A DISSERTATION ON PAIR CORRELATION FUNCTION IN STATISTICAL PHYSICS

PRESENTED BY

AKPOVOKA BARRY EGUOLOR

IDOGHOR A. EMMANUEL

DEPARTMENT OF PHYSICS

UNIVERSITY OF BENIN

BENIN CITY

EDO STATE.

AUGUST,2012.

i

TABLE OF CONTENTS

TITLE PAGE................................................................................................i

TABLE OF CONTENTS..............................................................................ii

ABSTRACT................................................................................................iv

1.1 INTRODUCTION.................................................................................1

1.2 DIFINITION.........................................................................................2

1.2.1 CORRELATION FUNCTION.............................................................2

1.2.2 PAIR CORRELATION FUNCTION....................................................5

1.3 DISTRIBUTION OF PARTICLES WITH CORRELATIONS................9

1.4 CALCULATION OF PAIR CORRELATION FUNCTION BY

COMPUTER SIMULATION METHOD.....................................................19

1.4.1 THE CLASSICAL-MAP HYPERNATTED CHAIN METHOD...........20

(CHNC METHOD)...................................................................................20

1.4.2 MONTE CARLO SIMULATION.......................................................21

1.4.3 RADIATION TECHNIQUES............................................................22

1.5 APPLICATIONS OF THE PAIR CORRELATION FUNCTION IN

STATISTICAL PHYSICS...........................................................................23

1.6 RELATIONSHIP BETWEEN THE RADIAL DISTRIBUTION

FUNCTION, ENERGY, COMPRESSIBILITY AND PRESSURE................24

REFERENCES.........................................................................................41

LIST OF FIGURES

FIG 1:CORRELATION CURVE........................................................................3

ii

FIG 2:CORRELATION CURVE FOR TWO MEASUREMENTS........................4

FIG. 3A.RADIAL DISTRIBUTION FUNCTION(RELATIVE POSITION)...........7

FIG 3B:RADIAL DISTRIBUTION FUNCTION IN A LIQUID MOLECULE

(SCHEMATIC CURVE)……………………………………….

………………………………...8

ABSTRACT

This work is aimed at citing out the distinctive properties,

importance and applications of the pair correlation function

iii

together with its relation to thermodynamic properties in

statistical physics.

iv

1.1 INTRODUCTION

The subject of atomic and molecules distribution functions

has enjoyed a long rich history.

This stems both from the experimental use of radiation

scattering to determine such functions at least at pair level[1,2]

as well as a wide range of theoretical developments motivated by

the presence of exact relations for thermodynamic properties in

term of those distribution functions[3,4]. This combination of

experimental measurements and theoretical insights has been an

indispensable component of condensed-matter physical

chemistry and physics. However, not surprisingly for a scientific

area so characterized by intrinsic complexity; some deep

problems of incomplete understanding still persist.

One of the basic problems concerns pair correlation

realizability. In its simplest version this concerns g(R), the pair

correlation function for a statistically homogenous single-

component many-body system comprising structureless

(Spherically symmetric) particles. In the large system limit, this

1

function is conventionally defined to approach unity as r ∞. By

definition, it cannot be negative.

g(R) 0

(1.0)

By virtue of exhaustive enumeration for system of 15 or

fewer sites subject to periodic boundary conditions, several

conclusion have formulated for the case of a constant pair

correlation beyond the exclusion ranges. These conclude (a) pair

correlation realizability over a nonzero density range, (b)

violation of the kirkwood superposition approximation for many

such realizations, and (c) inappropriateness of the so-called

“reverse monte Carlo” method that uses a candidate pair

correlation function as a means to suggest typical many-body

configurations.

1.2 DIFINITION

1.2.1 Correlation Function

2

A(t)

Time (t)

Let us consider a series of measurement of a quantity of a

random nature at different times.

Fig 1:Correlation curve

Although the value of A(t) is changing randomly, for two

measurements taken at time tI and tII that are close to each other

are good chance that A(tI) and A(tII) have similar values, their

values are correlated[5].

If two measurements taken at time tI and tII that are far

apart, there is no relationship between values of A(tI) and A(tII),

their values are uncorrelated.

The “level of correlation” plotted against time would start at

some value and then decay to a low value[5].

3

A(t)

Time, t

tcorr

Let us shift the data by time tcorr.

Fig 2:Correlation curve for two measurements

and multiply the values in new data set to the values of the

original data set.

Now let us average over the whole time range and get a

single number G(tcorr). If the two data sets are lined up, the peaks

and troughs are aligned and we will obtain a big value from this

multiple-and-integrate operation. As we increase tcorr the G(tcorr)

declines to a constant value.

The operation of multiplying two curves together and

integrating them over the x-axis is called an overlap integral,

4

since it gives a big value if the curves both have high and low

values in the same place.

The overlap integral is also called the Correlation function

G(tcorr) = <A(to)A(to + tcorr)>. This is not a function of time (since

we already integrated over time), this is function of the shift in

time or correlation time tcorr.

Decay of the correlation function is occurring on the

timescale of the fluctuation of the measured quantity undergoing

random fluctuations.

All of the above considerations can be applied to

correlations in space instead of time[5].

G(r) = <A(ro)A(ro + r)>

1.2.2 Pair Correlation Function

The pair correlation function gives the probability to find a

particle in the volume element dr located at r if at r = 0 there is

another particle. It gives the probability of finding the centre of a

particle a given distance from the center of another particle. For

5

short distances, this is related to how the particles are packed

together. For example, consider hard sphere (like marbles). The

spheres cannot overlap, so the closest distances two centers can

be is equal to the diameter of the spheres. However, several

sphere can be touching one sphere, then a few more can form a

layer around them, and so on.

Further away, these layers get more diffuse, and so for a

large distance, the probability of finding two spheres with a

given separation is essentially constant. In that case, its related

to the density, a more dense system has move spheres, thus its

more likely to find two of them with a given distance[9].

The pair correlation function is also called the radial

distribution function which is a special case of the pair

distribution function. The interaction between the molecules in a

liquid or gas cause correlation in their positions. The aim of

almost all modern theories of liquids is to calculate the radial

distribution function by means of statistical thermodynamical

reasoning. Alternatively, the radial distribution function can be

measure directly in computer simulations.

6

r

dr

The radial distribution function can be use in calculating the

energy, compressibility and pressure of a fluid, with a particular

application to a hard sphere fluid[6].

For instance, imaging we have placed ourselves on a certain

molecule in a liquid or gas. Now let us count the number of

molecules in a spherical shell of thickness dr at a distance r, i.e.

we count the number of molecules with a distance between r and

r + dr.

Fig. 3a.Radial distribution function(Relative position)

7

0.5

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 g(r) rrrr

1.0

1.5

2.0

2.5

3.0

3.5

r

g(R)

Simple liquid

Fig 3b:Radial distribution function in a liquid molecule(schematic curve)

Fig 3b implies that the radial distribution function as one

moves away from the origin (reference particle) become constant

with value = 1 i.e. the probability of finding a particle in the

volume element dr located at r if at r = 0 there is another

particle is 1 and as one move closer to the origin, a distance less

than the diameter the radial distribution function becomes zero.

1.3 DISTRIBUTION OF PARTICLES WITH

CORRELATIONS

8

Consider the distribution of particles with correlations.

Correlation may be due to molecular force. In quantum statistics,

the configuration in coordination space depends on the

configuration in momentum space, and thus correlation exists

between particles[7].

Consider a system composed of identical N particle. if r j (j =

1,2, . . .N) denotes the position of these particles, then the

number density at r is given by:

n(1) ( r⃗ )= ⟨n(r⃗ )⟩= ⟨n ⟩

But n ( r⃗ )=(r⃗ j−r⃗ )

n(1) ( r⃗ )=⟨( r⃗ j− r⃗) ⟩ (1.1)

Equation (1.1) above is for one-body density. For two-body

density we have

n(2) ( r⃗ ,r⃗¿ )=⟨ ( r⃗ j−r⃗ )δ(r⃗ k−r⃗1)⟩ (1.2)

Where ⟨ ⟩ means average. It may be a quantum mechanical

expectation values or a time average, or ensemble average, but

at the present moment it is sufficient to suppose that it is an

9

average in a certain sense. Equation (1.2) expresses the

correlation in density at positions r⃗∧r⃗1

The two-body density can be written in terms of one-body

density as

n(2)( r⃗ ,r⃗ 1)=n(1) ( r⃗ )n (1) ( r⃗1 )g ( r⃗ ,r⃗1−r⃗ ) (1.3)

Where g ( r⃗ , r⃗1−r⃗ ) is the pair distribution function for isotropic

substance like a gas or liquid, ⟨n ⟩ = n(1)(r) may be constant except

in the vicinity of container walls. Thus, do not depend on r and r1.

Equation (1.3) becomes

n(2)( r⃗ ,r⃗ 1)=⟨n ⟩2g (R ) (1.4)

Since n(1) ( r⃗ )n(1) (r⃗1 )= ⟨n ⟩ ⟨n ⟩=⟨n ⟩2 andR=|⃗r1−r⃗|

R is not a vector but the magnitude of the vector r⃗∧r⃗1

g(R) is called the radial distribution function which can be

obtained by X-ray scattering.

When the system is large, the correlation usually vanishes

for R ∞ since, correlation is a function of distance.

10

Now we integrate the two-body density with respect to r⃗∧r⃗1.

The sum with the restriction j K is given as the double sum

without the restriction subtracted by a simple sum. Integrating

over a volume V, we denote by Nv the number of particles in the

volume. We have from equation (1.2).

n(2) ( r⃗ , r⃗1¿)=⟨ ( r⃗j−r⃗ )δ( r⃗k−r⃗1)⟩

¿∑j ≠

N

∑k

δ ( r⃗j− r⃗ )δ(r⃗ k−r⃗1)

¿∑j=k

N

∑ δ ( r⃗j− r⃗ )δ ( r⃗k−r⃗ )−∑j=k

δ( r⃗j−r⃗ )

Integrating both sides with respect to r and r1 we have

∬n(2 )(r , r¿¿1)dr dr1=∬∑k∑j

δ ( r⃗j−r⃗ )δ (r⃗k−r⃗1)d r⃗ d r⃗ 1−∬∑j=k

δ (r⃗ j−r⃗❑)d r⃗ d r⃗ 1¿

¿ ⟨∬∑k∑j

δ ( r⃗j−r⃗ ) δ(r⃗ k−r⃗ )d r⃗ d r⃗1⟩−⟨∬∑j=k

δ ( r⃗ j−r⃗ )d r⃗ d r⃗ 1⟩

¿∬∑ ⟨n (r⃗ )⟩ ⟨n(r⃗ 1)⟩ d r⃗ d r⃗1−∫∑ ⟨n(r) ⟩d r⃗ d r⃗1

¿ ⟨ N v ⟩ ⟨N v ⟩− ⟨N v ⟩

¿ ⟨N2v ⟩−⟨N v ⟩

∴∬v

❑

n(2 )(r⃗ ,r⃗¿¿1)dr dr1= ⟨N2v ⟩−⟨N v ⟩¿ (1.5)

11

Where N v=∫v

❑

❑( r⃗j−r⃗ )d r⃗

When the volume v is far from the container walls i.e. when

the volume is small,

⟨ N v ⟩ =⟨n ⟩v (1.6)

{∫v

❑

∑ δ (r j−r )dr1=⟨∑j δ (r j−r )⟩}∫ d r⃗

dr=dxdydz=d3r

∴¿

Since n(1)r=⟨ ⟩

⟨n ⟩ been a constant

From equation (1.4)

n(2) (r,r1) = ⟨n ⟩2g(R)

Subtracting⟨n ⟩2 from both sides and integrating both sides

with respect to r, and r1

∬v

❑

¿¿

12

¿ ⟨n ⟩(2)∬v

❑

[ g(R)−1 ] dr dr1

¿ ⟨n ⟩2∫ dr∫ [g (R )−1 ] dR

∬v

❑

[n(2) ( r ,r 1)−⟨n ⟩2 ]dr dr1= ⟨n ⟩2V∫v

❑

[ g (R )−1 ]dR (1.7)

Now, we could also write

∬v

❑

[n(2) ( r ,r 1)−⟨n ⟩2 ]dr dr1=∬v

❑

n(2) ( r ,r1 )dr dr1−∬ ⟨n ⟩2dr dr1

From equation (1.5) we have

∬v

❑

n(2 )(r⃗ , r⃗¿¿1)d r⃗ d r⃗1= ⟨N 2v ⟩− ⟨N v ⟩ ¿

∴∬v

❑

¿¿

¿ ⟨N v2 ⟩−⟨N v ⟩−⟨n ⟩2V 2

¿ ⟨N v2 ⟩−⟨N v ⟩−⟨ N v ⟩2

∴∬v

❑

[n(2) ( r ,r1 )−⟨n ⟩2 ]dr dr1= ⟨N v2 ⟩− ⟨N v ⟩− ⟨N v ⟩2 (1.8)

Comparing (1.7) and (1.8)

⟨N v2 ⟩−⟨N v ⟩−⟨ N v ⟩2=⟨n ⟩2V∫

v

❑

[g (R )−1 ] dR

13

⟨N v2 ⟩−⟨N v ⟩2=⟨n ⟩2V∫

v

❑

[ g (R )−1 ]dR+⟨N v ⟩

Lets write the expression for

⟨ (N v− ⟨N v ⟩)² ⟩=⟨ (N v− ⟨N v ⟩)(N v− ⟨N v ⟩)⟩

=⟨N v2−N v ⟨N v ⟩−N v ⟨N v ⟩+ ⟨N v ⟩ ² ⟩

=⟨N v2−2N v ⟨N v ⟩+ ⟨N v ⟩ ² ⟩

=⟨N v2 ⟩−2 ⟨N v ⟩ ⟨ N v ⟩+ ⟨N v ⟩ ²

=⟨N v2 ⟩−2 ⟨N v ⟩ ²+ ⟨N v ⟩ ²

¿ ⟨N v2 ⟩−⟨N v ⟩ ²

From (1.9) we can write

⟨ (N v− ⟨NV ⟩ )2 ⟩= ⟨n ⟩2 v∫v

❑

[g (R )−1 ] dR+⟨N v ⟩

Divide both sides by ⟨ N v ⟩2

⟨ (N v− ⟨N v ⟩ )2 ⟩⟨N v ⟩2

= 1

⟨N v ⟩2¿

=1

⟨N v ⟩2¿

14

Since ⟨n ⟩2 v=⟨n ⟩ ⟨n ⟩ v= ⟨n ⟩ ⟨N v ⟩

¿⟨ N v ⟩⟨N v ⟩2

+⟨n ⟩ ⟨N v ⟩

⟨N v ⟩2∫v

❑

[ g (R )−1 ]dR

=1

⟨N v ⟩+

⟨n ⟩⟨N v ⟩∫v

❑

[g (R )−1 ] dR

∴⟨ (N v− ⟨N v ⟩)2 ⟩

⟨N v ⟩2= 1

⟨N v ⟩¿ (1.10)

Equation (1.10) is the mean square fluctuation. It tells us

the relationship between macroscopic quantity (on the left hand

side) and microscopic quantity (on the right hand side).

g(R )usually approaches 1 quite rapidly with increasing R. Thus,

when the volume V of the system is sufficiently large and v is a

small part if it, then the integral on the right-hand side of (1.10)

is independent of v (we assume that v is much larger than the

size of a molecule).

However, when v is comparable with the total volume V, the

above mentioned integral depends on v. If v is equal to V , then Nv

equals N and we have no fluctuation, so that the right hand side

of (1.10) much vanish.

15

In n(2)(r,r1), ⟨ δ (r j−r )∑ K (≠1 )δ (rk−r) ⟩ is a term which expresses

the probability density at r1,when the particle j = 1 is at r.

Therefore n(1)(r1)g(R) is the density at r¹. under the condition that

a particle is at r. For example, for classical independent

particles, when a particle is at r, the remaining N-1 particles are

in the total volume V and thus

⟨n ⟩ g (R )= N−1V

g (R )= N−1⟨n ⟩ v

= N−1N

=1− 1N

Therefore

⟨n ⟩∫v

❑

[ g (R )−1 ] dR=⟨n ⟩∫v

❑

[1− 1N

−1]dR

Since g (R )=1− 1N

⟨n ⟩∫v

❑

[ g (R )−1 ] dR=⟨n ⟩∫v

❑ −1N

dR

=−⟨n ⟩N

∫dR

= −⟨n ⟩V

N

16

Recall N ≈ N v= ⟨n ⟩V

∴ ⟨n ⟩∫v

❑

[ g (R )−1 ]dR=−⟨n ⟩ v⟨n ⟩V

⟨n ⟩∫v

❑

[ g (R )−1 ] dR=−vV

(1.11)

From equation (1.10)

¿¿

¿ 1

⟨N v ⟩ [1+[−vV ]]

¿ 1

⟨N v ⟩ [1+[−vV ]]

¿ 1

⟨N v ⟩[1−P ]

¿¿

Where v/V = P is the probability for a particle to be in the

volume v. By (1.10), the square of the relative fluctuation is thus

(1 – P) ∕ (Nv).

Another extreme case is a system of closely packed hard

spheres where fluctuation is nearly impossible and we must

have, ⟨n ⟩∫ [ g (R )−1 ] dR≈−1

17

Taking the Fourier transform of the deviation of the local

density n(r) from its mean value⟨n(r⃗ )⟩,we can calculate the mean

square of the Fourier component of the density fluctuation. Thus

obtain

{∫v

❑

[n (r )− ⟨n ⟩ ]exp (– if . r )dr ¿2} ¿ ⟨n ⟩ v {1+ ⟨n ⟩V∫ [ g (R )−1 ]exp (−if .R )dR } (1.12)

Equating (1.10, 12) are relations which connect

macroscopic quantities (the left hand side) to the microscopic

radial distribution function g(R) on the right-hand side of these

equation.

1.4 CALCULATION OF PAIR CORRELATION FUNCTION

BY COMPUTER SIMULATION METHOD

The radial distribution can be computed either via computer

simulation methods like the Monte Carlo method or via the

Ornstein-Zernike equation, using the approximative clouser

relations like the Percus-Yevick approximation or the

Hypernetted chian theory. It can also be determined

experimentally, by radiation scattering techniques.

18

1.4.1 THE CLASSICAL-MAP HYPERNETTED CHAIN

METHOD

(CHNC Method)

This is a method used in many-body theoretical physics for

interacting uniform electron liquid in two and three dimensions,

and for interacting hydrogen plasma. The method extends the

famous hypernetted-chain method (HNC) introduced by J.M. Van

Lecuwen et al to quantum fluids as well. The classical HNC,

together with the percus-yevick approximation, are the two

pillars which bear the burnt of most calculations in the theory of

interacting classical fluids. Also, HNC and PY have become

scheme in the theory of fluid, and hence they are of great

importance to the physics of many-particle systems.

The HNC and PY integral equation provides the pair

distribution functions of the particles in a classical fluid, even for

very high coupling strengths. The coupling strength is measure

by the ratio of the potential energy to the kinetic energy. In

classical fluid, the kinetic energy is proportional to the

temperature. In quantum fluid, the situation is very complicated

19

as one needs deal with the quantum operators, and matrix

elements of such operators, which appear in various perturbation

methods based on Feynman diagrams.

In the CHNC method, the pair distribution of the interacting

particles are calculated using a mapping which ensures that the

quantum mechanically correct non-interacting pair distribution

functions is recovered when the coulomb interaction are

switched off. The value of the method lies in its ability to

calculate the interacting pair distribution function g (R )at zero and

infinite temperatures.

Comparison of the calculated g (R )with results from Quantum

Monte Carlo show remarkable agreement even for very strongly

correlated systems.

1.4.2 MONTE CARLO SIMULATION

A trial wave function of the Slater-Jastrow type, with the

long-range part of the two body term modified to account for the

an isotropy of the system. The one-body term is optimized so that

20

the electronic density from vibrational and diffusion Monte Carlo

calculation in the local density approximation and the surface

energies to the results obtained using the Langreth-mehl and

perdew-wang generalized gradient approximation at high

densities. They agree with the results of the Fermi-hypernatted-

chain calculations. The pair correlation function at region near

the surface are tabulated, showing the anisotropy of the

exchange-correlation hole in regions of fast varying densities.

1.4.3 RADIATION TECHNIQUES

In other to obtain a pair distribution function from

crystalline material, 2g of the bulk N1 powder (99.9% Sigma

Aldrich) was measure at room temperature for 15 minutes.

Nickel in a strong neutron scatter which often used as a standard

sample for calibration purpose in neutron diffraction. The

sample was measured in the standard vanadium sample holder

with diameter 6mm. the data was normalized by scattering from

vanadium and background scattering from the empty container

was subtracted. The experimental pair distribution function was

21

obtained by a Fourier transform of the total scattering structure

function up to a high momentum transfer.

1.5 APPLICATIONS OF THE PAIR CORRELATION

FUNCTION IN STATISTICAL PHYSICS

1. The pair correlation function can be used to calculate some thermodynamic quantities like; energy, pressure and compressibility

2. It tells us about the structure of complex isotropic systems.3. It determines the thermodynamic quantities at the level of

the pair potential approximation.4. It can be measured in neutron and x-ray diffraction

experiments.

1.6 RELATIONSHIP BETWEEN THE RADIAL

DISTRIBUTION FUNCTION, ENERGY, COMPRESSIBILITY

AND PRESSURE

Once we know g (r ) ,we can derive all non-entropic

thermodynamic properties.

Energy

22

The simplest is the energy:

U=U∫¿+

32N K BT +

12N

VN ∫

0

∞

dr 4 πr2g (r )φcr ¿ (1.13)

Where

Uint = internal energies of the molecules. The second term

originates from the energy interactions. The average total

potential energy equal 12N times the average interaction of one

particular molecule with all others; the factor ½ serves to avoid

double counting. The distance contribution of all particles in a

spherical shell of thickness dr at a distance r to the average

interaction of one particular particule with all others is

4 π r2(N /V ) g(r )φ(r ). Integrating finally yields equation[6]..

φ (r ) is called the pair interaction between its constituent

molecules.

Compressibility

The isothermal compressibility KT is defined as

KT≡−1V ( dVdp )

T ,N (1.14)

23

From thermodynamics, it is known that KT can be linked to

spontaneous fluctuation in the number of particles in an open

volume V. )

⟨N ⟩ ρ KBT KT=¿ (see equation (1.5)

Where the pointy bracket indicate a long time average or an

average over many independent configurations commensurate

with the thermodynamic conditions. (In this case constant

temperature T and Volume V).

∫v

❑

d3 r1∫v

❑

d3r 2ρ2g ( r12)=⟨ N (N−1 ) ⟩=⟨ N2 ⟩−⟨N ⟩

Where r12 = 1r1 – r21

We can use this to link the compressibility to the radial

distribution function.

⟨N ⟩ ρ KBT KT=ρ∫v

❑

d3 r1ρ∫v

❑

d3 r2g (r 12)+ ⟨N ⟩−ρ∫v

❑

d3r 1ρ∫v

❑

d3 r2

¿ ρ∫v

❑

d3 r1ρ∫v

❑

d3 r2(g (r12 )−1)+ ⟨N ⟩

¿ ρ∫v

❑

d3 r1ρ∫R3

❑

d3 r (g (r❑)−1)+ ⟨N ⟩

Dividing through by ⟨N ⟩

24

⟨N ⟩ ρKBT KT

⟨N ⟩ =1

⟨N ⟩ [ NV .Vρ∫R 3

❑

d3r (g (r )−1)+ ⟨N ⟩ ]¿1

⟨N ⟩ [Nρ∫R 3

❑

d3 r (g (r )−1)+ ⟨N ⟩]¿

⟨N ⟩⟨N ⟩ [ ρ∫R3

❑

d3 r (g (r )−1)+1]¿1+ρ∫

R3

❑

d3 r (g (r )−1)

∴ ρKBT KT=1+ ρ∫R 3

❑

d3r (g (r )−1) (1.15)

Equation (1.15), is the so-called compressibility equation that

shows that the compressibility of a fluid is intimately connected

to the radial distribution function of its constituent molecules.

Pressure

We can now consider the pressure of a fluid. If the density of

fluid is not too high, correlations between three or more particles

may be ignored, in which case the radial distribution function is

given by

g (r )≈ exp(−β φ( r )) (1.16)

25

Where φ (r ) is the pair interaction potential. Also, for not too high

densities, the pressure of a fluid is to a good approximation given

by the first two terms in the viral equation.

PV=NKBT (1+B2 (T ) NV ) (1.17)

Where B2(T) is called the second viral coefficient. Our goal now is

to link B2(T) to the radial distribution function g (r ) or pair

interaction φ (r ). This may be achieved by differentiating the viral

equation with respect to v.

PV=NKBT (1+B2 (T ) NV )

ddV

(PV )=−NK BT B2(T ) ddV ( NV )

vdpdV

+P dVdV

=−NK BT B2(T ) NV 2

(1.18)

( dpdV )N , T

V +P=−NKBT B2(T ) NV 2

Recall, KT≡−1V ( δVdP )

T , N

(1.14)

26

−KTV ≡( δVdP )

−1KTV

≡( δVdP )(1.19)

Substitute equation (1.19) into (1.18)

−1KT

+P=−NKBT B2(T ) NV 2 (1.20)

Recall also,

PV=NKBT (1+B2 (T ) NV ) (1.17)

P=NKBT

V (1+B2 (T ) NV )

(1.21)

Substitute (1.21) into (1.20)

∴−1KT

+NKBT

V (1+B2 (T ) NV )=−NK BT B2(T ) N

V 2

−1KT

+NKBT

V+NKBT B2 (T ) N

V 2=−NKBT B2(T ) NV 2

−1KT

+NKBT

V=−NK BT B2(T ) N

V 2−NKBT B2(T ) NV 2

27

−1KT

+ρK BT=−2NKBT B2 (T ) NV 2 (1.22)

Since ρ=NV

Multiply equation (1.22) by KT

−1+ρKBT K T=−2K BT B2(T ) NV 2

KT

−1+ρKBT K T=−2 B2(T ) NV

ρKBT K T=1−2B2(T )NV

(1.23)

Since NK BK T

V≈1

Comparing the two equation (1.15) and (1.23), we can write the

second viral coefficient as a three dimensional integral over the

pair interaction φ(r )

We have

1+ρ∫R3

❑

d3 r (g (r )−1 )=1−2B2(T ) NV

ρ∫R3

❑

d3 r (g (r )−1 )=−2B2(T ) NV

28

Recall ρ=NV

∫R 3

❑

d3 r (g (r )−1 )=−2B2(T )

B2 (T )=−12∫R3

❑

d3 r (g (r )−1 )

B2 (T )=−12∫R3

❑

d3 r (e−βφ(r )−1 ) (1.24)

Since g (r )≈ exp(−β φ( r )) (1.16)

Equation (1.24) is important because it allows us to calculate the

pressure of a fluid knowing only the pair interactionφ (r ) between

its constituent molecules.

If we consider a small volume v << V of a large system in

thermal equilibrium. The fluctuation of the number of molecules

in v is related, in thermodynamics to the isothermal

compressibility [7].

KT=−1V ( dvdP )

T , N(1.14)

⟨ (N v− ⟨N v ⟩)2

⟨N v ⟩2 ⟩= KT KT

V(1.25)

29

Where K = KB = Boltzman constant

T = temperature

v = volume

Recall that,

⟨ (N v− ⟨N v ⟩)2

⟨N v ⟩2 ⟩= 1

⟨N v ⟩ {1+ ⟨n ⟩∫ [ g (R )−1 ]dR } (1.10)

Equating (1.10) and (1.25)

KT K T

v= 1

⟨N v ⟩ {1+ ⟨n ⟩∫ [ g (R )−1 ]dR }

Multiply through by v

KT KT=v

⟨N v ⟩ {1+⟨n ⟩∫ [ g (R )−1 ] dR }

¿ v

⟨N v ⟩+v ⟨n ⟩⟨N v ⟩∫ [g (R )−1 ] dR

¿ v

⟨N v ⟩+∫ [ g (R )−1 ] dR (1.26)

Since v ⟨n ⟩=⟨N v ⟩ (1.16)

⟨ N v ⟩=v ⟨n ⟩

30

v

⟨N v ⟩= vv ⟨n ⟩

= 1⟨n ⟩

equation (1.26) becomes

¿ 1⟨n ⟩

+∫ [g (R )−1 ] dR

KT KT=1

⟨n ⟩+∫ [ g (R )−1 ]dR (1.27)

Equation (1.27) is the well known Ornstein-Zernike relation or

compressibility equation. This relation applies to a quantum-

mechanical systems as well.

For usual liquids, compressibility is very small

(KT KT ⟨n ⟩ ≪1) which mean ⟨n ⟩∫ [ g (R )−1 ] dR=1

For classical gas subject to Boyles-Charles law

¿

KT KT=1

⟨n ⟩

−KTV ( dvdP )= 1

⟨n ⟩ (1.28)

Recall equation (1.6)thus, (1.28) becomes,

31

−KTv ( dvdP )= v

⟨ N v ⟩

−KT

v2 ( dvdP )= 1

⟨ N v ⟩

−⟨ N v ⟩ KT dv

v2=dp

Integrating both sides, we have

−⟨ N v ⟩ KT∫ dv

v2=∫dp

−⟨ N v ⟩ KT−v=p

⟨N v ⟩ KTv

=p

P= ⟨n ⟩KT (1.29)

Equation (1.29) is the Boyles-Charles law which is obtained if we

assume there is absence of correlation.

If we assume a classical gas of hard spheres, molecules

cannot come closer than their diameter D. for dilute gas we may

approximately put.

g(R){0(R<D)1(R>D)

32

Recall, equation (1.27)

KT KT=1

⟨n ⟩+∫ [ g (R )−1 ]dR

Putting g (R )=1, we have

∫(g (R )−1)dR

¿∫ (1−1 )dR=∫ dR (0 )=0

Putting g(R) = 0 we have,

¿∫ [0−1]dR=∫−dR=−∫ dR=v

Where v=4 πD3

3 (volume of a sphere)

equation (1.27) becomes

KT KT=1

⟨n ⟩ −4 πD3

3

−KTv ( dvdp )= 1

⟨n ⟩ −4πD3

3

Multiplying through by 1/v we have

−KTv ( dvdp )= 1

⟨n ⟩ v−4 πD

3

3 v

¿ 1⟨N v ⟩

−4 πD3

3v

33

¿V−4πD3 ⟨ N v ⟩3V ⟨ N v ⟩

(1.30)

Let q=4/3 πD3 ⟨N v ⟩

Equating (1.30) becomes

−KT

v2 ( dvdp )= v−qv ⟨N v ⟩

−KTv ( dvdp )= v−q

⟨N v ⟩

−KT ⟨N v ⟩dvv (v−q)

=dp

Integrating both sides we have

−KT ⟨ N v ⟩∫ dvv (v−q)

=∫ dp

Using integration by partial fraction on ∫ dvv (v−q), we have

∫ dvv (v−q)

=∫( Av + Bv−q )dv

¿∫ A (v−q )+B(v)v (v−q)

¿∫ Av−Aq+Bvv (v−q)

34

¿∫ Av+Bv−Aqv (v−q)

(A + B)v = 0A + B = 0−Aq=1

A=−1q

∴ A+B=−1q

+B=0

B=−1q

∴∫ [ Av + Bv−q ]dv=∫ [−1vq + 1

q (v−q) ]dv

∫ [ 1q (v−q)

− 1vq ]dv

∫ dvv (v−q)

=1q∫ [ 1

v−q−1v ]dv

∴−KT ⟨ N v ⟩

q[¿ (v−q )−¿v ]=p

−KT ⟨N v ⟩q

⌊∈( v−qv )¿¿

KT ⟨N v ⟩q

⌊∈( vv−q )¿¿ (1.31)

35

We have

¿( vv−q )=¿ [ 1

1−q/ v ]

Putting t=qv

We have

¿( 11−q/ v )=¿ [ 11−t ]

Expanding ¿ [ 11−t ] in series form and neglecting higher terms we

have

¿ [1+z ]=z− z2

2+ z3

3− z 4

4+. . .

∴∈[ 11−t ]=¿ [1+ t1−t ]=t (1−t)−1−

t2(1−t)−2

2+ .. .

But (1 – t)-1 = 1 + t + t2 + . . . ∴∈[1+ t

1−t ]=t (1+t+t¿¿2+ .. .)− t2

2(1+2t ¿¿2)+ .. .¿¿

Considering only lower terms we have

t+ t2− t 2

2

36

¿ [ 11−t ]=t (1+ t2 ) (1.32)

Substituting equation (1.32) into (1.31)

KT ⟨N v ⟩q

⌊ t (1+ t2 )¿¿

Recall t=qv

KT ⟨N v ⟩q

⌊ qv (1+ q

2v )¿¿

Also q=43 π D 3 ⟨ N v ⟩

KT ⟨N v ⟩v

⌊1+4 π D3 ⟨N v ⟩3.2 v

¿¿

KT ⟨N v ⟩v [1+2 π D3 ⟨N v ⟩

3 v ]=p

KT ⟨N v ⟩v [(1+ 2π D3 ⟨N v ⟩

3v )−1]=p

KT ⟨N v ⟩v

1

[1+ 2π D3 ⟨N v ⟩3v ]

−1=p

p=KT ⟨ N v ⟩

v ⌊1+(−2π D3 ⟨ N v ⟩3v )¿

¿

Since the series expansion of

37

[1+ 2 π D 3 ⟨ N v ⟩3 v ]

−1

=[1+(−2π D3 ⟨N v ⟩

3v ]And also neglecting higher terms

∴ p=KT ⟨N v ⟩

[v−2π D3 ⟨N v ⟩3 v ] (1.33)

let b=2 π D 3 ⟨ N v ⟩

3

Equation (1.32) becomes

p≈KT ⟨N v ⟩v−b

(1.34)

Equation (1.34) is the Vander waal’s equation that expresses the

pressure in a classical gas of hard sphere. Vander waal’s

equation is valid for quantum mechanicals because correlation

exist even when there are no molecular forces in which case

Boyles-Charles law is not applicable.

REFERENCES

38

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Function Realizability:lattice model implications,Journal of

Physical Chemistry, Department of Chemistry, Princeton

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England, 1962.

[4] McQuarrie, D.A. Statistical Mechanics,Harper and Row: New

York, 1976. Chapter 13.

[5] Zhigilei leonid:Introduction to Atomistic

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[6] Padding T. Johan:Statistical mechanics of liquids,Institut de

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39

[7] Toda M., R. Kubo, Saito N:Equilibrium Statistical Physics I,

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www.physics.emory.edu/~weeks/idl/gofr/html.

40