| 822 |
|
% |
| 823 |
|
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 824 |
|
q(\Delta t)} \right]. % |
| 825 |
< |
\label{introEquation:positionVerlet1} |
| 825 |
> |
\label{introEquation:positionVerlet2} |
| 826 |
|
\end{align} |
| 827 |
|
|
| 828 |
|
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
| 882 |
|
\] |
| 883 |
|
|
| 884 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 885 |
– |
|
| 886 |
– |
As a special discipline of molecular modeling, Molecular dynamics |
| 887 |
– |
has proven to be a powerful tool for studying the functions of |
| 888 |
– |
biological systems, providing structural, thermodynamic and |
| 889 |
– |
dynamical information. |
| 885 |
|
|
| 886 |
< |
\subsection{\label{introSec:mdInit}Initialization} |
| 886 |
> |
As one of the principal tools of molecular modeling, Molecular |
| 887 |
> |
dynamics has proven to be a powerful tool for studying the functions |
| 888 |
> |
of biological systems, providing structural, thermodynamic and |
| 889 |
> |
dynamical information. The basic idea of molecular dynamics is that |
| 890 |
> |
macroscopic properties are related to microscopic behavior and |
| 891 |
> |
microscopic behavior can be calculated from the trajectories in |
| 892 |
> |
simulations. For instance, instantaneous temperature of an |
| 893 |
> |
Hamiltonian system of $N$ particle can be measured by |
| 894 |
> |
\[ |
| 895 |
> |
T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
| 896 |
> |
\] |
| 897 |
> |
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
| 898 |
> |
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
| 899 |
> |
the boltzman constant. |
| 900 |
|
|
| 901 |
< |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 901 |
> |
A typical molecular dynamics run consists of three essential steps: |
| 902 |
> |
\begin{enumerate} |
| 903 |
> |
\item Initialization |
| 904 |
> |
\begin{enumerate} |
| 905 |
> |
\item Preliminary preparation |
| 906 |
> |
\item Minimization |
| 907 |
> |
\item Heating |
| 908 |
> |
\item Equilibration |
| 909 |
> |
\end{enumerate} |
| 910 |
> |
\item Production |
| 911 |
> |
\item Analysis |
| 912 |
> |
\end{enumerate} |
| 913 |
> |
These three individual steps will be covered in the following |
| 914 |
> |
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 915 |
> |
initialization of a simulation. Sec.~\ref{introSec:production} will |
| 916 |
> |
discusses issues in production run, including the force evaluation |
| 917 |
> |
and the numerical integration schemes of the equations of motion . |
| 918 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 919 |
> |
trajectory analysis. |
| 920 |
|
|
| 921 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 921 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 922 |
> |
|
| 923 |
> |
\subsubsection{Preliminary preparation} |
| 924 |
> |
|
| 925 |
> |
When selecting the starting structure of a molecule for molecular |
| 926 |
> |
simulation, one may retrieve its Cartesian coordinates from public |
| 927 |
> |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
| 928 |
> |
thousands of crystal structures of molecules are discovered every |
| 929 |
> |
year, many more remain unknown due to the difficulties of |
| 930 |
> |
purification and crystallization. Even for the molecule with known |
| 931 |
> |
structure, some important information is missing. For example, the |
| 932 |
> |
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 933 |
> |
be added. Moreover, in order to include electrostatic interaction, |
| 934 |
> |
one may need to specify the partial charges for individual atoms. |
| 935 |
> |
Under some circumstances, we may even need to prepare the system in |
| 936 |
> |
a special setup. For instance, when studying transport phenomenon in |
| 937 |
> |
membrane system, we may prepare the lipids in bilayer structure |
| 938 |
> |
instead of placing lipids randomly in solvent, since we are not |
| 939 |
> |
interested in self-aggregation and it takes a long time to happen. |
| 940 |
> |
|
| 941 |
> |
\subsubsection{Minimization} |
| 942 |
> |
|
| 943 |
> |
It is quite possible that some of molecules in the system from |
| 944 |
> |
preliminary preparation may be overlapped with each other. This |
| 945 |
> |
close proximity leads to high potential energy which consequently |
| 946 |
> |
jeopardizes any molecular dynamics simulations. To remove these |
| 947 |
> |
steric overlaps, one typically performs energy minimization to find |
| 948 |
> |
a more reasonable conformation. Several energy minimization methods |
| 949 |
> |
have been developed to exploit the energy surface and to locate the |
| 950 |
> |
local minimum. While converging slowly near the minimum, steepest |
| 951 |
> |
descent method is extremely robust when systems are far from |
| 952 |
> |
harmonic. Thus, it is often used to refine structure from |
| 953 |
> |
crystallographic data. Relied on the gradient or hessian, advanced |
| 954 |
> |
methods like conjugate gradient and Newton-Raphson converge rapidly |
| 955 |
> |
to a local minimum, while become unstable if the energy surface is |
| 956 |
> |
far from quadratic. Another factor must be taken into account, when |
| 957 |
> |
choosing energy minimization method, is the size of the system. |
| 958 |
> |
Steepest descent and conjugate gradient can deal with models of any |
| 959 |
> |
size. Because of the limit of computation power to calculate hessian |
| 960 |
> |
matrix and insufficient storage capacity to store them, most |
| 961 |
> |
Newton-Raphson methods can not be used with very large models. |
| 962 |
> |
|
| 963 |
> |
\subsubsection{Heating} |
| 964 |
> |
|
| 965 |
> |
Typically, Heating is performed by assigning random velocities |
| 966 |
> |
according to a Gaussian distribution for a temperature. Beginning at |
| 967 |
> |
a lower temperature and gradually increasing the temperature by |
| 968 |
> |
assigning greater random velocities, we end up with setting the |
| 969 |
> |
temperature of the system to a final temperature at which the |
| 970 |
> |
simulation will be conducted. In heating phase, we should also keep |
| 971 |
> |
the system from drifting or rotating as a whole. Equivalently, the |
| 972 |
> |
net linear momentum and angular momentum of the system should be |
| 973 |
> |
shifted to zero. |
| 974 |
> |
|
| 975 |
> |
\subsubsection{Equilibration} |
| 976 |
> |
|
| 977 |
> |
The purpose of equilibration is to allow the system to evolve |
| 978 |
> |
spontaneously for a period of time and reach equilibrium. The |
| 979 |
> |
procedure is continued until various statistical properties, such as |
| 980 |
> |
temperature, pressure, energy, volume and other structural |
| 981 |
> |
properties \textit{etc}, become independent of time. Strictly |
| 982 |
> |
speaking, minimization and heating are not necessary, provided the |
| 983 |
> |
equilibration process is long enough. However, these steps can serve |
| 984 |
> |
as a means to arrive at an equilibrated structure in an effective |
| 985 |
> |
way. |
| 986 |
> |
|
| 987 |
> |
\subsection{\label{introSection:production}Production} |
| 988 |
> |
|
| 989 |
> |
\subsubsection{\label{introSec:forceCalculation}The Force Calculation} |
| 990 |
> |
|
| 991 |
> |
\subsubsection{\label{introSection:integrationSchemes} Integration |
| 992 |
> |
Schemes} |
| 993 |
> |
|
| 994 |
> |
\subsection{\label{introSection:Analysis} Analysis} |
| 995 |
> |
|
| 996 |
> |
Recently, advanced visualization technique are widely applied to |
| 997 |
> |
monitor the motions of molecules. Although the dynamics of the |
| 998 |
> |
system can be described qualitatively from animation, quantitative |
| 999 |
> |
trajectory analysis are more appreciable. According to the |
| 1000 |
> |
principles of Statistical Mechanics, |
| 1001 |
> |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 1002 |
> |
thermodynamics properties, analyze fluctuations of structural |
| 1003 |
> |
parameters, and investigate time-dependent processes of the molecule |
| 1004 |
> |
from the trajectories. |
| 1005 |
> |
|
| 1006 |
> |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
| 1007 |
> |
|
| 1008 |
> |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
| 1009 |
> |
|
| 1010 |
> |
Structural Properties of a simple fluid can be described by a set of |
| 1011 |
> |
distribution functions. Among these functions,\emph{pair |
| 1012 |
> |
distribution function}, also known as \emph{radial distribution |
| 1013 |
> |
function}, are of most fundamental importance to liquid-state |
| 1014 |
> |
theory. Pair distribution function can be gathered by Fourier |
| 1015 |
> |
transforming raw data from a series of neutron diffraction |
| 1016 |
> |
experiments and integrating over the surface factor \cite{Powles73}. |
| 1017 |
> |
The experiment result can serve as a criterion to justify the |
| 1018 |
> |
correctness of the theory. Moreover, various equilibrium |
| 1019 |
> |
thermodynamic and structural properties can also be expressed in |
| 1020 |
> |
terms of radial distribution function \cite{allen87:csl}. |
| 1021 |
> |
|
| 1022 |
> |
A pair distribution functions $g(r)$ gives the probability that a |
| 1023 |
> |
particle $i$ will be located at a distance $r$ from a another |
| 1024 |
> |
particle $j$ in the system |
| 1025 |
> |
\[ |
| 1026 |
> |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1027 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
| 1028 |
> |
\] |
| 1029 |
> |
Note that the delta function can be replaced by a histogram in |
| 1030 |
> |
computer simulation. Figure |
| 1031 |
> |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
| 1032 |
> |
distribution function for the liquid argon system. The occurrence of |
| 1033 |
> |
several peaks in the plot of $g(r)$ suggests that it is more likely |
| 1034 |
> |
to find particles at certain radial values than at others. This is a |
| 1035 |
> |
result of the attractive interaction at such distances. Because of |
| 1036 |
> |
the strong repulsive forces at short distance, the probability of |
| 1037 |
> |
locating particles at distances less than about 2.5{\AA} from each |
| 1038 |
> |
other is essentially zero. |
| 1039 |
> |
|
| 1040 |
> |
%\begin{figure} |
| 1041 |
> |
%\centering |
| 1042 |
> |
%\includegraphics[width=\linewidth]{pdf.eps} |
| 1043 |
> |
%\caption[Pair distribution function for the liquid argon |
| 1044 |
> |
%]{Pair distribution function for the liquid argon} |
| 1045 |
> |
%\label{introFigure:pairDistributionFunction} |
| 1046 |
> |
%\end{figure} |
| 1047 |
> |
|
| 1048 |
> |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
| 1049 |
> |
Properties} |
| 1050 |
|
|
| 1051 |
+ |
Time-dependent properties are usually calculated using \emph{time |
| 1052 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
| 1053 |
+ |
at two different time |
| 1054 |
+ |
\begin{equation} |
| 1055 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
| 1056 |
+ |
\label{introEquation:timeCorrelationFunction} |
| 1057 |
+ |
\end{equation} |
| 1058 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1059 |
+ |
function is called \emph{auto correlation function}. One example of |
| 1060 |
+ |
auto correlation function is velocity auto-correlation function |
| 1061 |
+ |
which is directly related to transport properties of molecular |
| 1062 |
+ |
liquids. Another example is the calculation of the IR spectrum |
| 1063 |
+ |
through a Fourier transform of the dipole autocorrelation function. |
| 1064 |
+ |
|
| 1065 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1066 |
|
|
| 1067 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1095 |
|
The break through in geometric literature suggests that, in order to |
| 1096 |
|
develop a long-term integration scheme, one should preserve the |
| 1097 |
|
symplectic structure of the flow. Introducing conjugate momentum to |
| 1098 |
< |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
| 1098 |
> |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
| 1099 |
|
symplectic integrator, RSHAKE, was proposed to evolve the |
| 1100 |
|
Hamiltonian system in a constraint manifold by iteratively |
| 1101 |
< |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
| 1101 |
> |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
| 1102 |
|
method using quaternion representation was developed by Omelyan. |
| 1103 |
|
However, both of these methods are iterative and inefficient. In |
| 1104 |
|
this section, we will present a symplectic Lie-Poisson integrator |
| 1304 |
|
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1305 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1306 |
|
\] |
| 1307 |
< |
To reduce the cost of computing expensive functions in e^{\Delta |
| 1308 |
< |
tR_1 }, we can use Cayley transformation, |
| 1307 |
> |
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1308 |
> |
tR_1 }$, we can use Cayley transformation, |
| 1309 |
|
\[ |
| 1310 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1311 |
|
) |
| 1312 |
|
\] |
| 1313 |
< |
|
| 1146 |
< |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
| 1313 |
> |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1314 |
|
manner. |
| 1315 |
|
|
| 1316 |
|
In order to construct a second-order symplectic method, we split the |
| 1380 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1381 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1382 |
|
\] |
| 1383 |
< |
Moreover, \varphi _{\Delta t/2,V} can be divided into two sub-flows |
| 1384 |
< |
which corresponding to force and torque respectively, |
| 1383 |
> |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 1384 |
> |
sub-flows which corresponding to force and torque respectively, |
| 1385 |
|
\[ |
| 1386 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1387 |
|
_{\Delta t/2,\tau }. |
| 1388 |
|
\] |
| 1389 |
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1390 |
|
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 1391 |
< |
order inside \varphi _{\Delta t/2,V} does not matter. |
| 1391 |
> |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
| 1392 |
|
|
| 1393 |
|
Furthermore, kinetic potential can be separated to translational |
| 1394 |
|
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1418 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
| 1419 |
|
has been applied in a variety of studies. This section will review |
| 1420 |
|
the theory of Langevin dynamics simulation. A brief derivation of |
| 1421 |
< |
generalized Langevin Dynamics will be given first. Follow that, we |
| 1421 |
> |
generalized Langevin equation will be given first. Follow that, we |
| 1422 |
|
will discuss the physical meaning of the terms appearing in the |
| 1423 |
|
equation as well as the calculation of friction tensor from |
| 1424 |
|
hydrodynamics theory. |
| 1425 |
|
|
| 1426 |
< |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 1426 |
> |
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1427 |
|
|
| 1428 |
+ |
Harmonic bath model, in which an effective set of harmonic |
| 1429 |
+ |
oscillators are used to mimic the effect of a linearly responding |
| 1430 |
+ |
environment, has been widely used in quantum chemistry and |
| 1431 |
+ |
statistical mechanics. One of the successful applications of |
| 1432 |
+ |
Harmonic bath model is the derivation of Deriving Generalized |
| 1433 |
+ |
Langevin Dynamics. Lets consider a system, in which the degree of |
| 1434 |
+ |
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1435 |
+ |
Hamiltonian of the form |
| 1436 |
|
\begin{equation} |
| 1437 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1438 |
< |
\label{introEquation:bathGLE} |
| 1438 |
> |
\label{introEquation:bathGLE}. |
| 1439 |
|
\end{equation} |
| 1440 |
< |
where $H_B$ is harmonic bath Hamiltonian, |
| 1440 |
> |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
| 1441 |
> |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
| 1442 |
|
\[ |
| 1443 |
< |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1444 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
| 1443 |
> |
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1444 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1445 |
> |
\right\}} |
| 1446 |
|
\] |
| 1447 |
< |
and $\Delta U$ is bilinear system-bath coupling, |
| 1447 |
> |
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1448 |
> |
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
| 1449 |
> |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
| 1450 |
> |
coupling, |
| 1451 |
|
\[ |
| 1452 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1453 |
|
\] |
| 1454 |
< |
Completing the square, |
| 1454 |
> |
where $g_\alpha$ are the coupling constants between the bath and the |
| 1455 |
> |
coordinate $x$. Introducing |
| 1456 |
|
\[ |
| 1457 |
< |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
| 1458 |
< |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1459 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1460 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
| 1461 |
< |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1281 |
< |
\] |
| 1282 |
< |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
| 1457 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1458 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1459 |
> |
\] and combining the last two terms in Equation |
| 1460 |
> |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
| 1461 |
> |
Hamiltonian as |
| 1462 |
|
\[ |
| 1463 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1464 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1465 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1466 |
|
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1467 |
|
\] |
| 1289 |
– |
where |
| 1290 |
– |
\[ |
| 1291 |
– |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1292 |
– |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1293 |
– |
\] |
| 1468 |
|
Since the first two terms of the new Hamiltonian depend only on the |
| 1469 |
|
system coordinates, we can get the equations of motion for |
| 1470 |
|
Generalized Langevin Dynamics by Hamilton's equations |
| 1471 |
|
\ref{introEquation:motionHamiltonianCoordinate, |
| 1472 |
|
introEquation:motionHamiltonianMomentum}, |
| 1473 |
< |
\begin{align} |
| 1474 |
< |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 1475 |
< |
&= m\ddot x |
| 1476 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
| 1477 |
< |
\label{introEquation:Lp5} |
| 1478 |
< |
\end{align} |
| 1479 |
< |
, and |
| 1480 |
< |
\begin{align} |
| 1481 |
< |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
| 1482 |
< |
&= m\ddot x_\alpha |
| 1483 |
< |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
| 1484 |
< |
\end{align} |
| 1473 |
> |
\begin{equation} |
| 1474 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1475 |
> |
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
| 1476 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)}, |
| 1477 |
> |
\label{introEquation:coorMotionGLE} |
| 1478 |
> |
\end{equation} |
| 1479 |
> |
and |
| 1480 |
> |
\begin{equation} |
| 1481 |
> |
m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - |
| 1482 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1483 |
> |
\label{introEquation:bathMotionGLE} |
| 1484 |
> |
\end{equation} |
| 1485 |
|
|
| 1486 |
< |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
| 1486 |
> |
In order to derive an equation for $x$, the dynamics of the bath |
| 1487 |
> |
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1488 |
> |
transform which is particularly useful in solving linear ordinary |
| 1489 |
> |
differential equations, Laplace transform is the appropriate tool to |
| 1490 |
> |
solve this problem. The basic idea is to transform the difficult |
| 1491 |
> |
differential equations into simple algebra problems which can be |
| 1492 |
> |
solved easily. Then applying inverse Laplace transform, also known |
| 1493 |
> |
as the Bromwich integral, we can retrieve the solutions of the |
| 1494 |
> |
original problems. |
| 1495 |
|
|
| 1496 |
+ |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1497 |
+ |
transform of f(t) is a new function defined as |
| 1498 |
|
\[ |
| 1499 |
< |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
| 1499 |
> |
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1500 |
|
\] |
| 1501 |
+ |
where $p$ is real and $L$ is called the Laplace Transform |
| 1502 |
+ |
Operator. Below are some important properties of Laplace transform |
| 1503 |
+ |
\begin{equation} |
| 1504 |
+ |
\begin{array}{c} |
| 1505 |
+ |
L(x + y) = L(x) + L(y) \\ |
| 1506 |
+ |
L(ax) = aL(x) \\ |
| 1507 |
+ |
L(\dot x) = pL(x) - px(0) \\ |
| 1508 |
+ |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
| 1509 |
+ |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
| 1510 |
+ |
\end{array} |
| 1511 |
+ |
\end{equation} |
| 1512 |
|
|
| 1513 |
+ |
Applying Laplace transform to the bath coordinates, we obtain |
| 1514 |
|
\[ |
| 1515 |
< |
L(x + y) = L(x) + L(y) |
| 1515 |
> |
\begin{array}{c} |
| 1516 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
| 1517 |
> |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1518 |
> |
\end{array} |
| 1519 |
|
\] |
| 1520 |
< |
|
| 1520 |
> |
By the same way, the system coordinates become |
| 1521 |
|
\[ |
| 1522 |
< |
L(ax) = aL(x) |
| 1522 |
> |
\begin{array}{c} |
| 1523 |
> |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1524 |
> |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1525 |
> |
\end{array} |
| 1526 |
|
\] |
| 1527 |
|
|
| 1528 |
+ |
With the help of some relatively important inverse Laplace |
| 1529 |
+ |
transformations: |
| 1530 |
|
\[ |
| 1531 |
< |
L(\dot x) = pL(x) - px(0) |
| 1531 |
> |
\begin{array}{c} |
| 1532 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ |
| 1533 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ |
| 1534 |
> |
L(1) = \frac{1}{p} \\ |
| 1535 |
> |
\end{array} |
| 1536 |
|
\] |
| 1537 |
< |
|
| 1330 |
< |
\[ |
| 1331 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
| 1332 |
< |
\] |
| 1333 |
< |
|
| 1334 |
< |
\[ |
| 1335 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
| 1336 |
< |
\] |
| 1337 |
< |
|
| 1338 |
< |
Some relatively important transformation, |
| 1339 |
< |
\[ |
| 1340 |
< |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
| 1341 |
< |
\] |
| 1342 |
< |
|
| 1343 |
< |
\[ |
| 1344 |
< |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
| 1345 |
< |
\] |
| 1346 |
< |
|
| 1347 |
< |
\[ |
| 1348 |
< |
L(1) = \frac{1}{p} |
| 1349 |
< |
\] |
| 1350 |
< |
|
| 1351 |
< |
First, the bath coordinates, |
| 1352 |
< |
\[ |
| 1353 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
| 1354 |
< |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
| 1355 |
< |
}}L(x) |
| 1356 |
< |
\] |
| 1357 |
< |
\[ |
| 1358 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
| 1359 |
< |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
| 1360 |
< |
\] |
| 1361 |
< |
Then, the system coordinates, |
| 1537 |
> |
, we obtain |
| 1538 |
|
\begin{align} |
| 1363 |
– |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1364 |
– |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
| 1365 |
– |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
| 1366 |
– |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
| 1367 |
– |
}}\omega _\alpha ^2 L(x)} \right\}} |
| 1368 |
– |
% |
| 1369 |
– |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1370 |
– |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
| 1371 |
– |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
| 1372 |
– |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
| 1373 |
– |
\end{align} |
| 1374 |
– |
Then, the inverse transform, |
| 1375 |
– |
|
| 1376 |
– |
\begin{align} |
| 1539 |
|
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
| 1540 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1541 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
| 1554 |
|
(\omega _\alpha t)} \right\}} |
| 1555 |
|
\end{align} |
| 1556 |
|
|
| 1557 |
+ |
Introducing a \emph{dynamic friction kernel} |
| 1558 |
|
\begin{equation} |
| 1396 |
– |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
| 1397 |
– |
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1398 |
– |
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1399 |
– |
\end{equation} |
| 1400 |
– |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
| 1401 |
– |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
| 1402 |
– |
\[ |
| 1559 |
|
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1560 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 1561 |
< |
\] |
| 1562 |
< |
For an infinite harmonic bath, we can use the spectral density and |
| 1563 |
< |
an integral over frequencies. |
| 1564 |
< |
|
| 1409 |
< |
\[ |
| 1561 |
> |
\label{introEquation:dynamicFrictionKernelDefinition} |
| 1562 |
> |
\end{equation} |
| 1563 |
> |
and \emph{a random force} |
| 1564 |
> |
\begin{equation} |
| 1565 |
|
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 1566 |
|
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 1567 |
|
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 1568 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
| 1568 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), |
| 1569 |
> |
\label{introEquation:randomForceDefinition} |
| 1570 |
> |
\end{equation} |
| 1571 |
> |
the equation of motion can be rewritten as |
| 1572 |
> |
\begin{equation} |
| 1573 |
> |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
| 1574 |
> |
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1575 |
> |
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1576 |
> |
\end{equation} |
| 1577 |
> |
which is known as the \emph{generalized Langevin equation}. |
| 1578 |
> |
|
| 1579 |
> |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel} |
| 1580 |
> |
|
| 1581 |
> |
One may notice that $R(t)$ depends only on initial conditions, which |
| 1582 |
> |
implies it is completely deterministic within the context of a |
| 1583 |
> |
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1584 |
> |
uncorrelated to $x$ and $\dot x$, |
| 1585 |
> |
\[ |
| 1586 |
> |
\begin{array}{l} |
| 1587 |
> |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
| 1588 |
> |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
| 1589 |
> |
\end{array} |
| 1590 |
|
\] |
| 1591 |
< |
The random forces depend only on initial conditions. |
| 1591 |
> |
This property is what we expect from a truly random process. As long |
| 1592 |
> |
as the model, which is gaussian distribution in general, chosen for |
| 1593 |
> |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
| 1594 |
> |
still remains. |
| 1595 |
> |
|
| 1596 |
> |
%dynamic friction kernel |
| 1597 |
> |
The convolution integral |
| 1598 |
> |
\[ |
| 1599 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } |
| 1600 |
> |
\] |
| 1601 |
> |
depends on the entire history of the evolution of $x$, which implies |
| 1602 |
> |
that the bath retains memory of previous motions. In other words, |
| 1603 |
> |
the bath requires a finite time to respond to change in the motion |
| 1604 |
> |
of the system. For a sluggish bath which responds slowly to changes |
| 1605 |
> |
in the system coordinate, we may regard $\xi(t)$ as a constant |
| 1606 |
> |
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes |
| 1607 |
> |
\[ |
| 1608 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
| 1609 |
> |
\] |
| 1610 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1611 |
> |
\[ |
| 1612 |
> |
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1613 |
> |
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1614 |
> |
\] |
| 1615 |
> |
which can be used to describe dynamic caging effect. The other |
| 1616 |
> |
extreme is the bath that responds infinitely quickly to motions in |
| 1617 |
> |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
| 1618 |
> |
time: |
| 1619 |
> |
\[ |
| 1620 |
> |
\xi (t) = 2\xi _0 \delta (t) |
| 1621 |
> |
\] |
| 1622 |
> |
Hence, the convolution integral becomes |
| 1623 |
> |
\[ |
| 1624 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1625 |
> |
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1626 |
> |
\] |
| 1627 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1628 |
> |
\begin{equation} |
| 1629 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1630 |
> |
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1631 |
> |
\end{equation} |
| 1632 |
> |
which is known as the Langevin equation. The static friction |
| 1633 |
> |
coefficient $\xi _0$ can either be calculated from spectral density |
| 1634 |
> |
or be determined by Stokes' law for regular shaped particles.A |
| 1635 |
> |
briefly review on calculating friction tensor for arbitrary shaped |
| 1636 |
> |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1637 |
|
|
| 1638 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 1639 |
< |
So we can define a new set of coordinates, |
| 1639 |
> |
|
| 1640 |
> |
Defining a new set of coordinates, |
| 1641 |
|
\[ |
| 1642 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1643 |
|
^2 }}x(0) |
| 1644 |
< |
\] |
| 1645 |
< |
This makes |
| 1644 |
> |
\], |
| 1645 |
> |
we can rewrite $R(T)$ as |
| 1646 |
|
\[ |
| 1647 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
| 1647 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1648 |
|
\] |
| 1649 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1650 |
|
\[ |
| 1651 |
< |
\begin{array}{l} |
| 1651 |
> |
\begin{array}{c} |
| 1652 |
> |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1653 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1654 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1655 |
+ |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
| 1656 |
+ |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1657 |
+ |
= kT\xi (t) \\ |
| 1658 |
|
\end{array} |
| 1659 |
|
\] |
| 1660 |
< |
|
| 1435 |
< |
\begin{align} |
| 1436 |
< |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
| 1437 |
< |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
| 1438 |
< |
(t)q_\beta (0)} \right\rangle } } |
| 1439 |
< |
% |
| 1440 |
< |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
| 1441 |
< |
\right\rangle \cos (\omega _\alpha t)} |
| 1442 |
< |
% |
| 1443 |
< |
&= kT\xi (t) |
| 1444 |
< |
\end{align} |
| 1445 |
< |
|
| 1660 |
> |
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1661 |
|
\begin{equation} |
| 1662 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 1663 |
< |
\label{introEquation:secondFluctuationDissipation} |
| 1663 |
> |
\label{introEquation:secondFluctuationDissipation}. |
| 1664 |
|
\end{equation} |
| 1665 |
+ |
In effect, it acts as a constraint on the possible ways in which one |
| 1666 |
+ |
can model the random force and friction kernel. |
| 1667 |
|
|
| 1668 |
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1669 |
|
Theoretically, the friction kernel can be determined using velocity |
| 1839 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1840 |
|
)T_{ij} |
| 1841 |
|
\] |
| 1842 |
< |
where \delta _{ij} is Kronecker delta function. Inverting matrix |
| 1842 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 1843 |
|
$B$, we obtain |
| 1844 |
|
|
| 1845 |
|
\[ |
| 1883 |
|
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1884 |
|
we can easily find out that the translational resistance tensor is |
| 1885 |
|
origin independent, while the rotational resistance tensor and |
| 1886 |
< |
translation-rotation coupling resistance tensor do depend on the |
| 1886 |
> |
translation-rotation coupling resistance tensor depend on the |
| 1887 |
|
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1888 |
|
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1889 |
|
obtain the resistance tensor at $P$ by |
| 1923 |
|
\] |
| 1924 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1925 |
|
joining center of resistance $R$ and origin $O$. |
| 1709 |
– |
|
| 1710 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
