Different physical situation encountered in nature are described by three types of statistics-Maxwell-Boltzmann Statistics, Bose-Einstein Statistics and Fermi-Dirac Statistics
Information theory and statistical mechanicsChandanShah35
Focused on basic terminology used in Statistical Mechanics, Relation ship between Information Theory and Statistical Mechanics and few terms related to quantum mechanics
Different physical situation encountered in nature are described by three types of statistics-Maxwell-Boltzmann Statistics, Bose-Einstein Statistics and Fermi-Dirac Statistics
Information theory and statistical mechanicsChandanShah35
Focused on basic terminology used in Statistical Mechanics, Relation ship between Information Theory and Statistical Mechanics and few terms related to quantum mechanics
Struggling with your statistical physics exam? Live Exam Helper offers expert exam help services for all levels. Our experienced tutors and study aids can help you master the concepts and formulas of statistical physics, so you can ace your exam with confidence. Visit our website https://www.liveexamhelper.com/physics-exam-help.html to learn more about our services!
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
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Radioactivity as a phenomenon is often misunderstood: if one says ‘Radioactive’, most people will think about disastrous electrical plants, dangerous bombs and other forms of life-threatening details. In my native Germany, members of the Green party have been campaigning for a decade to put an end to nuclear energy. Only few think of the useful aspects of this unique actuality, although radiotherapy is most promising of tools in the fight against cancer, and radioactive dating allows us to identify the age of any historical item. But even fewer people see radioactivity as the natural process that it actually is: A spontaneous mechanism, in which one nucleus decays into another. As an aspiring Physicist and Engineer, Radioactivity is one my favourite topics in the realm of science. I am fascinated at how we are able to predict exactly how many Nuclei will decay in a certain amount of time, but not say for certain which Nuclei exactly will do so.
Homework # 7 Numerical Methods in Chemical Engineering .docxpooleavelina
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 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. (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.
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.
CYPRESS COLLEGE/THEATER 107
Diversity in American Theater/Fall 2019
ZOOT SUIT
WORKSHEET
1. Who is the “main character” of Zoot Suit and what is his goal in life?
2. What are the major obstacles facing the main character? [Both the internal
obstacles – issues relating to his or her personal character – and external obstacles –
real concerns that exist in the real world.]
3. Who are the main character’s allies? Who are his foes?
4. What does the Pachuco Zoot Suit character represent? Would you describe him as
real, imaginary or spiritual – or something else? Make a choice and support it with
exampl
JEE Mathematics/ Lakshmikanta Satapathy/ Theory of Probability part 9 which explains Random variables , its probability distribution, Mean of a random variable and Variance of a random variable
Struggling with your statistical physics exam? Live Exam Helper offers expert exam help services for all levels. Our experienced tutors and study aids can help you master the concepts and formulas of statistical physics, so you can ace your exam with confidence. Visit our website https://www.liveexamhelper.com/physics-exam-help.html to learn more about our services!
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Nuclear Decay - A Mathematical PerspectiveErik Faust
Radioactivity as a phenomenon is often misunderstood: if one says ‘Radioactive’, most people will think about disastrous electrical plants, dangerous bombs and other forms of life-threatening details. In my native Germany, members of the Green party have been campaigning for a decade to put an end to nuclear energy. Only few think of the useful aspects of this unique actuality, although radiotherapy is most promising of tools in the fight against cancer, and radioactive dating allows us to identify the age of any historical item. But even fewer people see radioactivity as the natural process that it actually is: A spontaneous mechanism, in which one nucleus decays into another. As an aspiring Physicist and Engineer, Radioactivity is one my favourite topics in the realm of science. I am fascinated at how we are able to predict exactly how many Nuclei will decay in a certain amount of time, but not say for certain which Nuclei exactly will do so.
Homework # 7 Numerical Methods in Chemical Engineering .docxpooleavelina
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 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. (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.
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.
CYPRESS COLLEGE/THEATER 107
Diversity in American Theater/Fall 2019
ZOOT SUIT
WORKSHEET
1. Who is the “main character” of Zoot Suit and what is his goal in life?
2. What are the major obstacles facing the main character? [Both the internal
obstacles – issues relating to his or her personal character – and external obstacles –
real concerns that exist in the real world.]
3. Who are the main character’s allies? Who are his foes?
4. What does the Pachuco Zoot Suit character represent? Would you describe him as
real, imaginary or spiritual – or something else? Make a choice and support it with
exampl
JEE Mathematics/ Lakshmikanta Satapathy/ Theory of Probability part 9 which explains Random variables , its probability distribution, Mean of a random variable and Variance of a random variable
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A baisc ideas of statistical physics.pptx
1. Basic ideas of statistical physics
Dr. J P SINGH
Associate Professor in Physics
PGGC-11, Chandigarh
2. Statistics is a branch of science which deals with
the collection, classification and interpretation
of numerical facts. When statistical concept are
applied to physics then a new branch of science
is called Physics Statistical Physics.
Basic ideas of statistical physics
3. Trial → experiment→ tossing of coin
Event → outcome of experiment
Exhaustive events The total number of
possible outcomes in any trial
For tossing of coin exhaustive events = 2
4. Favourable events number if possible
outcomes (events) in any trial
Number of cases favourable in drawing a king from a
pack of cards is 4.
Mutually exclusive events no two of them
can occur simultaneously.
Either head up or tail up in tossing of coin.
Equally likely events every event is equally
preferred.
Head up or tail up
5. Independent events if occurrence of one
event is independent of other
Tossing of two coin
Probability
The probability of an event =
ways
of
number
Total
occurs
event
the
which
in
cases
of
Number
6. If m is the number of cases in which an event occurs and
n the number of cases in which an event fails, then
Probability of occurrence of the event =
The sum of these two probabilities i.e. the total
probability is always one since the event may either
occur or fail.
Probability of failing of the event =
n
m
m
n
m
n
n
m
n
n
m
m
or
1
7. Tossing of two coins :
The following combinations of Heads up (H) and
Tails up(T) are possible :
2
1
,
2
1
2
1
H
H
2
1
,
2
1
2
1
T
T
4
1
2
1
2
1
2
1
H
H
4
1
2
1
2
1
2
1
T
T
4
1
2
1
2
1
2
1
H
H
4
1
2
1
2
1
2
1
H
H
n
P
P
P
P .
..........
2
1
8. Principle of equal a priori probability
The principle of assuming equal probability for
events which are equally likely is known as the
principle of equal a priori probability.
A priori really means something which exists in
our mind prior to and independently of the
observation we are going to make.
9. Distribution of 4 different Particles in
two Compartments of equal sizes
Particles must go in one of the compartments.
Both the compartments are exactly alike.
The particles are distinguishable. Let the four particles
be called as a, b, c and d.
The total number of particles in two compartments is 4
i.e.
2
1
4
i
i
n
10. The meaningful ways in which these four particles can
be distributed among the two compartments is shown
in table.
11. Macrostate
The arrangement of the particles of a system without
distinguishing them from one another is called
macrostate of the system.
In this example if 4 particles are distributed
in 2 compts, then the possible macrostates (4+1) =5
If n particles are to be distributed in 2 compts.
Then the no. of macrostates is = n+1
12. Microstate
The distinct arrangement of the particles of a system is
called its microstate.
For example, if four distinguishable particles
are distributed in two compartments, then
the no. of possible microstates (16)
If n particles are to be distributed in 2
compartments. The no. of microstates is
= 24
= 2n
=(Compts)particles
13. Thermodynamic probability or frequency
The numbers of microstates in a given macrostate is
called thermodynamics probability or frequency of that
macrostate.
For distribution of 4 particles in 2 identical
compartments
W(4,0) =1
W(3,1) =4
W(2,2) = 6
W(1,3) = 4
W(0,4) =1
14. W depends on the distinguishable or indistinguishable
nature of the particles. For indistinguishable particles,
W=1
Micro-
Comp 1
States
Comp 2
macrostate Frequency
W
probability
(4,0) 1
(3,1) 1
(2,2) 1
(1,3) 1
(0,4) 1
5
1
5
1
5
1
5
1
5
1
15. All the microstates of a system have equal a priori
probability.
Probability of a microstate =
microstate
of
no
Total .
1
Probability of a macrostate =
(no. of microstates in that microstate)
(Probability of one miscrostate)
n
W
W
W
2
1
2
1
16
1
4
= thermodynamic probability× prob. Of one
microstate
n
2
1
2
1
16
1
4
16. Constraints
Restrictions imposed on a system are called constraints.
Example
total no. particles in two compartments = 4
Only 5 macrostates (4.0), (3,1), (2,2),(1,3),(0,4) possible
The macrostates (1,2), (4,2), (0,1), (0,0) etc not possible
17. The macrostates / microstates which are allowed under
given constraints are called accessible states.
Accessible and inaccessible states
The macrostates/ microstates which are not allowed
under given constraints are called inaccessible states
Greater the number of constraints, smaller the number
of accessible microstates.
18. Distribution of n Particles in 2 Compartments
The (n+1) macrostates are
(0, n) (1, n, 1)… (n1, n2)…… (2, n2),….. (n 0),
Out of these macrostates, let us consider a particular
macrostate (n1, n2) such that
n1 + n2 = n
n particles can be arranged among themselves in
nPn = n! ways
19. These arrangements include meaningful as well as
meaningless arrangements.
Total number of ways = (no. of meaningful ways)
(no.of meaningless ways)
n1 particles in comp. 1 can be arranged in
= n1 ! meaningless ways.
n2 particles in comp. 2 can be arranged in
= n2 ! meaningless ways.
n1 particles in comp. 1 and n2 particles in comp. 2 can be
arranged in
= n1 ! n2 ! meaningless ways.
21. Deviation from the state of Maximum
probability
n
r
n
r
n
r
n
r
2
1
.
)!
(
!
!
)
,
(
The probability of the macrostate (r, n r) is
When n particles are distributed in two comp., the
number of macrostates = (n+1)
The macrostate (r, n r) is of maximum probability if r =
n/2, provided n is even.
The prob. of the most probable macrostate
2
,
2
n
n
22. n
n
n
n
P
2
1
!
2
!
2
!
max
Probability of macrostate is slightly deviate from most
probable state by x.(x<<n)
Then new macrostate will be
x
n
x
n
2
,
2
n
x
x
x
x
n
n
P
2
1
!
2
!
2
!
26. Thus we conclude that ass n increases the prob. of a
macrostate decreases more rapidly even for small
deviations w.r.t. the most probable state.
n1 > n2 > n3
(2x / n)
0.2 0.1 0 0.1 0.2
n3
n2
n1
27. Static and Dynamic systems
Static systems: If the particles of a system remain
at rest in a particular microstate, it is called static
system.
Dynamic systems: If the particles of a system
are in motion and can move from one microstate to
another, it is called dynamic system.
28. Equilibrium state of a dynamic system
A dynamic system continuously changes from one
microstate to another. Since all microstates of a
system have equal a priori probability, therefore, the
system should spend same amount of time in each of
the microstate.
If tobs be the time of observation in N microstates
The time spent by the system in a particular macrostate
N
t
t obs
m
Let microstate has frequency
)
,
( 2
1 n
n )
,
( 2
1 n
n
W
29. Time spend in macrostate )
,
( 2
1 n
n
microstate
of
No.
microstate
each
in
spend
time
Average
)
,
( 2
1 n
n
t
)
,
(
)
,
( 2
1
2
1 n
n
W
N
t
n
n
t obs
)
,
(
)
,
( 2
1
2
1 n
n
P
t
n
n
t obs
obs
t
n
n
t
n
n
P
)
,
(
)
,
( 2
1
2
1
That is the fraction of the time spent by a dynamic
system in the macrostate is equal to the probability
of that state
30. Equilibrium state of dynamic system
The macrostate having maximum probability is termed
as most probable state. For a dynamic system consisting
of large number of particles, the probability of deviation
from the most probable state decrease very rapidly.
So majority of time the system stays in the most
probable state. If the system is disturbed, it again tends
to go towards the most probable state because the
probability of staying in the disturbed state is very
small. Thus, the most probable state behaves as the
equilibrium state to which the system returns again and
again.
31. Distribution of n distinguishable
particles in k compartments of unequal
sizes
The thermodynamic prob. for macrostate )
....
,
,
( 3
2
1 k
n
n
n
n
!
!.....
!
!
)
.......
,
(
2
1
2
1
k
k
n
n
n
n
n
n
n
W
k
i
i
n
n
1
!
!
Let the comp. 1 is divided into no. of cells
Particle 1st can be placed in comp.1 in = no. of ways
1
g
1
g
Particle 2nd can be placed in comp.1 in = no. of ways
1
g
1
g
Particle can be placed in comp.1 in = no. of ways
th
n1
32. particles in comp. 1 can be placed in =
1
n 1
1
n
g
particles in comp. 2 can be placed in =
2
n 2
2
n
g
particles in comp. k can be placed in =
k
n k
n
k
g
total no. ways in which n particles in k comparmrnts
can be arranged in the cells in these compartments is
given by
k
n
k
n
n
n
g
g
g
g ......
.
. 3
2
1
3
2
1
i
n
i
k
i
g
1
33. Thermodynamic probability for macrostate is
k
n
k
n
n
k
k g
g
g
n
n
n
n
n
n
n
W )
....(
)
(
)
(
!
!.....
!
!
)
.......
,
( 2
1
1 2
2
1
2
1
k
i i
n
i
n
g
n
W
i
1 !
)
(
!