This is a problem all about binary choice systems!
Consider an N-spin 1-dimensional paramagnet in a magnetic field B where N>=2. Let σ(i) in {+1,-1} indicate the i-th spin, with σ(i)=+1 meaning the i-th spin is spin-up and σ(i)=-1 meaning the i-th spin is spin-down.
U(σ(i))=-μB∑_i σ(i)=-μB(N_(uarr)-N_(darr)),
where the number of spins that are spin-up (spin-down) is N_(uarr)(N_(darr)).
(a) If we have isolated the paramagnet with fixed energy U when we are in magnetic field B, what is the probability (in terms of U and N) that the first two neighboring spins are anti-aligned (e.g. one spin-up and one spin-down). What would this probability be if we were to or different.
Or different. [Hint: Taking your answer from the first bit and simply setting B to O will not work for the second [Hint: Taking your answer from the first bit and simply setting B to O will not w part. If you are stuck, try to think about what is different about the B=0 case.]
In our paramagnet, let y-=(N_(uarr))/(N) be the fraction of spins that are spin-up. In terms of N and y, the energy and entropy are given by
U=-NμB(2y-1);,S=-Nk_B(ylny+(1-y)ln(1-y)).
A graph of S-vs- y is shown which may be convenient for this problem. If our paramagnet is at temperature T then it turns out that the quantity y is given by
y=(1)/(2)(1+tanh((μB)/(k_BT))).
We can make heat engines/heat pumps/refrigerators with this system! ^(1)
We will assume that y>(1)/(2) throughout this problem (more than half our spins are aligned with the field which means U<0 and T>0). The paramagnet is initially at magnetic field B_(0) and temperature T_(0). Consider the following two-step thermodynamic process:
Reversibly and adiabatically decreasing the magnetic field strength from B_(0) to (B_(0))/(2) by thermally isolating the paramagnet.
Reversibly and "isomagnetically" (i.e. keeping B constant at (B_(0))/(2)) adding energy to the system.
(b) [re For the reversible, adiabatic process (1), determine the change in entropy ΔS and the temperature T_(1) at the end of step (1). Then, for the reversible, isomagnetic process (2), qualitatively determine whether the entropy increases or decreases and whether the temperature increases or decreases.
Polymers (like rubber) are very long molecules made up of a number of individual segments called monomers. We can think of the monomers as individual links in a long chain (the polymer). Consider a polymer model where each monomer can take one of two orientations: horizontal (which we will draw as '-') or vertical (which we will draw as '|'). This turns the polymer into another example of a binary choice system, which we will now use our statistical methods to analyze.
Let N_(h) and N_(v) the the number of monomers that are oriented horizontally and vertically, relet each vertical monomer contribute length zero. We also put the polymer in some field so that a vertically-aligned monomer contributes energy -εlon and a horizontally-aligned monomer contributes energy 0. This means L=N_(h)l and U=-N_(v)εlon for a polymer microstate containing N_(h) horizontal monomers and N_(v) vertical monomers. Consider an N=7-link polymer of fixed total energy U=-3εlon. A marker (drawn as a red dot *) is placed between the third and fourth monomers. Let x be the position of the marker. An example microstate is show
Consider an N-spin 1-dimensional paramagnet in a magnetic field B where N>=2. {+1,-1} indicate the-th spin, with = +1 meaning the i-th spin is spin-up and , - he f-th spin is spin-down
U(,)=-B
=B(NN),
()
st with fixed energy U when we are in magnetic field B, turn off the
In o
NB(2y1);
S=N(ylny+(1y)ln(1y)
y=+tanh
We will assume that y > 1/2 throughout this problem (more than half our spins are aligned with the field which means U < 0 and 7 > 0). The paramagnet is initially at magnetic field Bg and temperature Tg- Consider the following two-step thermodynamic process:
Reversibly and adiabatically decreasing the magnetic field strength from B0 to f0/2 b
2.Reversh systen
t B0/2) ddi
] For the mevereibe, eefebaffc process (1), determine the change in entropy S qualitatively determin and whether the teenperature
Polymers (like rubber) ai mber of individual segments called nonovere. We can think of the monomers as individual lints in a long chain (the polymer). Consider a polymer model where each monomer can take one of two orientations: horizontal (which we will draw as ) or vertical (which we will draw as 1). This turns the polymer into another
Let N and N the the mumber of moeoeners that are oriented horizontally and wertically. re
ner eontribute length zero. We also put the polymer in soone feld so that verticall ontribute nd a horizontally-alig ontributes ergy o.This r neans L - Ne and U - - -N.e for a polymer microstate containing N, horizon tal monomers and N._vertical monomers. Consider an N = 7-link polymer of fixod total energy
Y =7,
N = 4,
N. =3,
L) m 46,
U(+) = 3c,
cakly interacting subsystems of our isolated polymer
s to the right of
(e) What is the mmlipl from the 's start of the naximum entrop? As N
n themarker is a distano ss the macrostate with thi ger pcsitioe?
