

Newton’s Second Law on


I s the net force of blues on red and greens1

is the net force of blues on red and greens

The red is the MD cell.


T he net force of blues on red1

The net force of blues on red

is equal to that of red on greens.

The red is the MD cell.


T he net force of blues on green3

The net force of blues on green

is equal to that of red on greens.

The red is the MD cell.


T he net force of blues on green4

The net force of blues on green

is equal to that of red on greens.

The red is the MD cell.


T he net force of blues on green5

The net force of blues on green

is equal to that of red on greens.

The red is the MD cell.


Dynamical equations of periodic systems under constant external stress

Newton’s Second Law on


Recalling potential energy of the md cell

Recalling potential energy of the MD cell


Introducing main interaction tensor dyad

Introducing main interaction tensor/dyad


Remembering

Remembering

in


The left side of

The left side of

where and

and is used.


The left side of1

The left side of

where

Since in , any is non-negative, and

for any , there exists to cancel

it, then is zero.


Dynamical equations of periodic systems under constant external stress

First form of the period dynamics

directly from simplifying Newton’s Second Law


Now let us consider two states

Now let us consider two states

Only difference is translation between them.


Since they are indistinguishable

Since they are indistinguishable

We should take an unweighted average of the dynamical equations over all such states.


In all such states only different

In all such states, only different


What we really need is the unweighted average of

What we really need is the unweighted average of


What we really need is the unweighted average of1

What we really need is the unweighted average of

The total amount of such states can be represented by the volume of the MD cell .


What we really need is the unweighted average of2

What we really need is the unweighted average of

The total amount of such states can be represented by the volume of the MD cell .


Relative to in the following situation four cases should be considered for additional interactions

Relative to in the following situation, four cases should be considered for additional interactions.


R epresents the net force of blues on red and greens

represents the net force of blues on red and greens

The red is the MD cell.


Case 1 whenever passes through an md particle the old forces from blues on it should be deleted

Case 1, whenever passes through an MD particle, the old forces from blues on it should be deleted.

The amount of such states is


Case 2 whenever passes through an md particle the forces from it on yellows should be added

Case 2, whenever passes through an MD particle, the forces from it on yellows should be added.

The amount of such states is


Case 3 whenever is between two md particles the forces from left on right should be added

Case 3, whenever is between two MD particles, the forces from left on right should be added.

The amount of such states is


Dynamical equations of periodic systems under constant external stress

Case 4, whenever is between two MD particles, the forces from left in the greens on right MD particles should be added.

The amount of such states is


Dynamical equations of periodic systems under constant external stress

Period Dynamics

(first form)

(improved)

where the full interaction tensor


Dynamical equations of periodic systems under constant external stress

Now let us consider

forces associated with transport of momentum across geometrical planes,

even without collision or any other interactions


Dynamical equations of periodic systems under constant external stress

Forces only due to momentum transportation

But what is it?


Dynamical equations of periodic systems under constant external stress

Considering a single particle

without being acted by any regular force, but running with a constant velocity


Dynamical equations of periodic systems under constant external stress

Considering a single particle

without being acted by any regular force, but running with a constant velocity

Since it passes through many planes, is there any additional force acting on it?


Dynamical equations of periodic systems under constant external stress

Considering a single particle

without being acted by any regular force, but running with a constant velocity

Since it passes through many planes, is there any additional force acting on it?


Dynamical equations of periodic systems under constant external stress

As a matter of fact,

systems can be defined in two ways.


Dynamical equations of periodic systems under constant external stress

The first way,

systems are defined based on materials or particles.

For example, the above single particle, then no additional force should be considered, in order to satisfy Newton’s Laws.


Dynamical equations of periodic systems under constant external stress

The other way,

systems are defined based on space.

For example, still for the same single particle running process, we can define systems like red and green boxes, only based on space.


Dynamical equations of periodic systems under constant external stress

When the particle passes through

the plane between the red and green systems in

the momentum of the systems are changed. In order to satisfy Newton’s Laws, we can say there are forces between the two systems

This is the force purely associated with momentum transportation.


Dynamical equations of periodic systems under constant external stress

Our half systems are indeed defined based on space. Then such forces should be considered.


In all the previous translated only states

In all the previous translated-only states

During a unit time, particles passes

amount of states. Then the total

should be added into the

dynamical equation.


Dynamical equations of periodic systems under constant external stress

Period Dynamics

(first form)

(improved)

(further improved)

where the instantaneous internal stress


Dynamical equations of periodic systems under constant external stress

The last consideration

(further improved)

where the instantaneous internal stress

The periods should not depend on the instantaneous directions of particles microscopic motion, as they can be measured under constant external pressure and temperature.


Dynamical equations of periodic systems under constant external stress

The last unweighted average of the

(further improved)

where the instantaneous internal stress

over all particles’ moving directions.


Dynamical equations of periodic systems under constant external stress

Period Dynamics

(further improved)

(last)

where the internal stress


Dynamical equations of periodic systems under constant external stress

Period Dynamics

(first form)

(improved)

(further improved)

(last)


Dynamical equations of periodic systems under constant external stress

Summary


Dynamical equations of periodic systems under constant external stress

Applications

Applicable in any periodic system, especially in piezoelectric and piezomagnetic simulations.


Dynamical equations of periodic systems under constant external stress

Acknowledgements

The author wishes to thank

Prof. Ding-Sheng Wang, Institute of Physics,

Prof. Si-Yuan Zhang, Changchun Institute of Applied Chemistry, Prof. S. S. Wu, Jilin University,

Prof. En-Ge Wang, Beijing University, P.R. China,

Dr. Kenneth Edgecombe, Dr. HartmutSchmider,

Dr. Malcolm J. Stott, Dr. Kevin Robbie, Queen's University, and

Dr. XiaohuaWu, the Royal Military College of Canada,

for their helpful discussions and earnest encouragement.


Questions

Questions?

Thank you much very for your attention!

More details:

http://arxiv.org/pdf/cond-mat/0209372.pdf

and some printed copies available for picking up.

[email protected]


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