Physics 510: Statistical Physics (Winter 2004)
Contents
I.
Fundamentals of Statistical Mechanics
2
A.
Review of Hamiltonian Mechanics
2
B.
Ensembles and Averages
3
C.
Microcanonical ensemble
3
D.
Statistical Weight
4
E.
Entropy and Temperature
5
F.
Canonical Ensemble
7
1.
Gibbs distribution
7
2.
Free Energy and its Derivatives
8
3.
Equipartition theorem
9
4.
Gibbs paradox
10
G.
Other Ensembles
10
H.
Summary
11
I.
Review of Thermodynamics
12
J.
Fluctuations of basic thermodynamic variables
13
II.
Selected Applications
15
A.
Van der Waals gas
15
1.
Vapor—Liquid transition
16
2.
Fluctuations and stability
16
B.
Lattice Gas
17
C.
Quantum Statistics
18
1.
Bose Condensation
18
2.
Ideal Fermi Gas
18
3.
Photon and Phonon Gases
19
III.
Phase Transitions
21
A.
Ising Model
21
B.
Landau Theory
22
C.
Fluctuations of order parameter
22
D.
Zero modes and Berezinsky—Kosterlitz—Thouless transition
24
IV.
Nonequilibrium statistical physics
25
A.
Brownian motion
25
1.
Lagevin equation
25
2.
FluctuationDissipation Theorem
25
3.
Motion in a nonuniform potential
26
B.
Linear Response
26
C.
Di
ff
usion and FokkerPlanck Equation
26
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2
I. FUNDAMENTALS OF STATISTICAL MECHANICS
•
Equilibrium Statistical Mechanics
allows us to construct the thermodynamic description of a system starting
with classical or quantum mechanics.
The fundamental (mechanical) level is de
fi
ned by the
Hamiltonian
of
the system. On the macroscopic level, we are interested in the
equations of state
, i.e. relationships between
thermodynamic observables, such as temperature, density, magnetization, etc. Strictly speaking, the relationship
is statistical, not deterministic:
the equation of state determines the most likely values of the observable.
Statistical mechanics
also allows one to describe the
fl
uctuations around this most probable state.
•
Nonequilibrium
Statistical Mechanics.
The fundamental level is the same as in the previous case (classi
cal/quantum mechanics). On
the macroscopic level, one is interested in time evolution of the thermodynamic
parameters, for a given initial conditions and/or timedependent perturbation.
•
"Postmodern" Statistical Mechanics (aka Complex Systems)
uses the ideas of statistical physics in the
context of nonHamiltonian systems. Examples: modelling of granular materials, epidemics, economics, social
phenomena.
A. Review of Hamiltonian Mechanics
Microscopic variables
:
n
generalized coordinates
q
≡
(
q
1
, q
2
, ..., q
n
)
and momenta
p
≡
(
p
1
,
....
, p
n
)
. Their time
evolution is deterministic and given by Hamiltonian of the system
H
(
q
,
p
)
through Hamilton’s equations of motion:
˙
q
i
=
∂H
(
q
,
p
)
∂p
i
;
(1)
˙
p
i
=
−
∂H
(
q
,
p
)
∂q
i
.
(2)
•
Phase Space
is the
2
n
dimensional set of all possible coordinates and momenta
(
q
,
p
)
. In this space, the current
position
(
q
,
p
)
completely determines the future (and past) evolution of the system. Therefore, phase space
trajectories never cross.
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 Winter '04
 ANON
 mechanics, Statistical Mechanics, Entropy

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