6 |
|
|
7 |
|
\section{\label{introSec:theory}Theoretical Background} |
8 |
|
|
9 |
< |
The techniques used in the course of this research fall under the two main classes of |
10 |
< |
molecular simulation: Molecular Dynamics and Monte Carlo. Molecular Dynamic simulations |
11 |
< |
integrate the equations of motion for a given system of particles, allowing the researher |
12 |
< |
to gain insight into the time dependent evolution of a system. Diffusion phenomena are |
13 |
< |
readily studied with this simulation technique, making Molecular Dynamics the main simulation |
14 |
< |
technique used in this research. Other aspects of the research fall under the Monte Carlo |
15 |
< |
class of simulations. In Monte Carlo, the configuration space available to the collection |
16 |
< |
of particles is sampled stochastichally, or randomly. Each configuration is chosen with |
17 |
< |
a given probability based on the Maxwell Boltzman distribution. These types of simulations |
18 |
< |
are best used to probe properties of a system that are only dependent only on the state of |
19 |
< |
the system. Structural information about a system is most readily obtained through |
20 |
< |
these types of methods. |
9 |
> |
The techniques used in the course of this research fall under the two |
10 |
> |
main classes of molecular simulation: Molecular Dynamics and Monte |
11 |
> |
Carlo. Molecular Dynamic simulations integrate the equations of motion |
12 |
> |
for a given system of particles, allowing the researher to gain |
13 |
> |
insight into the time dependent evolution of a system. Diffusion |
14 |
> |
phenomena are readily studied with this simulation technique, making |
15 |
> |
Molecular Dynamics the main simulation technique used in this |
16 |
> |
research. Other aspects of the research fall under the Monte Carlo |
17 |
> |
class of simulations. In Monte Carlo, the configuration space |
18 |
> |
available to the collection of particles is sampled stochastichally, |
19 |
> |
or randomly. Each configuration is chosen with a given probability |
20 |
> |
based on the Maxwell Boltzman distribution. These types of simulations |
21 |
> |
are best used to probe properties of a system that are only dependent |
22 |
> |
only on the state of the system. Structural information about a system |
23 |
> |
is most readily obtained through these types of methods. |
24 |
|
|
25 |
< |
Although the two techniques employed seem dissimilar, they are both linked by the overarching |
26 |
< |
principles of Statistical Thermodynamics. Statistical Thermodynamics governs the behavior of |
27 |
< |
both classes of simulations and dictates what each method can and cannot do. When investigating |
28 |
< |
a system, one most first analyze what thermodynamic properties of the system are being probed, |
29 |
< |
then chose which method best suits that objective. |
25 |
> |
Although the two techniques employed seem dissimilar, they are both |
26 |
> |
linked by the overarching principles of Statistical |
27 |
> |
Thermodynamics. Statistical Thermodynamics governs the behavior of |
28 |
> |
both classes of simulations and dictates what each method can and |
29 |
> |
cannot do. When investigating a system, one most first analyze what |
30 |
> |
thermodynamic properties of the system are being probed, then chose |
31 |
> |
which method best suits that objective. |
32 |
|
|
33 |
|
\subsection{\label{introSec:statThermo}Statistical Thermodynamics} |
34 |
|
|
38 |
|
|
39 |
|
\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations} |
40 |
|
|
41 |
< |
Stochastic sampling |
41 |
> |
The Monte Carlo method was developed by Metropolis and Ulam for their |
42 |
> |
work in fissionable material.\cite{metropolis:1949} The method is so |
43 |
> |
named, because it heavily uses random numbers in the solution of the |
44 |
> |
problem. |
45 |
|
|
38 |
– |
detatiled balance |
46 |
|
|
40 |
– |
metropilis monte carlo |
41 |
– |
|
47 |
|
\subsection{\label{introSec:md}Molecular Dynamics Simulations} |
48 |
|
|
49 |
|
time averages |