Class 3
Department of Chemical Engineering
University at Buffalo, State University of New York

• Typical simulation size very small
– 100 - 1000 atoms
• Important extensive quantities small in magnitude
– when expressed in macroscopic units
• Small numbers are inconvenient
• Two ways to magnify them
– work with atomic-scale units
• ps, amu, nm or Å
– make dimensionless with characteristic values
• model values of size, energy, mass
2

• Scaling by model parameters
– size s
– energy e
– mass m
• Choose values for one atom/molecule pair potential arbitrarily
• Other model parameters given in terms of reference values
– e.g., e
2
/ e
1
= 1.2
• Physical magnitudes less transparent
• Sometimes convenient to scale coordinates differently
3

• Lennard-Jones potential in dimensionless form u * ( r *)

4




1 r *
12

1 r *
6



• Parameter independent!
• Dimensionless properties must also be parameter independent
– convenient to report properties in this form, e.g. P*( r
*,T*)
– select model values to get actual values of properties
– Basis of corresponding states
• Equivalent to selecting unit value for parameters
4

• Corresponding States Example
– Want pressure for methane at 0.0183 mol/cm 3 and 167 K
– LJ model parameters are s
= 0.3790 nm, e
/k = 142.1 K
– Dimensionless state parameters
• r
* = rs
3 = (0.0183 mol/cm 3 )(3.790

10 -8 cm) 3 (6.022

10 23 molecules/mole) = 0.6
•
T* = T/( e
/k) = (167 K)/(142.1 K) = 1.174
– From LJ equation of state
• P* = P s
3 / e
= 0.146
– Corresponding to a pressure
• P = 0.146 (142.1 K)(13.8 MPaÅ 3 /molecule )/(3.790Å) 3 = 5.3 MPa
• 53 bars
5

• All internal calculations are based on a consistent set of units
– Mass: amu or Dalton, 1/N
A grams = 1.66
– Length: Angstrom, 10 -10 meters

10 -24 grams
– Time: picosecond, 10 -12 seconds
• Conversion to other units is done for input or output by a
Device or Display
• Default action is to perform no conversion
– Temperature is given as k
B
T, in units of amu-A 2 /ps 2
• Default I/O unit system can be changed
– MKS (CGS, English, atomic, etc. to be defined)
– Lennard-Jones
– At beginning of simulation, use (for example)
• setUnitSystem(etomica.units.UnitSystem.MKS);
6

• A Unit class has the information needed to perform conversions
• Unit formed from two elements
– Prefix: milli, kilo, mega, nano, pico, etc.
– BaseUnit: meter, gram, second, etc.
– For example:
• import etomica.units;
• new Unit(Prefix.KILO, Gram.UNIT);
– Can construct without prefix specified (none by default)
• Display/Device units specified with setUnit method
• setUnit(new Unit(Second.UNIT));
• Lennard-Jones units use size and energy scales for conversions
– By default, s
= 3 Angstroms, e
/k = 300K
7

• Most often we simulate 2-dimensions systems
– Easier to visualize
• Some physical quantities have units defined exclusively for
3-dimensional systems
– Pressure: pounds per square inch, bars, mm Hg, etc.
– Volume: cubic centimeters, liters, gallons, etc.
• Etomica offers two approaches
– Report in appropriate 2-D units
• pounds per inch, Newtons per meter, square centimeters, etc.
– Ascribe an artificial “depth” to the simulation
• Volume = (depth)

(area)
• Presently
– LJ UnitSystem will report in appropriate 2-D units
– MKS UnitSystem ignores the problem (a bug!)
8

• Version now in use was completed in early June
• Much development/debugging of API proceeded over summer
– Etomica GUI hasn’t yet been brought up-to-date
– Progress now being made with it, and expected to become available
Real Soon Now
– In the meantime, please be patient!
• Etomica crashes in two known circumstances:
– Assembly of incompatible components
• Soft potential with hard integrator, for example
– Jamming of molecules with hard integrator
9

• The “equation of state” is the pressure-volume-temperature behavior of a fluid (or solid)
• Consider:
– What happens to the pressure as the volume decreases (density increases) at fixed temperature?
– What happens to the density as temperature increases at fixed pressure?
– Must it always happen this way?
• What is the law relating pressure, temperature, density?
– Is it exact, like the law F = ma?
10

• The “equation of state” is the pressure-volume-temperature behavior of a fluid (or solid)
• Consider:
– What happens to the pressure as the volume decreases (density increases) at fixed temperature?
• It increases, always.
– What happens to the density as temperature increases at fixed pressure?
• It decreases (hot expands), usually.
– Must it always happen this way?
• Yes and no.
• What is the law relating pressure, temperature, density?
– PV = nRT, the ideal gas law
– Is it exact, like the law F = ma?
• No! (nor is F = ma, for that matter).
11

• The ideal-gas law is, well, an idealization
– Appropriate at low density, and (less so) at high temperature
• Real materials deviate from ideal-gas behavior
– Repulsions between molecules tend to increase pressure above ideal
– Attractions tend to decrease pressure below ideal
• Here’s an empirical equation of state for the 2-D hard sphere model
PA
NkT r
1 0.356780
r 
0.021447
r 2
1 1.775171
r 
0.787808
r 2
– Note the adherence to ideal-gas behavior at low density
– It is hard to make good, general-purpose equations of state for real materials
12

• Consider a grid of equally spaced lines, separated by a distance d
• Take a needle of length l , and repeatedly toss it at random on the grid
• Record the number of “hits”, times that the needle touches a line, and “misses”, times that it doesn’t
– OK, do it!
13

• Consider a grid of equally spaced lines, separated by a distance d
• Take a needle of length l , and repeatedly toss it at random on the grid
• Record the number of “hits”, times that the needle touches a line, and “misses”, times that it doesn’t
• Buffon showed that the probability of a “hit” is
P

2 l
 d
• This “experiment” provides a means to evaluate pi
 
2 l
Pd
14

• Consider selecting many points at random in a unit square
1
1
• What is the fraction of the selections that lie in the unit circle?
15

• Fraction falling in circle is given by the ratio of their areas
P
 area of circle area of square


(1/ 2)
1
2
2


4
• This gives another experiment to measure pi
   number in circle number of trials
• Let’s try it!
16

• Random number generators
– subroutines that provide a new random deviate with each call
– basic generators give value on (0,1) with uniform probability
– uses a deterministic algorithm (of course)
• usually involves multiplication and truncation of leading bits of a number
X n

1

( aX n
 c ) mod m linear congruential sequence
• Returns set of numbers that meet many statistical measures of randomness
– histogram is uniform
– no systematic correlation of deviates
• no idea what next value will be from knowledge of present value (without knowing generation algorithm)
• but eventually, the series must end up repeating
Plot of successive deviates (X n
,X n+1
)
• Some famous failures
– be careful to use a good quality generator
Not so random!
17