Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

1
Nano Mechanics and MaterialsTheory, Multiscale
Methods and Applications

• by
• Wing Kam Liu, Eduard G. Karpov, Harold S. Park

2
4. Methods of Thermodynamics and Statistical
Mechanics
3
Definition and Features the Thermodynamic Method
Thermodynamics is a macroscopic, phenomenological
theory of heat.

• Basic features of the thermodynamic method
• Multi-particle physical systems is described
  by means of a small number of macroscopically
  measurable parameters, the thermodynamic
  parameters V, P, T, S (volume, pressure,
  temperature, entropy), and others.
• Note macroscopic objects contain 10231024
  atoms (Avogadros number 6x1023mol1).
• The connections between thermodynamic
  parameters are found from the general laws of
  thermodynamics.
• The laws of thermodynamics are regarded as
  experimental facts.
• Therefore, thermodynamics is a phenomenological
  theory.
• Thermodynamics is in fact a theory of
  equilibrium states, i.e. the states with
  time-independent (relaxed) V, P, T and S. Term
  dynamics is understood only in the sense how
  one thermodynamic parameters varies with a change
  of another parameter in two successive
  equilibrium states of the system.

4
Classification of Thermodynamic Parameters

• Internal and external parameters
• External parameters can be prescribed by
  means of external influences on the system by
  specifying external boundaries and fields.
• Internal parameters are determined by the
  state of the system itself for given values of
  the external parameters.
• Note the same parameter may appear as external
  in one system, and as internal in another system.
  
• Intensive and extensive parameters
• Intensive parameters are independent of the
  number of particles in the system, and they serve
  as general characteristics of the thermal atomic
  motion (temperature, chemical potential).
• Extensive parameters are proportional to the
  total mass or the number of particles in the
  system (internal energy, entropy).
• Note this classification is invariant with
  respect to the choice of a system.

5
Internal and External Parameters Examples
A same parameter may appear both as external and
internal in various systems
System A
System B
External parameter V
External parameter P, P
Mg/A Internal parameter P
Internal parameter V,
V Ah
6
State Vector and State Equation
Application of the thermodynamic method implies
that the system if found in the state of
thermodynamic equilibrium, denoted X, which is
defined by time-invariant state parameters, such
as volume, temperature and pressure The
parameters (V,T,P) are macroscopically
measurable. One or two of them may be replaced
by non-measurable parameters, such internal
energy or entropy. Note that only the mean
quantity of a state parameter A is
time-invariant, see the plot. A mathematical
relationship that involve a complete set of
measurable parameters (V,T,P) is called the
thermodynamic state equation Here, ? is the
vector of system parameters

7
Analysis of the State Equation System Parameters
The knowledge of an equation of state allows
evaluation of a group of the microscopic system
parameters, such as the compressibility,
expansion and pressure coefficients 1)
Isothermal compressibility coefficient 2)
Isobaric thermal expansion coefficient 3)
Isochoric pressure coefficient
8
Examples of the State Equation
Standard forms of the state equations are readily
available for ideal gases, real (Van der Waals)
gases, homogeneous liquid and homogeneous
isotropic solids. Ideal gas Van der Waals gas
(pair wise interaction V of gas molecules is
taken into account)
d effective diameter of the molecules W
pairwise potential
Homogeneous isotropic liquid or solid
9
The Postulate on Existence of Temperature
The temperature is introduced as a parameter,
which 1) serves as an intrinsic
characteristic of any equilibrium system (similar
to V and P) 2) determines thermodynamic
equilibrium between two systems in thermal
contact Thus, it is postulated that If two
adiabatically isolated systems in equilibrium are
brought into thermal contact with each other,
their states of equilibrium will not be altered
and the total system will be in equilibrium, only
if initial systems have the same
temperature. (Also known as the zeroth law of
thermodynamics) Any state of thermodynamic
equilibrium of an arbitrary system in entirely
determined by the set of external parameters and
temperature. Consequently, All internal
parameters of an equilibrium system are functions
of the external parameters and temperature.
10
The First Law of Thermodynamics
The first law of thermodynamics is essentially a
form of the energy conservation law, written in
relation to thermodynamic systems An
amount of heat absorbed by the system is equal to
the summary change of its internal energy and the
work done by the system over external bodies.
Note The internal energy U is defined solely
by the state of system, while the external
thermal energy Q and the mechanical work W may
depend both on the internal state of the system
and other factors.
11
The Second Law of Thermodynamics
The second law of thermodynamics specifies the
direction of thermodynamic processes. The
simplest form of the second law is given by
Clausius postulate Heat cannot flow
spontaneously from a colder to a hotter
system. This is equivalent to the following
(Kelvins postulate) It is impossible to devise
an engine (a perpetuum mobile of the second
kind) which, working in a cycle, would produce no
other effect than the transformation of heat
extracted from a reservoir completely into work.
12
Entropy
The most important consequence of the second law
of thermodynamics is that it asserts the
existence of a new function of state, namely the
entropy, S. The concept of entropy plays a
crucial role in statistical mechanics. For
quasistatic processes, entropy is an extensive
state function, which is defined by the
relation Based on the first law of
thermodynamics we obtain the fundamental
differential equation Alternative form of the
second law of thermodynamics All spontaneous
processes in adiabatically isolated systems, dQ
0, occur at a constant or growing entropy
13
The Third Law of Thermodynamics
The second law and the original definition of
entropy does not specify the absolute value of
entropy, whose value is provided up to a constant
value. The third law claims that All
thermodynamic processes at T 0 occur without a
change of the entropy. This allows
establishing an absolute scale for measuring the
entropy so that Also,
14
Thermodynamic Potentials
The method of thermodynamic potentials is a
powerful tool of thermodynamics. Thermodynamic
potentials are functions that uniquely describe
the state of the system. The relevant
thermodynamic parameters are found as derivatives
of the potentials. The simplest thermodynamic
potential is the internal energy, given by the
first law, Other thermodynamic potentials for
systems with constant number of particles
free energy Gibbs potential
enthalpy Exercise 3-1 (can be done in class)
derive the differentials of the free energy,
Gibbs potential and enthalpy, by utilizing the
definitions of these potentials and the first law
of thermodynamics.
15
4.1 Basic Results of the Thermodynamic Method
Differentiation of the thermodynamic potentials
gives the thermodynamic parameters
Also, system parameters can be evaluated using
their relationship with the above state
parameters (see reading assignment 1, Section
4.1.5)
16
Hamiltonian Mechanics
Equations of motion
Hamiltonian of the system and the generalized
momentum
Cartesian coordinates
17
Micromodel vs. Macromodel
In classical (deterministic) mechanics, the state
of the system is completely described by the
phase vector Q. Such a state is called a
microscopic state, and the corresponding model of
matter is called a micromodel. The microstate
is completely defined by specifying values of all
canonical variables, the components of the phase
vector, This approach is not tractable in
modeling macroscopic objects.
Thermodynamics provides a macromodel the state
of the a system is determined by a very limited
number of thermodynamic parameters, which are
sufficient for macroscopic characterization of
the system. The prescription of these
parameters, measured in a macroscopic experiment,
determines the macroscopic state the system. A
key point is that a single macroscopic state of
the system corresponds to a great number of
different microscopic states.
18
4.2 Statistics of Multiparticle Systems in
Thermodynamic Equilibrium
The macroscopic thermodynamic parameters, X
(V,P,T,), are macroscopically observable
quantities that are, in principle, functions of
the canonical variables, i.e.

• However, the specification of all the
  macroparameters X does not determine a unique
  microstate,
• Consequently, on the basis of macroscopic
  measurements, one can make only statistical
  statements about the values of the microscopic
  variables.

19
Statistical Description of Mechanical Systems
Statistical description of mechanical systems is
utilized for multi-particle problems, where
individual solutions for all the constitutive
atoms are not affordable, or necessary.
Statistical description can be used to reproduce
averaged macroscopic parameters and properties of
the system. Comparison of objectives of the
deterministic and statistical approaches
Deterministic particle dynamics Statistical mechanics
Provides the phase vector, as a function of time Q(t), based on the vector of initial conditions Q(0) Provides the time-dependent probability density to observe the phase vector Q, w(Q,t), based on the initial value w(Q,0)
20
Statistical Description of Mechanical Systems
From the contemporary point of view, statistical
mechanics can be regarded as a hierarchical
multiscale method, which eliminates the atomistic
degrees of freedom, while establishing a
deterministic mapping from the atomic to
macroscale variables, and a probabilistic mapping
from the macroscale to the atomic variables
Microstates Macrostates
Xk
(p,q)k
deterministic conformity
probabilistic conformity
21
Distribution Function
Though the specification of a macrostate Xi
cannot determine the microstate (p,q)i
(p1,p2,,ps q1,q2,,qs)i, a probability density
w of all the microstates can be found, or
abbreviated The probability of finding the
system in a given phase volume G The
normalization condition
22
Statistical Ensemble
Within the statistical description, the motion of
one single system with given initial conditions
is not considered thus, p(t), q(t) are not
sought. Instead, the motion of a whole set of
phase points, representing the collection of
possible states of the given system. Such a set
of phase points is called a phase space
ensemble. If each point in the phase space is
considered as a random quantity with a
particular probability ascribed to every possible
state (i.e. a probability density w(p,q,t) is
introduced in the phase space), the relevant
phase space ensemble is called a statistical
ensemble.
G volume in the phase space, occupied by the
statistical ensemble.
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Statistical Averaging
Statistical average (expectation) of an arbitrary
physical quantity F(p,q), is given most generally
by the ensemble average, The root-mean-square
fluctuation (standard deviation) The curve
representing the real motion (the experimental
curve) will mostly proceed within the band of
width 2?(F) For some standard equilibrium
systems, thermodynamic parameters can be
obtained, using a single phase space integral.
This approach is discussed below.
24
Ergodic Hypothesis and the Time Average
Evaluation of the ensemble average (previous
slide) requires the knowledge of the distribution
function w for a system of interest.
Alternatively, the statistical average can be
obtained by utilizing the ergodic hypothesis in
the form, Here, the right-hand side is the time
average (in practice, time t is chosen finite,
though as large as possible) This approach
requires F as a function of the generalized
coordinates. Some examples
More examples are given in Ref. 1.
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Law of Motion of a Statistical Ensemble
A statistical ensemble is described by the
probability density in phase space, w(p,q,t). It
is important to know how to find w(p,q,t) at an
arbitrary time t, when the initial function
w(p,q,0) at the time t 0 is given. In other
words, the equation of motion satisfied by the
function w(p,q,t) is needed.
The motion of of an ensemble in phase space may
be considered as the motion of a phase space
fluid in analogy to the motion of an ordinary
fluid in a 3D space. Liouvilles theorem claims
that Due to Liouvilles theorem, the following
equation of motion holds
26
Equilibrium Statistical Ensemble Ergodic
Hypothesis
For a system in a state of thermodynamic
equilibrium the probability density in phase
space must not depend explicitly on
time, Thus, the equation of motion for an
equilibrium statistical ensemble reads
A direct solution of this equation is not
tractable. Therefore, the ergodic hypothesis (in
a more general form) is utilized the probability
density in phase space at equilibrium depends
only on the total energy Notes the
Hamiltonian gives the total energy required the
Hamiltonian may depend on the values of external
parameters a (a1, a2,), besides the phase
vector X. This distribution function satisfies
the equilibrium equation of motion, because
Exercise Check the above equality.
27
Canonical Ensembles

• After adoption of the ergodic hypothesis, it then
  remains to determine the actual form of the
  function f(H). This function depends on the type
  of the thermodynamic system under consideration,
  i.e. on the character of the interaction between
  the system and the external bodies.
• We will consider canonical ensembles of two types
  of systems
• 1) Adiabatically isolated systems that have no
  contact with the surroundings and have a
  specified energy E.
• The corresponding statistical ensemble is
  referred to as the microcanonical ensemble, and
  the distribution function microcanonical
  distribution.
• 2) Closed isothermal systems that are in contact
  and thermal equilibrium with an external
  thermostat of a given temperature T.
• The corresponding statistical ensemble is
  referred to as the canonical ensemble, and the
  distribution function Gibbs canonical
  distribution.
• Both systems do not exchange particles with the
  environment.

28
Microcanonical Distribution
For an adiabatically isolated system with
constant external parameters, a, the total energy
cannot vary. Therefore, only such microstates X
can occur, for which This implies (d Diracs
delta function) and finally where O is the
normalization factor,
E, a (P,T,V,)
Within the microcanonical ensemble, all the
energetically allowed microstates have an equal
probability to occur.
29
Microcanonical Distribution Integral Over States
The normalization factor O is given by where G
is the integral over states, or phase
integral G(E,a) represents the
normalized phase volume, enclosed within the
hypersurface of given energy determined by the
equation H(X,a) E. Phase integral G is a
dimensionless quantity. Thus the normalization
factor O shows the rate at which the phase volume
varies due to a change of total energy at fixed
external parameters.
30
Microcanonical Distribution Integral Over States
The integral over states is a major calculation
characteristic of the microcanonical ensemble.
The knowledge of G allows computing thermodynamic
parameters of the closed adiabatic system
(These are the major results in terms of
practical calculations over microcanonical
ensembles.)
31
Microcanonical Ensemble Illustrative Examples
We will consider one-dimensional illustrative
examples of computing the phase integral, entropy
and temperature for microcanonical
ensembles Spring-mass harmonic
oscillator Pendulum (non-harmonic
oscillator) We will use the Hamiltonian
equations of motion to get the phase space
trajectory, and then evaluate the phase integral.
32
Harmonic Oscillator Hamiltonian
Hamiltonian general form Kinetic energy
Potential energy The
total Hamiltonian
Potential energy is a quadratic function of the
coordinate (displacement form the equilibrium
position)
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Harmonic Oscillator Equations of Motion and
Solution
Hamiltonian and equations motion Initial
conditions (m, m/s)
34
Harmonic Oscillator Total Energy
Total energy
35
Harmonic Oscillator Phase Integral
Phase integral
For the harmonic oscillator, phase volume grows
linearly with the increase of total energy.
36
Harmonic Oscillator Entropy and Temperature
Entropy
Temperature
We perturb the initial conditions (on 0.1 or
less) and compute new values The temperature is
computed then, as
37
Pendulum Total Energy
Total energy
38
Pendulum Phase Integral
Phase integral
For the pendulum, phase volume grows NON-linearly
with the increase of total energy at large
amplitudes.
39
Pendulum Entropy and Temperature
Entropy
Temperature
We perturb the initial conditions (on 0.1 or
less) and compute new values The temperature is
computed then, as
40
Summary of the Statistical Method Microcanonical
Distribution

1. Analyze the physical model justify applicability
   of the microcanonical distribution.
2. Model individual particles and boundaries.
3. Model interaction between particles and between
   particles and boundaries.
4. Set up initial conditions and solve for the
   deterministic trajectories (MD).
5. Compute two values of the total energy and the
   phase integral for the original and perturbed
   initial conditions.
6. Using the method of thermodynamic parameters,
   compute entropy, temperature and other
   thermodynamic parameters. If possible compare the
   obtained value of temperature with benchmark
   values.
7. If required, accomplish an extended analysis of
   macroscopic properties (e.g. functions T(E),
   S(E), S(T), etc.) by repeating the steps 4-7.

41
Canonical Distribution Preliminary Issues
One important preliminary issue related to the
use of Gibbs canonical distribution is the
additivity of the Hamiltonian of a mechanical
system. Structure of the Hamiltonian of an
atomic system Here, kinetic energy and the
one-body potential are additive, i.e. they can be
expanded into the components, each corresponding
to one particle in the system Two-body and
higher order potentials are non-additive
(function Q2 does not exist),
42
Canonical Distribution Preliminary Issues

• Thus, if the inter-particle interaction is
  negligible,
• the system is described by an additive
  Hamiltonian,
• Here, H is the total Hamiltonian, and hi is the
  one-particle Hamiltonian.
• For the statistical description, it is
  sufficient that this requirement holds for the
  averaged quantities only.
• The multi-body components, Wgt1, cannot be
  completely excluded from the physical
  consideration, as they are responsible for heat
  transfer and establishing the thermodynamic
  equilibrium between constitutive parts of the
  total system.
• A micromodel with small averaged contributions
  to the total energy due to particle-particle
  interactions is called the ideal gas.
• Example particles in a circular cavity.
  Statistically averaged value W2 is small

43
Canonical Distribution
Suppose that system under investigation S1 is in
thermal contact and thermal equilibrium with a
much larger system S2 that serve as the
thermostat, or heat bath at the temperature
T. From the microscopic point of view, both S1
and S2 are mechanical systems whose states are
described by the phase vectors (sets of canonical
variables X1 and X2). The entire system S1S2 is
adiabatically isolated, and therefore the
microcanonical distribution is applicable to
S1S2, Assume N1 and N2 are number of
particles in S1 and S2 respectively. Provided
that N1 ltlt N2, the Gibbs canonical distribution
applies to S1
Thermostat T N1 S1 N2
S2
44
Canonical Distribution Partition Function
The normalization factor Z for the canonical
distribution called the integral over states or
partition function is computed as
Before the normalization, this integral
represents the statistically averaged phase
volume occupied by the canonical ensemble. The
total energy for the canonical ensemble is not
fixed, and, in principle, it may occur arbitrary
in the range from ? to ? (for the infinitely
large thermostat, N2 ? ?).
45
Partition Function and Thermodynamic Properties
The partition function Z is the major
computational characteristic of the canonical
ensemble. The knowledge of Z allows computing
thermodynamic parameters of the closed isothermal
system (a ? V, external parameter)
Free energy (relates to mechanical
work) Entropy (variety of microstates) Pressure
Internal energy
These are the major results in terms of practical
calculations over canonical ensembles. Class
exercise check the last three above formulas
with the the method of thermodynamic potentials,
using the first formula for the free energy.
46
Free Energy and Isothermal Processes
Free energy, also Helmholtz potential is of
importance for the description of isothermal
processes. It is defined as the difference
between internal energy and the product of
temperature and entropy. Since free energy is
a thermodynamic potential, the function
F(T,V,N,) guarantees the full knowledge of all
thermodynamic quantities. Physical content of
free energy the change of the free energy dF of
a system at constant temperature, represents the
work accomplished by, or over, the system.
Indeed, Isothermal processes tend to a
minimum of free energy, i.e. due to the
definition, simultaneously to a minimum of
internal energy and maximum of entropy.
47
Canonical vs. Microcanonical Factorization of
the Partition Function
In terms of practical calculations, there exists
one major difference between the canonical and
microcanonical distributions For additive
Hamiltonians, the canonical distribution
factorizes, Note that this property does
not hold for the microcanonical distribution,
zi is the one-particle partition function
48
Factorization of the Partition Function
Computational Issues
Factorization of the canonical distribution is
crucial in terms of practical calculations over
real-life systems. In computing the partition
function Z, this property reduces the calculation
of a 6N-dimensional phase integral to a product
of N 6-dimensional integrals Here, zi is
the one-particle partition function, and hi is
the one-particle Hamiltonian, In case that
the system is comprised of identical particles,
calculation of Z requires evaluation of a single
6-dimensional integral z
The factorized canonical distribution can be very
effective computationally.
49
Analytical Example Non-Interactive Ideal Gas
The canonical distribution allows an exact
solution for the non-interactive ideal
gas Analytical results for this system are
useful, because they provide acceptable first
guess assessments for a wide class of systems.
One-particle partition function
Partition function (total system)
50
Non-Interactive Ideal Gas Thermodynamic
Parameters
Partition function Free energy (recall the
method of thermodynamic potentials) Entropy P
ressure Total internal energy (differs form the
earlier MD definition)
51
Numerical Example Interactive Gas
Repulsive interaction between the particles and
the wall is described by the wall function, a
one-body potential that depends on ri distance
between the particle i and the chambers
center) Interaction between particles is
modeled with the two-body Lennard-Jones potential
(rij distance between particles i and j) The
Hamiltonian
One particle is initially at rest. This
illustrates the concept of heat exchange between
the smaller subsystem, for which the canonical
distribution holds, and the external thermostat.
52
Interactive Gas Equations of Motion and Solution
The total potential Equations of
motion Parameters Initial conditions (nm,
nm/s)
53
Interactive Gas Temperature
Time averaged kinetic energy vs. time (five
particles)
Time averaged kinetic energy of particles is
approaching the value which corresponds to
temperature
Information on temperature allows computing the
partition function (integral over states), using
canonical distribution. Subsequently, the
partition function, computed at various
temperatures, can provide all the remaining
thermodynamic parameters. For sufficiently long
simulations, the value of temperature does not
depend on the choice of a subsystem (particle).
54
Interactive Gas Partition Function and Free
Energy
Hamiltonian (can be viewed as additive in the
statistical sense, due to smallness of the time
averaged pair-wise interaction) One-particle
partition function (value at given T and V
pR2) Partition function (total system) Free
energy
55
Interactive Gas Thermodynamic Parameters
In order to compute the thermodynamic quantities,
it necessary to evaluate 2 values of the
partition function 1) Z for the
initially computed temperature T, 2) for
a perturbed temperature T ?T (?T/T lt
0.1). Note the simulation needs to be run once
only (not two times). Entropy Pressure
Internal energy Ideal gas benchmark Other
parameters can be computed using the method of
thermodynamic potentials
56
Phase Integral, Free Energy and Entropy vs.
Temperature
Assume that we observe the same isothermal system
at various temperatures of thermostat. The
following trends are available
Partition function Free
energy Entropy
Partition function, and therefore the phase
volume occupied by this canonical ensemble, grows
exponentially vs. temperature. Free energy
decreases linearly the work done by the system
does not depend on temperature. Entropy decays
vs. temperature. Physical implication (according
to the second law) temperature cannot grow
spontaneously in an isothermal system, once
thermal equilibrium with the thermostat is
established.
57
Specifics of Calculations for Liquids and Solids
In liquids, the energy due to pair-wise
interaction between particles is close to the
kinetic energy (per particle). However,
interaction, wij, between separate constitutive
parts (subdomains) i and j is still weak, if
compared with the total kinetic energy of the
smaller domain. Indeed, the kinetic energy
depends on the subdomain volume, while wij
depends on the surface area. Therefore, the
Hamiltonian can be expanded into hi
Hamiltonians of the sufficiently large
subdomains. Partition function (z partition
functions for N identical subdomains, n
number of subdomain particles) For reasonably
small subdomains, numerical evaluation of the
liquids partition function can be effective. A
similar approach is also applicable to solids.
Example
Note In case of large wij, the micro-canonical
distribution should be utilized.
58
Summary of the Statistical Method Canonical
Distribution

1. Analyze the physical model justify applicability
   of the Gibbs canonical distribution.
2. Model individual particles and boundaries.
3. Model interaction between particles and between
   particles and boundaries.
4. Set up initial conditions and solve for the
   deterministic trajectories (MD).
5. Compute the averaged kinetic energy and
   temperature, or assume T given.
6. Based on the canonical distribution, compute two
   values of the partition function for the
   original and perturbed temperatures.
7. Compute the free energy and other thermodynamic
   parameters, using the method of thermodynamic
   potentials. If possible compare the obtained
   value of internal energy with a benchmark value.
8. If required, accomplish an extended analysis of
   macroscopic properties (e.g. dependences P(T),
   P(N), S(T), etc.) by repeating the steps 4-7.

59
4.3 Numerical Heat Bath Techniques

• Berendsen thermostat
• Adelman-Doll thermostatting GLE
• Phonon heat bath
• Time-history kernel and transform techniques
• Random force and lattice normal modes

60
Finite Temperatures
A heat bath technique is required to represent a
peripheral region at finite temperatures Berendse
n thermostat for a standard Langevin equation
Berendsen et al., JCP 81(8), 1984
61
Finite Temperatures
Adelman-Dolls thermostatting GLE for gas-solid
interface
Adelman, Doll et al., JCP 64(6), 1976
Almost exactly what we seek, however, the update
is needed
gas/solid interface -gt solid/solid interface
62
Phonon Heat Bath
Phonon heat bath represents energy exchange due
to correlated motion of lattice atoms along an
imaginary atomic/continuum (solid-solid)
interface Phonon heat bath is a configurational
method
atom next to the interface
Karpov, Liu, preprint.
63
Time History Kernel (THK)
The time history kernel shows the dependence of
dynamics in two adjacent cells. Any time history
kernel is related to the response function.
64
Elimination of Degrees of Freedom T 0
Equations for the DoF ngt0 are no longer
required. We have taken them into account
implicitly.
65
Bridging Scale at T 0 Impedance Boundary
Conditions
The MD domain is too large to solve, so that we
eliminate the MD degrees of freedom outside the
localized domain of interest. Collective atomic
behavior of in the bulk material is represented
by an impedance force applied at the formal
MD/continuum interface
MD degrees of freedom outside the localized
domain are solved implicitly
FE Reduced MD Impedance BC
MD
FE

Due to atomistic nature of the model, the
structural impedance is evaluated computed at
the molecular scale.
66
Dynamic Response Function 1D Illustration
Assume first neighbor interaction only
Displacements
Velocities
Illustration Transfer of a unit pulse due to
collision (movie)
67
Discrete Fourier Transform (DFT)
Discrete functional sequences
DFT of infinite sequences
p wavenumber, a real value between ? and ?
DFT of periodic sequences
Here, p integer value between N/2 and N/2
Discrete convolution
68
Numerical Laplace Transform Inversion
Most numerical algorithms for the Laplace
transform inversion utilize series decompositions
of the sought originals f(t) in terms of
functions whose Laplace transform is tabulated.
The expansion coefficients are found numerically
from F(s). Examples

• Weeks algorithm
  (J
  Assoc Comp Machinery 13, 1966, p.419)
• Sin-series expansion (J Assoc Comp Machinery
  23, 1976, p.89)
• For an odd function f gives

69
Elimination of Degrees of Freedom at T gt 0
Domain of interest
Heat bath
-2 -1 0 1
2
Compare with
Equations for the DoF ngt0 are no longer
required. We have taken them into account
implicitly - mechanical response is
described by the THK - thermal contact is
described by the randon force R
70
Random Force Term
( g the lattice response function )
Is there a more effective way to compute R(t) ?
71
Gibbs Distribution and Lattice Hamiltonian
Gibbs distribution
Lattice Hamiltonian
Normal modes decomposition
72
Random Force Term via Lattice Normal Modes
Random force
Distribution of the normal amplitudes and phases