Introduction to statistical mechanics.

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Lecture 1

• Introduction to statistical mechanics.
• The macroscopic and the microscopic states.
• Equilibrium and observation time.
• Equilibrium and molecular motion.
• Relaxation time.
• Local equilibrium.
• Phase space of a classical system.
• Statistical ensemble.
• Liouvilles theorem.
• Density matrix in statistical mechanics and its
  properties.
• Liouvilles-Neiman equation.

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Introduction to statistical mechanics.
From the seventeenth century onward it was
realized that material systems could often be
described by a small number of descriptive
parameters that were related to one another in
simple lawlike ways. These parameters referred
to geometric, dynamical and thermal properties of
matter. Typical of the laws was the ideal gas
law that related product of pressure and volume
of a gas to the temperature of the gas.
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Bernoulli (1738)
Joule (1851)
Krönig (1856)
Clausius (1857)
C. Maxwell (1860)
L. Boltzmann (1871)
J. Loschmidt (1876)
H. Poincaré (1890)
T. Ehrenfest
J. Gibbs (1902)
Planck (1900)
Einstein (1905)
Langevin (1908)
Smoluchowski (1906)
Pauli (1925)
Compton (1923)
Bose (1924)
Thomas (1927)
Debye (1912)
Dirac (1927)
Fermi (1926)
Landau (1927)
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Microscopic and macroscopic states
The main aim of this course is the investigation
of general properties of the macroscopic systems
with a large number of degrees of dynamically
freedom (with N 1020 particles for example).
From the mechanical point of view, such systems
are very complicated. But in the usual case only
a few physical parameters, say temperature, the
pressure and the density, are measured, by means
of which the state of the system is
specified. A state defined in this cruder manner
is called a macroscopic state or thermodynamic
state. On the other hand, from a dynamical point
of view, each state of a system can be defined,
at least in principle, as precisely as possible
by specifying all of the dynamical variables of
the system. Such a state is called a microscopic
state.
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Averaging
The physical quantities observed in the
macroscopic state are the result of these
variables averaging in the warrantable
microscopic states. The statistical hypothesis
about the microscopic state distribution is
required for the correct averaging. To find the
right method of averaging is the fundamental
principle of the statistical method for
investigation of macroscopic systems. The
derivation of general physical lows from the
experimental results without consideration of the
atomic-molecular structure is the main principle
of thermodynamic approach.
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Zero Low of Thermodynamics
One of the main significant points in
thermodynamics (some times they call it the zero
low of thermodynamics) is the conclusion that
every enclosure (isolated from others) system in
time come into the equilibrium state where all
the physical parameters characterizing the system
are not changing in time. The process of
equilibrium setting is called the relaxation
process of the system and the time of this
process is the relaxation time. Equilibrium means
that the separate parts of the system
(subsystems) are also in the state of internal
equilibrium (if one will isolate them nothing
will happen with them). The are also in
equilibrium with each other- no exchange by
energy and particles between them.
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Local equilibrium
Local equilibrium means that the system is
consist from the subsystems, that by themselves
are in the state of internal equilibrium but
there is no any equilibrium between the
subsystems. The number of macroscopic parameters
is increasing with digression of the system from
the total equilibrium
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Classical phase system
Let (q1, q2 ..... qs) be the generalized
coordinates of a system with i degrees of freedom
and (p1 p2..... ps) their conjugate moment. A
microscopic state of the system is defined by
specifying the values of (q1, q2 ..... qs, p1
p2..... ps). The 2s-dimensional space
constructed from these 2s variables as the
coordinates in the phase space of the system.
Each point in the phase space (phase point)
corresponds to a microscopic state. Therefore the
microscopic states in classical statistical
mechanics make a continuous set of points in
phase space.
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Phase Orbit
If the Hamiltonian of the system is denoted by
H(q,p), the motion of phase point can be along
the phase orbit and is determined by the
canonical equation of motion
(i1,2....s) (1.1)
(1.2)
Therefore the phase orbit must lie on a surface
of constant energy (ergodic surface).
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? - space and ?-space
Let us define ? - space as phase space of one
particle (atom or molecule). The macrosystem
phase space (?-space) is equal to the sum of ? -
spaces. The set of possible microstates can be
presented by continues set of phase points. Every
point can move by itself along its own phase
orbit. The overall picture of this movement
possesses certain interesting features, which are
best appreciated in terms of what we call a
density function ?(q,pt). This function is
defined in such a way that at any time t, the
number of representative points in the volume
element (d3Nq d3Np) around the point (q,p) of
the phase space is given by the product ?(q,pt)
d3Nq d3Np. Clearly, the density function
?(q,pt) symbolizes the manner in which the
members of the ensemble are distributed over
various possible microstates at various instants
of time.
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Function of Statistical Distribution
Let us suppose that the probability of system
detection in the volume d??dpdq?dp1....
dps dq1..... dqs near point (p,q) equal dw (p,q)
?(q,p)d?. The function of statistical
distribution ? (density function) of the system
over microstates in the case of nonequilibrium
systems is also depends on time. The statistical
average of a given dynamical physical quantity
f(p,q) is equal
(1.3)
The right phase portrait of the system can be
described by the set of points distributed in
phase space with the density ?. This number can
be considered as the description of great (number
of points) number of systems each of which has
the same structure as the system under
observation copies of such system at particular
time, which are by themselves existing in
admissible microstates
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Statistical Ensemble
The number of macroscopically identical systems
distributed along admissible microstates with
density ? defined as statistical ensemble. A
statistical ensembles are defined and named by
the distribution function which characterizes it.
The statistical average value have the same
meaning as the ensemble average value.
An ensemble is said to be stationary if ? does
not depend explicitly on time, i.e. at all times
(1.4)
Clearly, for such an ensemble the average value
ltfgt of any physical quantity f(p,q) will be
independent of time. Naturally, then, a
stationary ensemble qualifies to represent a
system in equilibrium. To determine the
circumstances under which Eq. (1.4) can hold, we
have to make a rather study of the movement of
the representative points in the phase space.
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Liovilles theorem and its consequences
Consider an arbitrary "volume" ? in the relevant
region of the phase space and let the "surface
enclosing this volume increases with time is
given by
(1.5)
where d??(d3Nq d3Np). On the other hand, the net
rate at which the representative points flow
out of the volume ? (across the bounding surface
?) is given by
(1.6)
here v is the vector of the representative points
in the region of the surface element d?, while
is the (outward) unit vector normal to this
element. By the divergence theorem, (1.6) can be
written as
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(1.7)
where the operation of divergence means the
following
(1.8)
In view of the fact that there are no "sources"
or "sinks" in the phase space and hence the
total number of representative points must be
conserved, we have , by (1.5) and (1.7)
(1.9)
or
(1.10)
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The necessary and sufficient condition that the
volume integral (1.10) vanish for arbitrary
volumes ? is that the integrated must vanish
everywhere in the relevant region of the phase
space. Thus, we must have
(1.11)
which is the equation of continuity for the swarm
of the representative points. This equation means
that ensemble of the phase points moving with
time as a flow of liquid without ?sources? or
?sinks?.
Combining (1.8) and (1.11), we obtain
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(1.12)
The last group of terms vanishes identically
because the equation of motion, we have for all i,
(1.13)
From (1.12), taking into account (1.13) we can
easily get the Liouville equation
(1.14)
where ?,H the Poisson bracket.
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Further, since ???(qi,pit), the remaining terms
in (1.12) may be combined to give the total
time derivative of ?. Thus we finally have
(1.15)
Equation (1.15) embodies the so-called
Liouvilles theorem.
According to this theorem ?(q0,p0t0)?(q,pt) or
for the equilibrium system ?(q0,p0) ?(q,p), that
means the distribution function is the integral
of motion. One can formulate the Liouvilles
theorem as a principle of phase volume
maintenance.
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Density matrix in statistical mechanics
The microstates in quantum theory will be
characterized by a (common) Hamiltonian,
which may be denoted by the operator. At time t
the physical state of the various systems will be
characterized by the correspondent wave functions
?(ri,t), where the ri, denote the position
coordinates relevant to the system under study.
Let ?k(ri,t), denote the (normalized) wave
function characterizing the physical state in
which the k-th system of the ensemble happens to
be at time t naturally, k1,2....N. The time
variation of the function ?k(t) will be
determined by the Schredinger equation
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(1.16)
Introducing a complete set of orthonormal
functions ?n, the wave functions ?k(t) may be
written as
(1.17)
(1.18)
here, ?n denotes the complex conjugate of ?n
while d? denotes the volume element of the
coordinate space of the given system. Obviously
enough, the physical state of the k-th system can
be described equally well in terms of the
coefficients . The time variation of these
coefficients will be given by
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(1.19)
where
(1.20)
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(for all k) (1.21)
We now introduce the density operator as
defined by the matrix elements (density matrix)
(1.22)
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Equation of Motion for the Density Matrix ?mn(t)
Thus, we are concerned here with a double
averaging process - once due to the probabilistic
aspect of the wave functions and again due to the
statistical aspect of the ensemble!!
The quantity ?mn(t) now represents the
probability that a system, chosen at random from
the ensemble, at time t, is found to be in the
state ?n. In view of (1.21) and (1.22) we have
(1.23)
Let us determine the equation of motion for the
density matrix ?mn(t).
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(1.24)
Here, use has been made of the fact that, in view
of the Hermitian character of the operator,
HnlHln. Using the commutator notation,
Eq.(1.24) may be written as
(1.25)
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This equation Liouville-Neiman is the
quantum-mechanical analogue of the classical
equation Liouville.

• Some properties of density matrix
• Density operator is Hermitian, ?? -
• The density operator is normalized
• Diagonal elements of density matrix are non
  negative
• Represent the probability of physical values