Homework # 7 Numerical Methods in Chemical Engineering Problem 1. (100 points) Nanoparticle self-assembly Interactions between nanoparticles can lead to self-assembled arrays that resemble the structure of crystals. In this problem, you will use Metropolis Monte Carlo simulations to describe the distribution of a set of non-charged nanoparticles. We will approximate the interaction between two particles using a Lenard-Jones potential: 𝑈(𝑅𝑎𝑏) = 4𝜀 [( 𝜎 𝑅𝑎𝑏 ) 12 − ( 𝜎 𝑅𝑎𝑏 ) 6 ] 𝑤𝑖𝑡ℎ 𝑅𝑎𝑏= |𝑟𝑎 − 𝑟𝑏| where ra and rb are the positions of particle a and b respectively, and the values for σ and ε are provided below. You may assume that the sample at equilibrium follows a Boltzmann distribution with probability of finding the assemble in a state q, 𝑃(𝑞) ∝ exp(−𝑈𝑡𝑜𝑡(𝒒)/(𝑘𝑏𝑇)), where 𝑈𝑡𝑜𝑡(𝑞) = ∑ ∑ 𝑈(𝑅𝑎𝑏) 𝑁 𝑏=𝑎+1 𝑁 𝑎=1 and q is the state vector containing the position of all the nanoparticles in the system. For simplicity, we will consider the self-assembly of a group of N spherical nanoparticles of radius R in two dimensions, and the state vector has the following structure, q = [x1 y1 x2 y2 …. xN yN] T . i. (25 points) Write a function that computes the value of Utot for a given state vector q. ii. function z = my_fun(x) iii. z = zeros(size(x,1),3); % allocate output iv. z(:,1) = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ... v. 4*x(:,2).^2 - 3*x(:,2); vi. z(:,2) = x(:,1).*x(:,2) + cos(3*x(:,2)./(2+x(:,1))); vii. z(:,3) = tanh(x(:,1) + x(:,2)); where P(Σ) is a given probability distribution (e.g., the Boltzmann distribution P(Σ) = Z exp[-βE(Σ)]) andN Σ  is the number of configurations Σ  (e.g., the number of configurations generated with a particular arrangement of [x1 y1 x2 y2 …. xN yN spins viii. (40 points) Write a function that performs a Monte Carlo routine with an N_MC Monte Carlo steps, a constant temperature T, and an N number of nanoparticles in the initial state q_0. For each MC step, you should attempt moves of one nanoparticle at a time for all the particles. The function should output a matrix containing the values of the state vector at each MC step in different columns, and a vector containing Utot for each MC step. The Metropolis Monte Carlo algorithm can be described as follows: function mypi = approxpi(n) % Input: n = number of points to generate % Default is n = 1e6 % Larger values of n should perform better if nargin < 1 n = 1e6; end % Generate uniformly distributed points in % [0, 1] x [0, 1] xy = rand(n, 2); % Compute distance from (0.5, 0.5) r = sqrt((xy(:,1)-0.5).^2+(xy(:,2)-0.5).^2); % Count fraction of points within 1/2 unit of (0.5, 0.5) frac = sum(r <= 0.5) / n; % Since square has side 1, circle has radius (1/2) % and should have area of pi*(1/2)^2 % frac is approximately pi/4 so pi is approximately 4*frac mypi = 4*frac; Step (1): Pick a configuration Σn Step (2): Pick a trial configuration Σt (usually a configuration similar to Σn) and compute the pr

1. Homework # 7 Numerical Methods in Chemical Engineering
Problem 1. (100 points) Nanoparticle self-assembly Interactions
between nanoparticles can lead to self-assembled arrays that
resemble the structure of crystals. In this problem, you will use
Metropolis Monte Carlo simulations to describe the distribution
of a set of non-charged nanoparticles. We will approximate the
interaction between two particles using a Lenard-Jones
potential: �(���) = 4� [( � ��� ) 12 − ( � ��� ) 6 ]
���ℎ ���= |�� − ��| where ra and rb are the positions of
particle a and b respectively, and the values for σ and ε are
provided below. You may assume that the sample at equilibrium
follows a Boltzmann distribution with probability of finding the
assemble in a state q, �(�) ∝ exp(−����(�)/(���)), where
����(�) = ∑ ∑ �(���) � �=�+1 � �=1 and q is the state
vector containing the position of all the nanoparticles in the
system. For simplicity, we will consider the self-assembly of a
group of N spherical nanoparticles of radius R in two
dimensions, and the state vector has the following structure, q =
[x1 y1 x2 y2 …. xN yN] T .
i. (25 points) Write a function that computes the value of Utot
for a given state vector q.
ii. function z = my_fun(x)
iii. z = zeros(size(x,1),3); % allocate output
iv. z(:,1) = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ...
v. 4*x(:,2).^2 - 3*x(:,2);
vi. z(:,2) = x(:,1).*x(:,2) + cos(3*x(:,2)./(2+x(:,1)));
vii. z(:,3) = tanh(x(:,1) + x(:,2));
where P(Σ) is a given probability distribution (e.g., the
Boltzmann distribution P(Σ) = Z exp[-βE(Σ)]) andN Σ is the
number of configurations Σ (e.g., the number of configurations
generated with a particular arrangement of [x1 y1 x2 y2 …. xN
yN spins
viii. (40 points) Write a function that performs a Monte Carlo

2. routine with an N_MC Monte Carlo steps, a constant
temperature T, and an N number of nanoparticles in the initial
state q_0. For each MC step, you should attempt moves of one
nanoparticle at a time for all the particles. The function should
output a matrix containing the values of the state vector at each
MC step in different columns, and a vector containing Utot for
each MC step.
The Metropolis Monte Carlo algorithm can be described as
follows:
function mypi = approxpi(n)
% Input: n = number of points to generate
% Default is n = 1e6
% Larger values of n should perform better
if nargin < 1
n = 1e6;
end
% Generate uniformly distributed points in
% [0, 1] x [0, 1]
xy = rand(n, 2);
% Compute distance from (0.5, 0.5)
r = sqrt((xy(:,1)-0.5).^2+(xy(:,2)-0.5).^2);
% Count fraction of points within 1/2 unit of (0.5, 0.5)
frac = sum(r <= 0.5) / n;
% Since square has side 1, circle has radius (1/2)
% and should have area of pi*(1/2)^2
% frac is approximately pi/4 so pi is approximately 4*frac
mypi = 4*frac;
Step (1): Pick a configuration Σn
Step (2): Pick a trial configuration Σt (usually a configuration
similar to Σn) and compute the probability ratio R=p(Σt)/p(Σn) .
Pick a random number p with value between 0 and 1. Make
Σn+1=Σt if p≤R . Otherwise, make Σn+1=Σt.
Step (3): Go to (2) replacing Σn by Σn+1 .
Step (3) is repeated N times, where N is a sufficiently large

3. number. Note that, according to step (2), the probability of
accepting a trial configuration Σt by making Σn+1=Σt from a
configuration Σn is
ix. (35 points) Use your function to generate histograms
containing the distribution of nanoparticle distances, Rab, for a
set of 16 nanoparticles. Assume that the particles are initially
arranged in a square grid of 4x4 nanoparticles, with center-to-
center distance between the particles equal to 2R.
Generate graphs for T = 100 K, 500 K, 1000 K. Use a minimum
of 100,000 MC steps. Discuss your results. Physical parameters:
R = 5 nm, σ = 5 nm, ε/kb = 5000 K.
专业 matlab
紐約時間12/5 2pm
我要matlab的code
matlab文件
我功課是要交code不是word
我要的是長這樣的東西
[email protected]

4. 12.2pm New York time
I need the code for matlab
Matlab files
My homework is to hand in code, not word
This is my old homework（Modestino_HW5(2)）
Homework # 7
Numerical Methods in Chemical Engineering
Submission due: Monday, December 6th, 2019 at 12:25 pm
through NYU Classes
Problem 1. (100 points) Nanoparticle self-assembly
Interactions between nanoparticles can lead to self-assembled
arrays that resemble the structure of
crystals. In this problem, you will use Metropolis Monte Carlo

5. simulations to describe the distribution of
a set of non-charged nanoparticles. We will approximate the
interaction between two particles using a
Lenard-Jones potential:
�(���) = 4� [(
�
���
)
12
− (
�
���
)
6
] ���ℎ ���= |�� − ��|
where ra and rb are the positions of particle a and b
respectively, and the values for σ and ε are provided
below. You may assume that the sample at equilibrium follows
a Boltzmann distribution with probability
of finding the assemble in a state q, �(�) ∝
exp(−����(�)/(���)), where
����(�) = ∑ ∑ �(���)

6. �
�=�+1
�
�=1
and q is the state vector containing the position of all the
nanoparticles in the system. For simplicity, we
will consider the self-assembly of a group of N spherical
nanoparticles of radius R in two dimensions, and
the state vector has the following structure, q = [x1 y1 x2 y2 ….
xN yN]
T.
i. (25 points) Write a function that computes the value of Utot
for a given state vector q.
ii. (40 points) Write a function that performs a Monte Carlo
routine with an N_MC Monte Carlo
steps, a constant temperature T, and an N number of
nanoparticles in the initial state q_0. For
each MC step, you should attempt moves of one nanoparticle at
a time for all the particles. The
function should output a matrix containing the values of the
state vector at each MC step in

7. different columns, and a vector containing Utot for each MC
step.
iii. (35 points) Use your function to generate histograms
containing the distribution of
nanoparticle distances, Rab, for a set of 16 nanoparticles.
Assume that the particles are initially
arranged in a square grid of 4x4 nanoparticles, with center-to-
center distance between the
particles equal to 2R. Generate graphs for T = 100 K, 500 K,
1000 K. Use a minimum of 100,000
MC steps. Discuss your results.
Physical parameters: R = 5 nm, σ = 5 nm, ε/kb = 5000 K.