1

Lecture 1Lecture 1

Introduction to statistical mechanics. Introduction to statistical mechanics. The macroscopic and the microscopic states.The macroscopic and the microscopic states. Equilibrium and observation time. Equilibrium and observation time. Equilibrium and molecular motion. Equilibrium and molecular motion. Relaxation time. Relaxation time. Local equilibrium. Local equilibrium. Phase space of a classical system. Phase space of a classical system. Statistical ensemble. Statistical ensemble. Liouville’s theorem. Liouville’s theorem. Density matrix in statistical mechanics and its Density matrix in statistical mechanics and its

properties. properties. Liouville’s-Neiman equation.Liouville’s-Neiman equation.

2

Introduction to statistical mechanics.Introduction to statistical mechanics.

From the seventeenth century onward it was realized that material systemsmaterial systems could often be described by a small number of descriptive small number of descriptive parametersparameters that were related to one another in simple lawlike ways.

These parameters referred to geometric, geometric, dynamical and thermal propertiesdynamical and thermal properties of matter.

Typical of the laws was the ideal gas lawideal gas law that related product of pressure pressure and volumevolume of a gas to the temperaturetemperature of the gas.

3

C. Maxwell (1860) L. Boltzmann (1871)

H. Poincaré (1890)

J. Loschmidt (1876)

T. Ehrenfest J. Gibbs (1902)

Bernoulli (1738)

Krönig (1856)Joule (1851)

Clausius (1857)

Planck (1900)Einstein (1905)

Compton (1923)Bose (1924)

Fermi (1926)

Pauli (1925)

Dirac (1927)

Thomas (1927)Debye (1912)

Landau (1927)

Smoluchowski (1906)Langevin (1908)

4

Microscopic and macroscopic statesMicroscopic 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 statea 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 statea microscopic state.

5

Averaging Averaging

The physical quantities observed in the macroscopic state are the result of these variables averaging averaging in the warrantable microscopic states. The statistical hypothesis about the microscopic state distribution is required for the correct averagingcorrect averaging.

To find the right method of averagingright 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.

6

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.

7

Local equilibrium Local equilibrium

Local equilibriumLocal equilibrium means that the system is consist from the subsystemssubsystems, 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

8

Classical phase systemClassical phase system

Let (q(q11, q, q

22 ..... q ..... qss)) be the generalized coordinates of a system

with ii degrees of freedom and (p(p11 p p

22..... p..... pss)) their conjugate

moment. A microscopic state of the system is defined by specifying the values of (q(q

11, q, q22 ..... q ..... q

ss, p, p11 p p22..... p..... p

ss).).

The 2s-dimensional space constructed from these 2s variables as the coordinates in the phase spacephase space of the system. Each point in the phase space (phase point)(phase point) corresponds to a microscopic state. Therefore the microscopic states in classical statistical mechanics make a continuous set of points in phase space.

9

Phase Orbit Phase Orbit

If the Hamiltonian of the system is denoted by HH(q,p),(q,p), the motion of phase point can be along the phase orbitphase orbit and is determined by the canonical equation of motion

ii q

p

H

ii p

q

H

(i=1,2....s) (1.1)

P

Constant energy surface

Phase Orbit

H(q,p)=E

Epq ),(H (1.2)

Therefore the phase orbit must lie on a surface of constant energy (ergodic surfaceergodic surface).

10

- space and -space - space and -space

Let us define - space - space as phase space of one particle (atom or molecule). The macrosystem phase space (-space-space) is equal to the sum of - spaces - spaces.

The set of possible microstates can be presented by continues set of phase points. Every point can move by itself along it’s 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 functiondensity function (q,p;t).(q,p;t).

This function is defined in such a way that at any time tt, the number of representative points in the ’volume element’’volume element’ (d(d3N3Nq dq d3N3Np)p) around the point (q,p)(q,p) of the phase space is given by the product (q,p;t) d(q,p;t) d3N3Nq dq d3N3Npp.

Clearly, the density functiondensity function (q,p;t)(q,p;t) symbolizes the manner in which the members of the ensemble are distributed over various possible microstates at various instants of time.

11

Function of Statistical Distribution Function of Statistical Distribution

Let us suppose that the probability of system detection in the volume dddpdqdpdqdpdp

11.... dp.... dpss dq dq

11..... dq..... dqss near point (p,q)(p,q) equal dw (p,q)= dw (p,q)= (q,p)d(q,p)d..

The function of statistical distributionfunction 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)f(p,q) is equal

pqddtpq

pqddtpqqpff

NN

NN

33

33

);,(

);,(),(

(1.3)

The right ’’phase portrait’’’’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

12

Statistical Ensemble Statistical Ensemble

The number of macroscopically identical systems distributed along admissible microstates with density defined as statistical ensemblestatistical ensemble. A statistical ensembles are defined and named by the distribution function which characterizes it. The statistical average valuestatistical 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

t

0

Clearly, for such an ensemble the average valueaverage value <f><f> of any physical quantity f(p,q)f(p,q) will be independent of timeindependent 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.

(1.4)

13

Lioville’s theorem and its consequences Lioville’s 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

dt

(1.5)

where dd(d(d3N3Nq dq d3N3Np).p). On the other hand, the net rate at which the representative points ‘’flow’’‘’flow’’ out of the volume (across the bounding surface ) is given by

)dσ(ρσ

nν (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

n

14

ddiv )( v

N

ii

ii

i

pp

qq

div3

1

)()()(

v

(1.7)

(1.8)

where the operation of divergence means the following:

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)

div d( )

v t d (1.9)

t

div d( )v 0 (1.10)

or

dt

15

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

tdiv ( )v 0

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.

(1.11)

Combining (1.8) and (1.11), we obtain

N

ii

ii

i

pp

qq

div3

1

)()()(

v

16

0

3

1

3

1

N

i ii

iN

ii

ii

i p

p

q

qp

pq

qt

(1.12)

The last group of terms vanishes identically because the equation of motion, we have for all ii,

q

q

q p

q p

q p

q p

p

pi

i

i i

i i

i i

i i i

2 2H H( , ) ( , )

(1.13)

From (1.12), taking into account (1.13) we can easily get the Liouville equation

03

1

Hpq ρ, t

ρ

p

ρ

q

ρ

t

ρ N

ii

ii

i

(1.14)

where {,H} the Poisson bracket.

17

Further, since (q(qii,p,pii;t);t),, the remaining terms in (1.12) may be

combined to give the «total» time derivative of . Thus we finally have

d

dt

t, H 0 d

dt

t, H 0 (1.15)

Equation (1.15) embodies the so-called Liouville’s theorem.

According to this theorem (q0,p0;t0)=(q,p;t) or for the equilibrium

system (q0,p0)= (q,p), that means the distribution function is the integral

of motion. One can formulate the Liouville’s theorem as a principle of phase volume maintenance.

0

t

q

p

t=0

0

t

q

p

t=0

18

Density matrix in statistical mechanicsDensity 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

tt the physical state of the various systems will be characterized by the

correspondent wave functions (ri,t), where the rri,i, denote the position

coordinates relevant to the system under study.

H

Let k(ri,t), denote the (normalized) wave function characterizing the

physical state in which the k-thk-th system of the ensemble happens to be at

time tt ; naturally, k=1,2....Nk=1,2....N. The time variation of the function k(t) will

be determined by the Schredinger equation

19

H

k kt i t( ) ( )

knk

nnt a t( ) ( )

Introducing a complete set of orthonormal functions nn, the

wave functions kk(t)(t) may be written as

(1.16)

(1.17)

a t t dnk

nk( ) ( ) (1.18)

here, nn** denotes the complex conjugate of nn

while dd

denotes the volume element of the coordinate space of the given system. Obviously enough, the physical state of the k-k-

thth system can be described equally well in terms of the coefficients . The time variation of these coefficients will be given by

20

i a t i t d t d

a t d

H a t

nk

nk

nk

n mk

mm

nm mk

( ) ( ) ( )

( )

( )

* *

*

=

=m

H

H

H H mnm n d *

(1.19)

(1.20)

where

The physical significance of the coefficients is evident from eqn. (1.17). They are the probability amplitudes for the

k-thk-th system of the ensemble to be in the respective states nn; to

be practical the number represents the probability that a measurement at time t t finds the k-th-th system

of the ensemble to be in particular state nn. Clearly, we must

have

a tnk ( )

a tnk ( )

2

21

a tnk

n

( )2

1

N

k

kn

kmmn tata

Nt

1

* )()(1

)(

(for all k) (1.21)

We now introduce the density operatordensity operator as defined by the matrix elements (density matrix)(density matrix)

( )t

(1.22)

clearly, the matrix element mnmn(t)(t) is the ensemble average of

the quantity aamm(t)a(t)ann

**(t)(t) which, as a rule, varies from member to

member in the ensemble. In particular, the diagonal element

mnmn(t)(t) is the ensemble average of the probability, the

latter itself being a (quantum-mechanical) average.

a tnk ( )

2

22

Equation of Motion for the Density Matrix mn(t)

a tnk

n

( )2

1

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!!

nnn

1(1.23)

Let us determine the equation of motion for the density matrix mnmn(t(t).

The quantity mnmn(t)(t) now represents the probabilityprobability that a

system, chosen at randomat random from the ensemble, at time tt, is

found to be in the state nn. In view of (1.21) and (1.22) we

have

N

k

kn

kmmn tata

Nt

1

* )()(1

)(

23

i tN

i a t a t a t a t

NH a t a t a t H a t

H t t H

mn mk

nk

mk

nk

k

N

mll

lk

nk

mk

nll

lk

k

N

ml mll

mn

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

)

* *

* * *

ln ln

1

1

1

1

=

=

= (H H (1.24)

Here, use has been made of the fact that, in view of the

Hermitian character of the operator, HH**nlnl=H=H

lnln. Using the

commutator notation, Eq.(1.24) may be written as

H

, t

i

H 0 (1.25

)

24

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 nnn

1 0