Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
choices. Each choice will give rise to the same physics but,
usually, very different equations in the intervening steps. Here
we make the following gauge choice, A~ = (By, 0, 0) (3.62) Lets
place our particle in a cubic box with sides of length L. Solutio
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
other one. This is the true ground state of the system. In
contrast, the point which is locally, but not globally, a minimum
corresponds to a metastable state of the system. In order for
the system to leave this state, it must first fluctuate up and over
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
energy will give us the second virial coefficient B2(T). We can be
somewhat more precise about what it means to be at low
density. The exact form of the integral R d 3 rf(r) depends on
the potential, but for both the LennardJones potential (2.22)
and the
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
energy of the particle is E = E 0 + ~ 2k 2 z 2m where E 0 is the
energy of the harmonic oscillator, E 0 = n + 1 2 ~c n Z
These discrete energy levels are known as Landau levels. They
are highly degenerate. To see this, note that kx is quantized in
unit of
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
which case they form a black hole. 101 The factor in
brackets in (3.57) is an interesting mix of fundamental constants
associated to quantum theory, relativity and gravity. In this
combination, they define a mass scale called the Planck mass,
M2 pl = ~c G
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
expanding sin x/x, writing resulting sum as a product of roots,
and then equating the x 2 term). After all this work, we finally
have the result we want. The low temperature expansion of
fn(z) is an expansion in 1/ log z = 1/, fn(z) = (log z) n (n + 1) 1
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
background magnetic field B~ , the kinetic energy picks up an
extra term Espin = BBs (3.58) where B = e~/2mc is the Bohr
magneton. (It has nothing to do with the chemical potential. Do
not confuse them!) Since spin up and spin down electrons have
differ
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
Chapter 4 of Pippards book. This motivates a mathematically
concise statement of the second law due to Caratheodory.
Second Law `a la Caratheodory: Adiabatic surfaces exist. Or,
more prosaically: if you want to be able to return, there are
places you cann
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
with dQ = 0 which is equivalent to dS = 0. In fact, for the
simplest systems such as the ideal gas which require only two
variables p and V to specify the state, we do not need the
second law to infer to the existence of an adiabatic surface. In
that case
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
compression Isothermal expansion expansion Figure 28: The
Carnot cycle in cartoon. 4.3.1 The Carnot Cycle Kelvins
statement of the second law is that we cant extract heat from a
hot reservoir and turn it entirely into work. Yet, at first glance,
this appe
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
shown p p p 1 2 V1 V2 V Figure 26: to the right. Start in state
(p1, V1), take the lower path to (p2, V2) and then the upper
path back to (p1, V1). The energy is unchanged because H dE =
0. But the total work done is nonzero: H pdV 6= 0. By the first
law
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
prejudices spread by schools around the world. To define area
and volume with precision, their definitions must have two
properties: the values must be additive, i.e., for finite and
infinite sets of objects, the total area and volume must be the
sum of t
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
Christoph Schiller June 1990October 2016 free pdf file
available at www.motionmountain.net 3 how to describe
motion kinematics 77 space, are used in weather forecasting.
The phase space diagram is also called state space diagram. It
plays an essential rol
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
there are complications that will not be discussed here. In
addition, the problems mentioned in the definition of length of
fractals also reappear for area if the surface to be measured is
not flat. A typical example is the area of the human lung:
dependi
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
equatorial plane 39.4, 42.8 and 43.8, respectively leads to
zerotwist structures. In these ideal configurations, the rope will
neither rotate in one nor in the other direction under vertical
strain ( Jakob Bohr). Two sides of a hollow cube with side
leng
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
(6) therefore allows us to determine the depth of a well, given
the time a Challenge 130 s stone takes to reach its bottom. The
equation also gives the speed with which one hits the ground
after jumping from a tree, namely = 2 . (7) A height of 3 m
yields
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
f(r) (2.27) To understand what this is telling us, we need to
compute R d 3 rf(r). Firstly lets look at two trivial examples:
Repulsion: Suppose that U(r) > 0 for all separations r with U(r =
) = 0. Then f = e U 1 < 0 and the pressure increases, as
wed ex
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
of variational methods that we use in Lagrangian mechanics.
We write the variation of the free energy as F = Z d d r 2am m
+ 4bm3 m + 2cm m = Z d d r 2am + 4bm3 2c2m m
where to go from the first line to the second we have integrated
by parts. (We need to
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
we need only use the fact that the energy is a function of
temperature only: E = E(T). The isothermal parts of the Carnot
cycle are trivially the same and we reproduce (4.7) and (4.8).
The adiabatic parts cannot be solved exactly without knowledge
of E(T)
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
permanent dipole moments, p1 and p2, have a potential energy
which scales as p1p2/r3 . Neutral atoms dont have permanent
dipoles, but they can acquire a temporary dipole due to
quantum fluctuations. Suppose that the first atom has an
instantaneous dipole
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
~r1 ~r2. (You might worry that the limits of integration change
in the integral over ~r, but the integral over f(r) only picks up
contributions from atomic size distances and this is only actually
a problem close to the boundaries of the system where it i
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
A small amount of heat is written dQ and a small amount of
work is written dW. The first law of thermodynamics in
infinitesimal form is then dE = dQ + dW (4.4) Although we
introduced the first law as applying to all types of work, from
now on the discussi
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
Ivor = Carnot if Ivor is reversible. This means that for all
reversible engines operating between TH and TC have the same
efficiency. Or, said another way, the ratio QH/QC is the same for
all reversible engines. Moreover, this efficiency must be a
functio
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
Evaluating in the limit of vanishing magnetic field, we find that
the magnetization is inversely proportional to the temperature,
(B = 0) = N2 B kBT This 1/T behaviour is known as Curies law.
(Pierre, not Marie). 103 The above results hold in the high
tem
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
reference system to define temperature. The standard choice is
the ideal gas equation of state (which, as we have seen, is a
good approximation to real gases at low densities), T = pV N kB
110 4.2 The First Law The first law is simply the statement of
th
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
continue to expand. As it does so, both the pressure and
temperature will decrease. Isothermal contraction CD at
constant temperature TC. We now start to restore the system to
its original state. We do work on the system by compressing the
115 TH CT CT T
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
positive axis. If R contains no zero of Z(z, V, T) for all z R then
is a an analytic function of z for all z R. In particular, all
derivatives of are continuous. In other words, there can be no
phase transitions in the region R even in the V limit. The
l
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
the state space of the system. dE is exact; dW and dQ are not.
112 4.3 The Second Law Once or twice I have been
provoked and have asked company how many of them could
describe the Second Law of Thermodynamics, the law of
entropy. The response was cold: i
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
LandauGinzburg Theory Landaus theory of phase transition
focusses only on the average quantity, the order parameter. It
ignores the fluctuations of the system, assuming that they are
negligible. Here we sketch a generalisation which attempts to
account f
Hyderabad Institute of Arts, Science & Technology, Hyderabad
ECON 103

Fall 2016
, the chemical potential, and X = N, the particle number.
Another very common example is y = M, the magnetization,
and X = H, the applied magnetic field. For our purposes, it wont
matter what variables y and X are: just that they exist. We need
to choose