Boundary Conditions
Monte Carlo and Molecular Dynamics simulations aim to provide
information about the properties of a macroscopic sample (by simulating
atoms and molecules and then using the theory of statistical mechanics to
bridge from microscopic to macroscopic information)
The number of atoms that can be conveniently handled in presentday computers ranges from a few hundred to a few million.
Most simulations probe the structural and thermodynamic properties of a
system of a few thousand particles. For such small systems the choice of
boundary conditions can strongly affect the measured properties.
For free boundaries, the fraction of all
molecules that are at the surface is
proportional to N-1/3. For example, in a
cubic crystal of 1000 atoms, 49% of
the atoms are at the surface. For a
crystal of 1,000,000 atoms, only 6% of
them are at the surface.
What do you want to simulate?
Boundary Conditions
For most studies, you don’t want to study a nano-sized droplet of water!
Instead, you would like to study a “piece” of a large region of liquid water.
Why? Because the properties of water (and other molecules) at an
interface are very different from in the bulk
For example, water's surface is acidic. The hydronium ion prefers to sit at
the water surface rather than deep inside the liquid. Whereas H2O
molecules typically form four hydrogen bonds to their neighbours, H3O+ can
only form three. The three hydrogens can bind to other water molecules,
but the oxygen atom, where most of the positive charge resides, can no
longer act as a good 'acceptor' for hydrogen bonds.
So, hydronium acts somewhat like an amphiphile - a molecule with a watersoluble part and a hydrophobic part, like a soap molecule. The ions gather
at the air-water interface with the hydrogen atoms pointing downwards to
make hydrogen bonds and the oxygen pointing up out of the liquid. In 2005
Saykally and his colleague Poul Petersen used a spectroscopic technique to
get indirect evidence for this surface excess of acidic hydronium ions.
Boundary Conditions
For most studies, you don’t want to study a nano-sized droplet of water!
Instead, you would like to study a “piece” of a large region of liquid water.
Why? Because the properties of water (and other molecules) at an
interface are very different from in the bulk
Boundary Conditions
Snapshots (side view) of the
solution–air interface of 1.2 M
aqueous sodium halides and
density profiles (number densities)
of water oxygen atoms and ions
plotted vs. distance from the center
of the slabs in the direction normal
to the interface, normalized by the
bulk water density. From top to
bottom the systems are NaF, NaBr,
and NaI. The colors of the density
profiles correspond to the coloring
of the atoms in the snapshots (blue
for water and green for Na+ in all of
the plots, black for F, orange for Br,
and magenta for I). The
water density is scaled differently
from those of the ions so that it can
be easily displayed on the same
plots.
Phys. Chem. Chem. Phys., 2008, 10, 4778 - 4784
Boundary Conditions
Say you want to study ion solvation
WITHOUT interface effects. But you can
only afford to simulation a few thousand
atoms. What should you do?
Use periodic boundary conditions and
the minimum image convention
Finite size effects: phase transitions
Ehrenfest classification scheme: group phase transitions based on the
degree of non-analyticity involved.
Under this scheme, phase transitions are labeled by the lowest
derivative of the free energy that is discontinuous at the transition. Firstorder phase transitions exhibit a discontinuity in the first derivative of
the free energy with a thermodynamic variable.
The various solid/liquid/gas transitions are classified as first-order
transitions because they involve a discontinuous change in density (which
is the first derivative of the free energy with respect to chemical potential.)
Second-order phase transitions are continuous in the first derivative but
exhibit discontinuity in a second derivative of the free energy. These
include the ferromagnetic phase transition in materials such as iron,
where the magnetization, which is the first derivative of the free energy
with the applied magnetic field strength, increases continuously from
zero as the temperature is lowered below the Curie temperature. The
magnetic susceptibility, the second derivative of the free energy with
the field, changes discontinuously.

Finite size effects: phase transitions
But there is a problem: with a finite number of variables all state functions
are analytic (no discontinuities).
You have to do simulations with
increasingly larger systems and
look for where the quantity of
interest begins to diverge.
J. Phys. A: Math. Gen. 35 (2002) 33–42
U 
CV   
T V
Molecular Dynamics Simulations
Molecular Dynamics simulations are in many respects similar to real
experiments.
When we perform a real experiment, we proceed as follows:
1)Prepare sample of the material for study
2)Connect the sample to a measuring device (e.g. thermometer)
3)Measure property of interest over a certain time interval (the longer
we measure or the more times we repeat the measurement, the more
accurate our measurement becomes due to statistical noise)
For MD we follow the same approach. 1) first we prepare a sample by
selecting a system of N atoms 2) we then solve Newton’s equations of
motion for this system until the system properties no longer change with
time (we equilibrate the system); 3) after equilibration we perform the
actual measurement.
In fact, some of the most common mistakes that can be made when
performing an MD simulation are very similar to the mistakes that can
be made in real experiments (e.g. sample not prepared correctly,
measurement too short, system undergoes an irreversible change during
the experiment, or we do not measure what we think)
Molecular Dynamics Simulations
If we solve Newton’s equations, we (in principle) conserve energy. But this
is not compatible with sampling configurations from the Boltzmann
distribution. So how do we make sure we are visiting configurations in
accordance with their Boltzmann weight?
N
2 1
2
1
T
m
v

i i
2
3k B N i1
One way to do this is to use the Anderson thermostat. In this method, the
system is coupled to a heat bath that imposes the desired temperature.
The coupling to a heat bath is represented by stochastic impulsive forces
that act occasionally on
randomly selected particles. These stochastic
collisions with the heat bath can be considered as Monte Carlo moves that
transport the system from one constant-energy shell to another.
Between stochastic collisions, the system evolves at constant energy
according to Newtonian mechanics. The stochastic collisions ensure that
all accessible constant-energy shells are visited according to their
Boltzmann weight.
Molecular Dynamics Simulations
We take the distribution between time intervals for two successive
stochastic collisions to be P(t;v) which is of Poisson form
P(t;v)  v exp vt
Where P(t;v)dt is the probability that the next collision will take place in
the time interval (t, t+dt)
A constant temperature simulation consists of
1)Start with an initial set of positions and momenta and integrate the
equations of motion for a time Δt.

2)A number of particles are selected to undergo a collision with the heat
bath. The probability that a particle is selected in a time step of length Δt is
vΔt.
3)If particle i has been selected to undergo a collision, its new velocity is
drawn from a Maxwell-Boltzmann distribution corresponding to the
desired temperature T. All other particles are unaffected by this collision.
Statistical mechanics of non-equilibrium
systems: linear response theory and the
fluctuation-dissipation theorem
1968 Nobel Prize in chemistry:
Lars Onsager
Up to this point, we have used statistical mechanics to examine equilibrium
properties (the Boltzmann factor was derived assuming equilibrium).
What about non-equilibrium properties?
Example: relaxation rate by which a system reaches equilibrium from a
prepared non-equilibrium state (e.g. begin with all reactions, no products,
and allow a chemical reaction to proceed)
Our discussion will be limited to systems close to equilibrium. In this
regime, the non-equilibrium behavior of macroscopic systems is described
by linear response theory.
The cornerstone of linear response theory is the fluctuation-dissipation
theorem. This theorem connects relaxation rates to the correlations
between fluctuations that occur spontaneously at different times in
equilibrium systems.
Systems close to equilibrium
What do we mean by “close” to equilibrium?
We mean that the deviations from equilibrium are linearly related to the
perturbations that remove the system from equilibrium.
For example, consider an aqueous electrolyte solution. At equilibrium,
there is no net flow of charge; the average current <j> is zero.
At some time t = t1, an electric field of strength E is applied, and the
charged ions begin to flow. At time t = t2, the field is then turned off. Let
j(t) denote the observed current as a function of time.
The non-equilibrium behavior, j(t) ≠ 0, is linear if j(t) is proportional to E.
Another example: we know that the presence of a gradient in chemical
potential is associated with mass flow. Thus, for small enough gradients,
the current or mass flow should be proportional to the gradient.
The proportionality must break down when the gradients become large.
Then quadratic and perhaps higher order terms are needed.
Systems close to equilibrium
A final example: to derive the selection rules for Raman spectroscopy we
assume that the applied electric field (namely shining light on a molecule)
inducing a dipole in the molecule in a manner such that the induced dipole
is proportional (linearly related) to the applied field strength. The
proportionality constant is called the polarizability of the molecule.
Onsager’s regression hypothesis
and time correlation functions
When left undisturbed, a non-equilibrium system will relax to its
thermodynamic equilibrium state. When not far from equilibrium, the
relaxation will be governed by a principle first proposed in 1930 by Lars
Onsager in his regression hypothesis:
The relaxation of macroscopic non-equilibrium disturbances is governed
by the same laws as the regression of spontaneous microscopic
fluctuations in an equilibrium system.
This hypothesis is an important consequence of a profound theorem in
mechanics: the fluctuation-dissipation theorem.
To explore this hypothesis, we need to talk about correlations of
spontaneous fluctuations, which is done using time correlation functions.
A(t)  A(t)  A
Define A(t) as the instantaneous deviation
(fluctuation) of A from its equilibrium average
Onsager’s regression hypothesis
and time correlation functions
Of course,
A  0
However, if one looks at equilibrium correlations between fluctuations at
different times, one can learn something about the system

The correlation between A(t) and an instantaneous or spontaneous
fluctuation at time zero is C(t)  A(0) A(t)  A(0)A(t) 
Look at the limiting
 behavior: at short times,
C(0)  A
while at 
long times A(t) is uncorrelated to
2
A(0) , thus
C(t)  A(0) A(t)  0 as t  

This decay of correlations with increasing time is the “regression of
spontaneous 
fluctuations” referred to inOnsager’s hypothesis

A
2
Onsager’s regression hypothesis
and time correlation functions
By invoking the ergodic hypothesis, we can view the equilibrium average as
a time average, and write
T
“sliding window”
1
A(0) A(t)  lim
T  T
 d A( ) A(  t)
0
We can now state Onsager’s regression hypothesis. Imagine at time t=0, we
prepare a system in a non-equilibrium state and allow it to relax to
equilibrium. In the linear regime, the relaxation obeys

A (t) C(t)

A (0) C(0)
where
A (t)  A (t)  A
C(t)  A(0) A(t)
In a system close to equilibrium, we cannot distinguish between spontaneous
fluctuations and deviations from equilibrium that are externally prepared.

Since we cannot distinguish,
the relaxation of A(0) A(t) should indeed
coincide with the decay
to equilibrium of A(t)
Application: (self-) diffusion
Fick’s 1st Law: the diffusive flux goes from regions of high concentration to
regions of low concentration, with a magnitude that is proportional to the
concentration gradient
J  D 
J is the flux: the amount of substance that will flow
through a small area in a small time interval
D is the diffusion constant
ϕ is the concentration

Fick’s 2nd Law: use the 1st law and mass conservation to get

 D  2
t
Solutions to this partial differential equation describe
how concentration gradients relax in terms of a
parameter, the diffusion constant.

Application: (self-) diffusion
The constant D is called a transport coefficient. To learn how this transport
coefficient is related to the microscopic dynamics, let us consider the
correlation function
C(r,t)   (r,t)  (0,0)
Where
(r,t) is the instantaneous density at position r and time t
N
(r,t)  r  r j (t)
According to Onsager’s regression hypothesis,
C(r,t) also obeys Fick’s 2nd law

C(r,t)
2

D

C(r,t)

t
Notice that
P(r,t)

(r,t) (0,0)
j1

is proportional to
= the conditional probability distribution that a particle
is at r at time t given that the particle was at the origin
at time zero.
Application: (self-) diffusion
Why is this? First, recall that C(r,t) 
As a result, we have

(0,0) (r,t)  (0,0)(r,t)  
2
P(r,t)
2
 D  P(r,t)
t
2
Now, consider R (t)  r1 (t)  r1 (0)
which is the mean squared
displacement of a tagged solute molecule in a time t.
2

Clearly,
R2 (t) 

Hence,
d
R 2 (t) 
dt

P(r,t)
 dr r t 
2
2
dr
r
P(r,t)

2
2
dr
r
D
P(r,t)  6  dr DP(r,t)

Application: (self-) diffusion
P(r,t)
or,
is normalized for any value of time, giving
R 2 (t)  6Dt
d
R 2 (t)  6D
dt
This formula was first derived by Einstein
Notice that ballistic motion looks like


We can write
r1 (t)  r1 (0) 
t
 d v( )
0
Hence R 2 (t)  r (t)  r1 (0)
1
2

where v(t) is the particle velocity
t
t

0
0
 d  d v( )  v( )
Take
 d/dt on both sides

R(t)  t
t
d
R 2 (t)  2  d v(t)  v( )
dt
0
Application: (self-) diffusion
t
2  d v(t)  v( )  2 v(t)  r(t)  r(0)  2 v(0)  r(0)  r(t)
0
0
t
t
t
0
0
 2  d v(0)  v( )  2  d v(0)  v( )  2  d v(0)  v( )
t
d
giving
R 2 (t)  2  d v(0)  v( )
dt
0
Since the left-hand side goes to 6D in the limit of large times, we have

D

1
3
 d v(0)  v( )
0
This equation relates a transport coefficient to an integral of an autocorrelation
function. Such relationships are known as Green-Kubo formulas.