Part 3 Additional Topics

1
Part 3Additional Topics

• PHYS 4315
• R. S. Rubins, Fall 2009

2
Kinetic Theory

• PHYS 4315
• R. S. Rubins, Fall 2009

3
Kinetic Theory Introduction

• Features of kinetic theory
• Kinetic theory goes beyond the limitations of
  classical thermodynamics by taking into account
  the structures of materials.
• However, there is a price to be paid in
  complexity, since general relationships may be
  obscured.
• Unlike both classical thermo and statistical
  mechanics, kinetic theory may be used to describe
  non-equilibrium situations.
• Kinetic theory has been particularly useful in
  describing the properties of dilute gases.
• It gives a deeper insight into concepts such as
  pressure, internal energy and specific heat, and
  explains transport processes, such as viscosity,
  heat conduction and diffusion.

4
Classical Dilute Gas Assumptions 1

• Molecules are classical particles with
  well-defined positions and momenta.
• A macroscopic volume contains an enormous number
  of molecules.
• At STP (0oC and 1 atm), 1 kmole of a gas occupies
  22.4 m3, which is a density of 3 x 1025
  molecules/m3.
• Molecular separations are much larger than both
  their dimensions and the range of intermolecular
  forces.
• At STP, the molecular separation is roughly 3 x
  10 9 m.
• The Lennard-Jones or 6-12 potential
• V K (d/r)12 (d/r)6,
• where K and d are empirical constants is
  negligible for molecular separations of 3 x 10 9
  m.

5
Classical Dilute Gas Assumptions 2

• The mean free path the average distance a
  molecule moves between collisions is of the order
  10 7 m.
• The only interaction between particles occur
  during collisions so brief that their durations
  may be neglected.
• The container is assumed to have an idealized
  surface.
• The collisions are elastic, so that momentum and
  KE are conserved.
• The molecules are assumed to be distributed
  uniformly in both position and velocity
  direction.
• Molecular chaos exists that is, the velocity of
  a molecule is uncorrelated with its position.

6
Kinetic Theory Pressure

• Macroscopic pressure (classical thermo)
• Balancing force F PA.
• Microscopic pressure (kinetic theory)
• Impulse ?F??t ?ptot,
• where, for a single particle,
• ?p 2mv cos?.
• The piston makes an irregular (Brownian)
  motion about its equilibrium position, because of
  the random collisions of molecules with the
  piston.

Macroscopic pressure
?
?
Microscopic pressure
7
Molecular Flux
A
x
vave?t

• Simplified calculation
• Assume that the molecules move equally in the x,
  y, and z directions i.e. n/6 move in the x
  direction, where n is the number of molecules per
  unit volume.
• In time ?t, a total of Anvave?t/6 molecules will
  strike the wall.
• Thus, the molecular flux F, the number of
  molecules striking unit area of the wall in unit
  time, is F nvave/6.
• An exact calculations, including integrations
  over all directions and speeds gives F nvave/4.

8
Molecular Effusion

• F nvave/4, where (from stat. mech.) vave
  v(8kT/mp).
• For an ideal gas, PV NkT, so that n N/V
  P/kT.
• Thus, F P/v(2pmkT).
• Imagine two containers containing the same ideal
  gas at (P1,T1) and (P2,T2), connected by a
  microscopic hole of diameter D ltlt L, where L is
  the mean free path, which is of the order 107 m
  at STP.
• The number of molecules passing through the hole
  is so small that the pressure and the temperature
  on each side of the hole is unchanged over the
  time of the experiment.
• The condition for equilibrium is the absence of a
  net flux i.e.
• F1 F2 or P1/vT1 P2/vT2.

9
Experimental Problem
Macroscopic hole (D gtgt L) At equilibrium, P1
P2, T1 T2. Microscopic hole (D ltlt L) At
equilibrium, P1/vT1 P2/vT2.
T2 300 K

• Under effusion conditions,
• P1/vT1 P2/vT2 ,
• so that
• P1 P2/ v(T1/T2) ,
• or
• P1 0.07 P2.

vacuum
P2
?P
Hg
Liquid He
He vapor at pressure P1
Liquid He at T1 1.3 K
10
Boltzmanns Transport Equation

• Boltzmanns H theorem
• dH/dt 0, where H(t) ?d3v f(p,t) log
  f(p,t) .
• Since H ? Hmin, the entropy has the form S
  const. H.
• The equilibrium (Maxwell) distribution is given
  by

11
Superfluidity in Liquid Helium

• PHYS 4315
• R. S. Rubins, Fall 2009

12
Boiling Points

• The two helium isotopes have the lowest boiling
  points of all known substances, 3.2 K for 3He and
  4.2 K for 4He.
• Both isotopes would apparently remain liquid down
  to absolute zero to solidify helium would
  require a pressure of about 25 atmospheres.
• Two factors produce the reluctance of helium to
  condense
• i. the low mass of the atoms
• ii. The extremely weak forces between atoms.
• The low atomic mass ensures a high zero-point
  energy, a result that may be deduced from the
  uncertainty principle.

13
Zero-point Energy 1

• The uncertainty in momentum of a particle in a
  cavity is
• ?p ? h/R.
• Its zero-point energy is thus
• E0 ? (?p)2/2m
• or
• E0 ? h2/2mR2.
• The large zero-point energy must be added to the
  potential energy of the liquid to give the
  liquids total energy.

14
Zero-point Energy 2

• Because He atoms are
• so light, the zero-point
• Energy is comparable to
• the PE, the minimum of
• the total energy occurs
• at a relativity high atomic
• volume.
• For other inert gases, the
• zero-point energy is of
• negligible magnitude.

15
Phase Diagrams

• The large zero-point energy of liquid eliminates
  the solid-vapor curve present for a normal
  material.
• The ? line occurs only for 4He, and is associated
  with the ?-point transition to superfluid
  behavior near 2 K.

16
The ? Specific Heat Transition in Liquid 4He

• If liquid helium, which
• liquifies below 4.2 K, is
• cooled by lowering the
• pressure above it,
• bubbles of vapor form
• within the liquid, which
• boils vigorously.
• However, below 2.17 K,
• the ? point, the liquid
• becomes very still, as
• the transition from a
• Normal fluid (He I) to a
• superfluid (He II) occurs.
• In 3He, a transition to a
• superfluid occurs near
• 3 mK.

17
Macroscopic Properties of Superfluid He II 1

• Zero viscosity
• Measurements showing the zero resistance to flow
  of He II were made in 1964.
• This was done by showing that the flow velocity
  through channels of widths between 0.1 µm and 4
  µm were independent of the pressure gradient
  along the channel.

18
Two-fluid Model of He II

• Zero viscosity 2
• Experiments showed an apparent contradiction,
  that He II was both viscous and non-viscous at
  the same time.
• This result was the source of the two-fluid model
  of He II, introduced by Tisza in 1938.
• This is a quantum effect the liquid does not
  consist of two distinct fluids, one normal and
  the other superconducting.
• In Andronikashvilis 1946 experiment, a series of
  equally spaced metal disks, suspended by a
  torsion fibre, were made to oscillate in liquid
  He.
• The results confirmed that He II consists of a
  normal viscous fluid of density ?n and a
  superfluid of density ?s, and allowed the ratios
  ?s/? and ?n/? to be measured as functions of
  temperature, where ? ?n ?s.

19
Andronikashvilis Experiment
20
Macroscopic Properties of Superfluid He II 2

• Infinite thermal conductivity
• This makes it impossible to establish a
  temperature gradient in a bulk liquid.
• In a normal liquid, bubbles are formed when the
  local temperature in a small region in the body
  of the liquid is higher than the surface
  temperature.
• Unusually thick adsorption film
• The unusual flow properties of He II result in
  the covering of the exposed surface of a
  partially immersed object being covered with a
  film about 30 nm (or 100 atomic layers) thick,
  near the surface, and decreasing with height.

21
Flow of He II over Beaker Walls

• The temperature is the same throughout the
  system, and the superfluid acts as a siphon,
  flowing through the film to equalize the levels
  in the two bulk liquids.
• By observing the rate at which the beaker level
  changes, the superfluid velocity has been found
  to be about 20 cm/s.

22
Thermomechanical Effect 1

• If a temperature gradient is set up between two
  bulk volumes connected by a superleak, through
  which only the superfluid can flow, the
  superfluid flows to the higher temperature side,
  in order to reduce the temperature gradient.
• This example of the thermomechanical effect,
  shows that heat transfer and mass transfer cannot
  be separated in He II.

23
Thermomechanical Effect 2

• At equilibrium, GA GB i.e. ?G 0.
• Now, dG S dT V dP 0
• ? ?P (S/V) ?T (s/v) ?T,
• where s and v are the values per kg.
• Now ? 1/v, so that,
• ?P s ? ?T.

24
The Fountain Effect

• In this celebrated experiment of Allen and Jones
  (1938), the superleak is heated by a flashlight.
  
• In order to equalize temperatures, the superfluid
  flows through the superleak with sufficient speed
  to produce a fountain rising 30 cm or more.
• According to Landaus theory (1941), the normal
  fluid consists of the excited quantum states.
  The fine channels in the superleak filter out the
  excited states

25
Bose-Einstein Condensation

• PHYS 4315
• R. S. Rubins, Fall 2009

26
About BEC

• In 1924, Einstein applied Satyendra Boses
  explanation of blackbody radiation to matter,
  predicting the phenomenon known as Bose-Einstein
  condensation (BEC).
• BEC is a quantum mechanical phase-transition,
  thought to be responsible for superfluidity in
  liquid helium.
• Not until 1995 was it observed in isolated atoms,
  in 87Rb (NIST), 23Na (MIT) and 7Li (Rice U.).
  Since then, BEC has been observed around the
  world, and 1H (MIT) and 4He France.
• Samples typically contain of the order of 105 -
  106 atoms, in which several thousand form the
  condensate, with transition temperatures in the
  range 300 600 nK.

27
BEC Scientific Entanglements

• BEC belongs to atomic
• physics, condensed matter
• physics and stat. mech.
• It could not have been
• produced without the tools
• of optics and laser physics,
• the manipulation of
• magnetism and fluid
• dynamics, and the use of
• new techniques in low
• temperature physics.
• BEC is a deep entanglement
• of fields, giving rise to a
• totally new field of physics.
• See Physics Today, December 2006

28
Bosons and Fermions

• Identical particles follow either Bose-Einstein
  or Fermi-Dirac statistics.
• Bosons have integer angular momentum quantum
  numbers (e.g. photons, atoms with an even no. of
  neutrons.).
• They have symmetrical wavefunctions
• i.e. if two particles (1 and 2) are in the
  states a and b, then
• ?sym ?a(1) ?b(2) ?a(2) ?b(1) ? ?a(1)
  ?a(2) if a b.
• Fermions have half-integer angular momentum
  quantum nos. (e.g. electrons, nucleons, atoms
  with an odd no. of neutrons.).
• They have antisymmetrical wavefunctions
• i.e. if two particles (1 and 2) are in the
  states a and b, then
• ?anti ?a(1) ?b(2) ?a(2) ?b(1) ? 0 if
  a b.

29
Boson and Fermion Gases Below 1 mK

• In these Rice University
• images of atomic clouds,
• those of 7Li (a boson with
• 4 neutrons) continue to
• collapse as the
• temperature is lowered.
• Since identical fermions
• cannot occupy the
• same space (the Pauli
• exclusion principle), the
• atomic cloud of 6Li (a
• fermion with 3 neutrons)
• shows a smaller collapse.

30
BEC Photo from Rice University

• Cloud of about 70,000 7Li atoms, with about 1200
  in the BEC peak at the center, at about 600 nK.

31
BEC a Phase Transition in an Ideal Gas

• Like the ferromagnetic transition at the Curie
  point of iron (1043 K), BEC is a phase
  transition, but unlike the ferromagnetic
  transition, which occurs because of the strong
  interaction between iron atoms, BEC occurs in an
  ideal gas, for which interatomic forces are
  negligible.
  

32
BEC Atoms Each in the Same Wave Function

• The de Broglie
• wavelength ?dB h/mv,
• becomes for a quantum
• gas
• ?dB h/(2pmkT)1/2.
• Thus ?dB increases as
• T is lowered, and a
• phase transition to
• a BEC state occurs
• when ?dB reaches the
• atomic separation.

33
Interference Between BEC Waves

• Like the interference patterns that may be
  produced by the coherent light from lasers, BEC
  waves show interference phenomena.
• However, unlike laser beams, which are in
  non-equilibrium states, a BEC wave is an
  equilibrium state.

34
Loading a Magnet Trap for Li7 (Rice U.)

• The apparatus is contained in an ultra-high
  vacuum at room temperature.
• Hot Li7 atoms, emitted from an oven at 800 K,
  form an atomic beam.
• The atomic beam is slowed by an oppositely
  directed laser beam, and deflected by a second
  laser beam towards a magnetic and optical trap.
• Another laser beam collimates the deflected
  atomic beam, and optically pumps it, so that each
  atom is in the same magnetic state.
• Once in the trap, the atomic beam is contained by
  a set of six laser beams.

35
Magnetic Trap (Rice U.)

• If the magnetic moment of an atom is parallel to
  the magnetic field, it will be attracted to a
  local minimum of the field, which occurs at the
  center of the magnet distribution.
• If the direction of the magnetic moment is
  reversed, the center of the distribution becomes
  a local maximum, which causes that atom to leave.
• The magnetic field at the minimum must not be
  zero, otherwise the atomic moments may
  spontaneously reverse their directions.
• In practice, the field at the minimum was 0.1 T.
• Atoms in the trap may be lost by collisions in
  which the moment direction is reversed.

36
Laser Cooling 1

• Laser cooling is achieved by using the Doppler
  effect to reduce vrms.
• Two opposing laser beams of equal intensity are
  each tuned to the low frequency side of an
  optical transition.
• The beam opposing the atoms motion is
  blue-shifted to higher frequencies, so that the
  force on it is increased.
• The beam in the same direction as the atoms
  motion is red-shifted to lower frequencies, so
  that the force on it is decreased.

37
Laser Cooling 2

• The net effect of the two opposing laser beams is
  to reduce the magnitude of the velocity component
  of each atom along the axis of the two beams.
• Three orthogonal pairs of lasers are used to slow
  the motions of atoms moving in all directions.
• Using laser cooling for Rb87, the NIST group in
  Boulder, achieved temperatures of 10 µK, which
  are still ten to a hundred times too high for
  observing BEC.
• The effect of reducing vrms on the temperature of
  the sample may be calculated using the
  equipartition theorem i.e.
• ½ mvrms2 (3/2)kT.

38
Evaporative Cooling 1

• This method is analogous to the cooling of a hot
  liquid by evaporation.
• The fastest moving atoms move furthest from the
  minimum, to a position of highest energy (see the
  upper atom shown in the figure).
• Magnetic resonance is used to reverse the moments
  of the most energetic atoms, causing them to
  leave the trap, which is now an energy maximum.
• Slowly reducing the radio frequency removes
  progressively cooler atoms.
• At the end, only about 1 of the atoms remain in
  the trap, and the temperature is reduced by a
  factor of about 100, giving a temperature of the
  order of 100 nK.

39
Photographing the Condensate (NIST) 1

• False color images show the velocity distribution
  just before the appearance of
• BEC (right), just after it (center), and for a
  nearly pure condensate (right).
• To increase the sample size, the magnetic trap is
  turned off.
• The excited- state (thermal) atoms move out
  faster, leaving the condensate
• near the center of the trap.
• These photographs were taken after the atoms had
  moved for about 0.05 s.
• The thermal cloud is almost circular, while the
  condensate cloud is elliptical.

40
Photographing the Condensate (NIST) 2

• The right frame has a horizontal dimension of 40
  50 µm, equivalent to about 1500 atoms forming a
  single wave.
• The shape of the peak is related to the
  elliptical shape of the trap, giving a vivid
  demonstration of the uncertainty principle ?px?x
  ? h.
• The temperature within the condensate may be of
  the order of 1 nK.

41
Information Theory

• PHYS 4315
• R. S. Rubins, Fall 2009

42
Lack of Information

• Entropy S is a measure of the randomness (or
  disorder) of a system.
• A quantum system in its single lowest state is in
  a state of perfect order S0 (3rd law).
• A system at higher temperatures may be in one of
  many quantum states, so that there is a lack of
  information about the exact state of the system.
• The greater the lack of information, the greater
  is the disorder.
• Thus, a disordered system is one about which we
  lack complete information.
• Information theory (Shannon, 1948) provides a
  mathematical measure of the lack of information,
  which may be linked to the entropy.

43
Missing Information H

• For an experiment with n possible outcomes p1,
  p2, pn, Shannon introduced a function H(p1, p2,
  pn), which quantitatively measures the missing
  information associated with the set of
  probabilities.
• Three conditions are needed to specify H to
  within a constant factor.
• 1. H is a continuous function of pi.
• 2. If all the pi are equal, then pi 1/n, and H
  is a monotonically increasing function of n,
  since the number of possibilities increases with
  n.
• 3. If the possible outcomes of an experiment
  depend on the outcomes of n subsidiary
  experiments, then H is the sum of the
  uncertainties of the subsidiary experiments.
• With these assumptions, H was found to be
  proportional to the entropy S k ?r pr ln pr.

44
Example of Sum of Uncertainties

• Single experiment, using H K(pr lnpr).
• H(1/2,1/3,1/6) K(1/2) ln(1/2) (1/3)
  ln(1/3) (1/6) ln(1/6)
• K(1/2)ln2 (1/3)ln3 (1/6)ln6
  K(2/3)ln2 (1/2)ln3 1.01 K.
  
• Two successive experiments
• H(1/2,1/2) K(1/2) ln(1/2) (1/2) ln(1/2)
  K ln2.
• (1/2)H(2/3,1/3) K( (1/3)ln2 (1/3)ln3
  (1/6)ln3
• Thus, H(1/2,1/2) (1/2)H(2/3,1/3) K1
  (1/3)ln2 (1/3) (1/6)ln3
• 
  K(2/3)ln2 (1/2)ln3 1.01 K.

45
Shannons Calculation 1
The simplest choice of continuous function
(Condition 1)is
For simplicity, consider the case of equal
probabilities i.e. pi 1/n.
Since H is a monotonically increasing function of
n (Condition 2),
For two successive experiments (Condition 3)
46
Shannons Calculation 2
Since H(1/n,1/n) n f(1/r),
.
Letting R 1/r and S 1/s,
47
Shannons Calculation 3
Since g(R) g(S) g(R,S)
and
,
so that
Thus,
so that
Since R 1/r,
.
48
Shannons Calculation 4
For r 1, the result is certain, so that H(r)
f(1/r) 0. Thus, C 0, so that
Now d/dn( A ln n) must be positive (Condition
2), so that A must be negative. Letting K
A, and p pi 1/n,
Thus, the missing information function H is given
by
.
With K replaced by Boltzmanns constant k, H
equals S (entropy).
49
Black-Hole Thermodynamics

• PHYS 4315
• R. S. Rubins, Fall 2009

50
Quantum Fluctuations of the Vacuum

• The uncertainty principle applied to
  electromagnetic fields indicates that it is
  impossible to find both E and B fields to be zero
  at the same time.
• The quantum fluctuations of the vacuum so
  produced cannot be detected by normal
  instruments, because they carry no energy.
• However, they may be detected by an accelerating
  detector, which provides a source of energy.
• The accelerating observer would measure a
  temperature of the vacuum (the Unruh
  temperature), given by
• TU ah/2pc.
• Notes
• i. For an acceleration of 1019 m/s2, TU 1 K.
• ii. TU 0 if either h 0 or c 8, which is the
  classical result.

51
Zeroth Law of Black-Hole Mechanics

• Zeroth law
• The horizon of a stationary black hole has a
  uniform surface gravity ?.
• Thermodynamic analogy
• An object in thermal equilibrium with a heat
  reservoir has a uniform temperature T.
• Relationship between ? and T
• Analogous to the Unruh effect , Hawking showed
  that black holes emit Hawking radiation at a
  temperature TH, given by
• TH h?/2pc,
• where ? may be thought of as the magnitude of
  the acceleration needed by a spaceship to just
  counteract the gravitational acceleration just
  outside the event horizon.

52
Entropy of a Black Hole

• Black holes must carry entropy, because the 2nd
  law of thermodynamics requires that the loss of
  entropy of an object falling into a black hole
  must at least be compensated by the increase of
  entropy of the black hole.
• The expression for the entropy of a black hole,
  obtained by Beckenstein, and later confirmed by
  Hawking is
• SBH kAc3/4Gh,
• where k is Boltzmanns constant, A is the
  area of the black holes horizon, and BH could
  stand for black hole or Beckenstein-Hawking.
• A system of units with c1 gives SBH kA/4Gh,
  while one in which c1, h1, k1 and G1 gives
  SBH A/4, showing that a black-holes entropy is
  proportional to the area of its horizon.

53
First Law of Black-Hole Mechanics

• 1st law
• dM (?/8p) dA O dJ F dQ,
• where M is the mass, O is the angular
  velocity, J is the angular momentum,F is the
  electric potential, Q is the charge, and the
  constants c, h, k, and G are all made equal to
  unity.
• Thermodynamic analogy
• dU T dS P dV
• Relationship between (?/8p)dA and TdS
• Since TH ?/2p and SBH A/4,
• (?/8p) dA (2pTH)(1/8p)(4dSBH) THdSBH
• i.e. the first term is just the product of
  the black-hole temperature and its change of
  entropy.

54
Second Law of Black-Hole Mechanics

• 2nd law
• The area A of the horizon of a black hole is a
  non-decreasing function of time i.e. ?A 0.
• Thermodynamic analogy
• The entropy of an isolated system is a
  non-decreasing function of time i.e. ?S 0.
• Hawking radiation
• If the quantum fluctuations of the vacuum
  produces a particle-antiparticle pair near the
  horizon of a black hole, and the antiparticle
  drops into the hole, the particle will appear to
  have come from the black hole, which loses
  entropy.
• This leads to a generalized 2nd law
• ?Soutside (A/4) 0.