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Hello, I need help doing this assignment.
Here's the link for the relevant pictures and det
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Hello, I need help doing this assignment.
Here's the link for the relevant pictures and details: http://umdberg.pbworks.com/w/page/50457651/Polymers%20and%20entropy
One way of understanding entropy is to say that since a system with random motion moves through all microstates with equal probability, if most microstates correspond to a particular macrostate, that's the state that the system will move towards.This is basically the second law of thermodynamics.
Perhaps the simplest example of this that lets us work out the math of this is a set of objects that take two states. The microstate is the specification of the state of each object; the macrostate is the specification of how many of each state is present. A simple physical example of this is the flipping of a set of fair coins that can come up either heads (H) or tails (T).
Part 1: Coin tosses
Consider a set of N coins. If we toss each coin, each has two ways of coming down, H or T. Since the first coin can come down 2 ways, and the second coin can come down 2 ways, etc., the number of different ways (microstates) that the N coins can come down is 2 x 2 x ... (N times) = 2^{N}. While this is interesting, this is not the number we want. Rather, we want to know if we choose a particular macrostate (a given number of heads and tail) how many microstates correspond to that macrostate. That is, how many different ways could you get a string of coin flips that came up with that particular number of heads and tails?
A. For 4 coins, count explicitly how many different ways there are to get each of the following macrostates:
4H, 0T
3H, 1T
2H, 2T
1H, 3T
0H, 4T.
B. Now suppose that you had N coins. Create an mathematical expression that would allow you to calculate how many different ways you could create a string of flips that would give M heads and (NM) tails. Consider a set of N coins that have M heads and NM tails showing. How many different ways could you choose a sequence of the coins? (Hint: You could choose the first one in N different ways. You could then choose one of the remaining N1 in N1 different ways; etc.) Since we don't care what order we get the heads or tails in, you have to divided by the number of ways of permuting the heads and the tails. This result is called _{N}C_{M}, the number of ways of choosing M objects out of a set of N without respect to order. (What you are to do for this part of the problem is justify the expression for the number of combinations in terms of the relevant factorials by describing the choosing and arranging process.)
C. Use a spreadsheet to draw bar graphs of the number of microstates of coin flipping to get M heads out of N flips, _{N}C_{M,} as a function of M for N = 10, 20, and 30. (You probably want to use the FACT(N) function which gives the value of N factorial (N!). An example of such a bar graph for is shown at the right. We see that 3H, 3T is the most likely result and 6H or 6T only have one way of getting them.
Once you have these bar graphs, fill out the following table that shows: the fraction of the total that correspond to the 5050 macrostate; the halfwidth of the peak (about how far down you have to go on each side of the middle for the number to fall to half  just eyeball it); and the ratio of the half width compared to N. The values for 6 are given in the table below.


N  Total number of different  Fraction of microstates  Half width  Half width / N 
6  2^{6} = 64  20/64 = 0.31  ~3  3/6 = 0.5 
10 




20 




30 




Does the peak get wider or narrower as the number of total flips goes up?
Part 2: Polymer folding
Consider a polymer like DNA. One very simple model of such a polymer is to assume that the polymer forms a onedimensional chain consisting of N >> 1 links, each having a particular length a. Each of the links in the chain may be freely oriented to the right or left, with no energy difference between these two orientations. The likelihood that each link in the chain orients to the left or the right is precisely 50/50, just like a coin toss.
Suppose that n_{R} is the number of elements oriented to the right and n_{L} is the number of elements oriented to the left, such that N = n_{L} + n_{R}.
A. Refer to the figure at the right, in which one possible conformation of polymer links is illustrated (but where the individual links have been distributed vertically for clarity). For the example drawn, what are the values of N, n_{R}, and n_{L}? For the example drawn, what is the value of Lin terms of the link length a?
B. Write down a general expression for the endtoend extension of such a chain,L,in terms of the parameters n_{R},n_{L}, and a. Of course, for the particular configuration drawn, your general expression must reduce to L = 6a. 

C. Write down an expression for the number of arrangements W as a function of the total number of links N and the number of links pointing left or right, n_{L} and n_{R}. Explain your reasoning. (Hint: Refer back to your analysis in part 1.)
D.What would the state of minimum and maximum entropy of this polymer look like?
E. Can you use your results from parts AD of this problem (and the second law of thermodynamics) to predict what you think the natural state of such a polymer would most likely look like?
Hello, I need help doing this assignment.
Here's the link for the relevant pictures and details: http://umdberg.pbworks.com/w/page/50457651/Polymers%20and
%20entropy
One way of understanding entropy is to say that since a system with random motion moves through all microstates
with equal probability, if most microstates correspond to a particular macrostate, that's the state that the system will
move towards.This is basically the second law of thermodynamics.
Perhaps the simplest example of this that lets us work out the math of this is a set of objects that take two states. The
microstate is the specification of the state of each object; the macrostate is the specification of how many of each
state is present. A simple physical example of this is the flipping of a set of fair coins that can come up either heads
(H) or tails (T).
Part 1: Coin tosses
Consider a set of N coins. If we toss each coin, each has two ways of coming down, H or T. Since the first coin can
come down 2 ways, and the second coin can come down 2 ways, etc., the number of different ways (microstates)
that the N coins can come down is 2 x 2 x ... (N times) = 2N. While this is interesting, this is not the number we want.
Rather, we want to know if we choose a particular macrostate (a given number of heads and tail) how many
microstates correspond to that macrostate. That is, how many different ways could you get a string of coin flips that
came up with that particular number of heads and tails?
A. For 4 coins, count explicitly how many different ways there are to get each of the following macrostates: 4H, 0T 3H, 1T 2H, 2T 1H, 3T 0H, 4T. B. Now suppose that you had N coins. Create an mathematical expression that would allow you to calculate how
many different ways you could create a string of flips that would give M heads and (NM) tails. Consider a set of N
coins that have M heads and NM tails showing. How many different ways could you choose a sequence of the
coins? (Hint: You could choose the first one in N different ways. You could then choose one of the remaining N1 in N1 different ways; etc.) Since we don't care what order we get the heads or tails in, you have to divided by the number
of ways of permuting the heads and the tails. This result is called NCM, the number of ways of choosing M objects out
of a set of N without respect to order. (What you are to do for this part of the problem is justify the expression for the
number of combinations in terms of the relevant factorials by describing the choosing and arranging process.) C. Use a spreadsheet to draw bar graphs of the number of microstates of coin flipping to get M heads out of N flips, NCM, as a function of
and 30. (You probably want to use the FACT(N) function which gives the value of N factorial (N!). An example of such a bar graph for N=
right. We see that 3H, 3T is the most likely result and 6H or 6T only have one way of getting them. Once you have these bar graphs, fill out the following table that shows: the fraction of the total that correspond to the 5050 macrostate;
the peak (about how far down you have to go on each side of the middle for the number to fall to half  just eyeball it); and the ratio of th
compared to N. The values for 6 are given in the table below. N Total number of different
ways the result can come out Fraction of microstates
that correspond to 5050
(the most common macrostate) Half width
(eyeball it) H 6 26 = 64 20/64 = 0.31 ~3 3 10 20 30 Does the peak get wider or narrower as the number of total flips goes up?
Part 2: Polymer folding
Consider a polymer like DNA. One very simple model of such a polymer is to assume that the polymer forms a onedimensional chain consisting of N >> 1 links, each having a particular length a. Each of the links in the chain may be
freely oriented to the right or left, with no energy difference between these two orientations. The likelihood that each
link in the chain orients to the left or the right is precisely 50/50, just like a coin toss.
Suppose that nR is the number of elements oriented to the right and nL is the number of elements oriented to the left,
such that N = nL + nR. A. Refer to the figure at the right, in which one possible conformation of polymer links is illustrated (but where the individual links have
distributed vertically for clarity). For the example drawn, what are the values of N, nR, and nL? For the example drawn, what is the valu
of the link length a? B. Write down a general expression for the endtoend extension of such a chain,L,in terms of the parameters nR,nL, and a. Of course, f
particular configuration drawn, your general expression must reduce to L = 6a. C. Write down an expression for the number of arrangements W as a function of the total number of links N and the
number of links pointing left or right, nL and nR. Explain your reasoning. (Hint: Refer back to your analysis in part 1.)
D.What would the state of minimum and maximum entropy of this polymer look like?
E. Can you use your results from parts AD of this problem (and the second law of thermodynamics) to predict what
you think the natural state of such a polymer would most likely look like?
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