A

Definite Integrals Frequently Metin Statistical Physics

A.1 Gamma-Function or Euler Integral of Second Kind

Γ(n) =

∞∫0

xn−1e−xdx; n > 0. (A.1)

Write down this integral for Γ(n + 1) and integrate up by parts once. As aresult, we get the following recurrent formula

Γ(n+ 1) = nΓ(n). (A.2)

For any integer number (n > 0) from (A.2) it follows that

Γ(n+ 1) = n · (n− 1) · (n− 2) . . . 1Γ(1). (A.3)

Inasmuch as the integral entering here is

Γ(1) =

∞∫0

e−xdx = 1, (A.4)

then (A.3) takes the formΓ(n+ 1) = n! (A.5)

For a semi-integer n = 2k+12 from (A.2) it follows that

Γ(

32

)=

12Γ(

12

); Γ(

52

)=

12· 32Γ(

12

);

Γ(

72

)=

12· 32· 52Γ(

12

).

(A.6)

364 A Definite Integrals Frequently Met in Statistical Physics

In the general case

Γ(

2k + 12

)=

(2k − 1)!!2k

Γ(1/2); k = 1, 2, 3, . . . (A.7)

Here the double factorial (2k− 1)!! is the product of subsequent odd numbersfrom 1 to (2k − 1), i.e. (2k − 1)!! = 1 · 3 · 5 · . . . · (2k − 1). According to thedefinition

Γ(1/2) =

∞∫0

x−1/2e−xdx. (A.8)

To calculate this integral introduce the replacement x1/2 = t. Then we get

Γ(1/2) = 2

∞∫0

e−t2dt. (A.9)

In as much as the latter integral equals√π/2, then

Γ(1/2) =√π. (A.10)

As a result, for an arbitrary semi-integer argument (A.7) takes the form

Γ(

2k + 12

)=

(2k − 1)!!2k

√π; k = 0, 1, 2, 3, . . . (A.11)

A.2 Integral of Type

In =

∞∫0

xne−ax2dx; a > 0, n ≥ 0 is an integer number. (A.12)

Carry out the replacement ax 2 = y. Then we have

In =12a−

n+12

∞∫0

e−yyn−1

2 dy =12a−

n+12 Γ

(n+ 1

2

). (A.13)

Hence for n = 0

I0 =12a−1/2Γ(1/2) =

12

√π

a, (A.14)

for even n, i.e. n = 2k

I2k =12a−

2k+12 Γ

(2k + 1

2

)(A.15)

A.3 Integral of Type 365

or [(see A.11)]

I2k =(2k − 1)!!

2k+1

√π

a2k+1, k ≥ 1 (A.16)

for odd n, i.e. n = (2k + 1)

I2k+1 =12

1ak+1

Γ (k + 1) =k!

2ak+1, k ≥ 0. (A.17)

Adduce values of the integral In in particular cases:

I1 =12a

; I2 =14

√π

a3/2; I3 =

12a2

; I1 =38

√π

a5/2. (A.18)

If in an integral (A.12) the boundaries change in the limits from −∞ to +∞,for odd n we get

∞∫−∞

xne−ax2dx = 0, (A.19)

and for even n ∞∫−∞

xne−ax2dx = 2

∞∫0

xne−ax2dx. (A.20)

A.3 Integral of Type

Kn =

∞∫0

xndxex − 1

, (A.21)

n is an integer or a semi-integer positive number. Transpose a certain part ofthe integrand:

1ex − 1

=e−x

1 − e−x= e−x(1 + e−x + e−2x + · · · ) = e−x

∞∑k=0

e−kx (A.22)

and substitute this series into (A.21). As a result we have

Kn =

∞∫0

xn∞∑

k=0

e−(k+1)xdx. (A.23)

If we introduce the replacement of variables (k + 1)x = t, then

Kn =∞∑

k=0

(k + 1)−(n+1)x

∞∫0

tne−tdt (A.24)

366 A Definite Integrals Frequently Met in Statistical Physics

and we getKn = Γ(n+ 1)ζ(n+ 1), (A.25)

where Γ(n) is the gamma-function [see (A.1)], and

ζ(n) =∞∑

k=1

1kn

(A.26)

is the Riemann function.All information about the Γ(n)- function are given in Appendix A.Adduce some values of ζ(n) function:

ζ(2) = π2

6 ; ζ(3) = 1, 202; ζ(4) = π4

90 ;

ζ(5) = 1, 037; ζ(3/2) = 2, 612; ζ(5/2) = 1, 341.(A.27)

If we take into account these values, we can find necessary values of the integralof the Kn type

K1/2 = Γ(3/2)ζ(3/2) = 2, 33;K1 = Γ(2)ζ(2) =π2

6; (A.28)

K3/2 = Γ(5/2)ζ(5/2) = 1, 78;K2 = Γ(3)ζ(3) = 2, 4; (A.29)

K3 = Γ(4)ζ(4) =π4

15. (A.30)

A.4 Integral of Type

Mn =

∞∫0

xnexdx(ex − 1)2

; n > 1. (A.31)

In order to calculate this integral, we expand a certain part of the integrandfunction into an infinitive series

(ex − 1)−2 = e−2x(1 − e−x)−2 = e−2x(1 + 2e−x + 3e−2x + · · · )or

(ex − 1)−2 = e−2x∞∑

m=0

(m+ 1)e−mx . (A.32)

Substituting (A.32) into (A.31), we get

Mn =∞∑

m=0

(m+ 1)

∞∫0

xne−(m+1)xdx. (A.33)

A.5 Integral of Type 367

Introducing the replacement of variables (m+ 1)x = t, we get

Mn =∞∑

m=0

(m+ 1)−n

∞∫0

tne−tdt;n > 1. (A.34)

If we take into account definitions of Γ(n) (A.1) and ζ(n) (A.26) functions,the integral takes the form

Mn = Γ(n+ 1)ζ(n). (A.35)

In particular cases

M3/2 = Γ(5/2)ζ(3/2) = 3, 48;M2 = Γ(3)ζ(2) =π2

3; (A.36)

M5/2 = Γ(7/2)ζ(5/2) = 4, 45;M3 = Γ(4)ζ(3) = 7, 21; (A.37)

M4 = Γ(5)ζ(4) =4π4

15. (A.38)

A.5 Integral of Type

Ln =

∞∫0

xndxex + 1

. (A.39)

To compute this integral, expand the expression (ex + 1)−1 into a series inpowers of e−x:

(ex + 1)−1 = e−x(1 + e−x)−1 = e−x∞∑

k=0

(−1)ke−kx. (A.40)

Having substituted the latter expression into (A.39), we get

Ln =∞∑

k=0

(−1)k

∞∫0

xne−(k+1)xdx. (A.41)

If we introduce the replacement of variables (k + 1)x = t, the integral takesthe form

Ln =∞∑

k=0

(−1)k

(k + 1)n+1

∞∫0

tne−tdt = Γ(n+ 1)∞∑

k=0

(−1)k

(k + 1)n+1. (A.42)

Transpose the series entering into this expression. To do this in (A.42) addand subtract a series consisting of k. As a result we get:

368 A Definite Integrals Frequently Met in Statistical Physics

∞∑k=0

(−1)k

(k + 1)n+1=∞∑

k=0

1(k + 1)n+1

− 2∞∑

m=0

1(2m+ 2)n+1

(A.43)

or∞∑

k=0

(−1)k

(k + 1)n+1= (1 − 2−n)

∞∑k=0

1(k + 1)n+1

= (1 − 2−n)ζ(n + 1). (A.44)

Substituting (A.44) into (A.42), we finally get

Ln = (1 − 2−n)Γ(n+ 1)ζ(n+ 1). (A.45)

Note that at n = 0 expression (A.45) turns into uncertainty, inasmuch asζ(1) = ∞.

However, at n = 0 integral (A.39) can be calculated immediately. Indeed,at n = 0

L0 =

∞∫0

dxex + 1

=

∞∫1

dyy(y + 1)

= − ln(

1 + y

y

)∣∣∣∣∞

1

= ln 2. (A.46)

In particular cases from (A.45) we find

L1/2 = (1 − 2−1/2)Γ(3/2)ζ(3/2) = 0.673, (A.47)

L1 =12Γ(2)ζ(2) =

π2

12, (A.48)

L2 = (1 − 2−2)Γ(3)ζ(3) = 1.8, (A.49)

L3 = (1 − 2−3)Γ(4)ζ(4) =7π4

120. (A.50)

B

Jacobian and Its Properties

The Jacobian, defined as

∂(u, υ)∂(x, y)

=

∣∣∣∣∣∣∣

∂u

∂x

∂u

∂y∂υ

∂x

∂υ

∂y

∣∣∣∣∣∣∣, (B.1)

possesses the following properties:

1.∂(u, υ)∂(x, y)

=(∂u

∂x

)y

(∂υ

∂y

)x

−(∂u

∂y

)x

(∂υ

∂x

)y

. (B.2)

2.∂(u, υ)∂(x, y)

= −∂(υ, u)∂(x, y)

;∂(u, υ)∂(x, y)

= −∂(u, υ)∂(y, x)

. (B.3)

3.∂(u, y)∂(x, y)

=(∂u

∂x

)y

. (B.4)

4.∂(u, υ)∂(x, y)

=∂(u, υ)∂(t, s)

∂(t, s)∂(x, y)

. (B.5)

5.ddt∂(u, υ)∂(x, y)

=∂(

dudt , υ

)∂(x, y)

+∂(u, dυ

dt

)∂(x, y)

. (B.6)

These properties of the Jacobian are frequently used when finding thermody-namic relationships and calculating thermodynamic coefficients.

Bibliograpy

1. L.D. Landau, E.M. Lifshitz, Statistical Physics of A Course of TheoreticalPhysics, vol 5 (Elsevier, Butterworth-Heinemann, 1980) [L.D. Landau, E.M.Lifshitz, Statistical Physics, vol 5 (Moscow, Fizmatlit, 2002) (in Russian)]

2. L.D. Landau, E.M. Lifshitz, Mechanics of A Course of Theoretical Physics, vol1 (Elsevier, Oxford, 1996)

3. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, of ACourse of Theoretical Physics, vol. 3 (Butterworth-Heinemann, Oxford, 2003)[L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, vol3 (Fizmatlit, Moscow, 2002) (in Russian)]

4. R. Kubo, Thermodynamics (North-Holland Publishing Company, Amsterdam,1968) [R. Kubo, Thermodynamics (Mir, Moscow, 1970) (in Russian)]

5. R. Kubo, Statistical Mechanics (North-Holland, Amsterdam, 1965) [R. Kubo,Statistical Mechanics (Mir, Moscow, 1970) (in Russian)]

6. I.P. Bazarov, Thermodynamics (Pergamon Press, Oxford, 1964) [I.P. Bazarov,Thermodynamics (Vishaya School, Moscow, 1991) (in Russian)]

7. K. Huang, Statistical Mechanics (Wiley, New York, 1987) [K. Huang, StatisticalMechanics (Mir, Moscow, 1966) (in Russian)]

8. E. Fermi, Thermodynamics. PH, 1973. [E. Fermi, Thermodynamics. Kharkov(1973) (in Russian)]

9. J.W. Gibbs, Elementary Principles in Statistical Mechanics (Yale Unimpressed,New Heaven, 1902) [J.W. Gibbs, Elementary Principles in Statistical Mechanics(Moscow, Nauka, 1946) (in Russian)]

10. J.W. Gibbs, Thermodynamics. Statistical Mechanics (Nauka, Moscow, 1982) (inRussian)]

11. F. Reif, Statistical Physics (McGraw-Hill, Moscow, 1970) [F. Reif, StatisticalPhysics (Nauka, Moscow, 1972) (in Russian)]

12. J.S. Blakemore, Semiconductor Statistics (Pergamon Press, Oxford, 1962) [J.S.Blakemore, Semiconductor Statistics (IL, Moscow, 1964) (in Russian)]

13. A.I. Anselm, Fundamentals of Statistical Physics and Thermodynamics (Nauka,Moscow, 1973) (in Russian)]

14. A.I. Anselm, Introduction to Semiconductor Theory (Prentice Hall, EnglewoodCliffs, NJ, 1982) [A.I. Anselm, Introduction to Semiconductor Theory (Nauka,Moscow, 1978) (in Russian)]

372 Bibliograpy

15. Yu.B. Rumer, M.Sh. Ryvkin, Thermodynamics, Statistical Physics and Kinet-ics (Mir Pub, Moscow, 1980) [Yu.B. Rumer, M.Sh. Ryvkin. Thermodynamics,Statistical Physics and Kinetics (Nauka, Moscow, 1977) (in Russian)]

16. B.M. Askerov, Electron Transport Phenomena in Semiconductors (World Sci-entific, Singapore, 1994) [B.M. Askerov, Electron Transport Phenomena inSemiconductors (Nauka, Moscow, 1985) (in Russian)]

17. Smirnov, A Course of Higher Mathematics, vol 3 (Pergamon Press, Oxford,1964), part 2. [Smirnov,A Course of Higher Mathematics (Nauka, Moscow, 1974)(in Russian)]

18. O. Madelung, Festkorpertheorie. I, II. (Springer, Heidelberg, 1988)[O. Madelung, Solid State Theory (Nauka, Moscow, 1980) (in Russian)]

19. K. Seeger, Semiconductor Physics (Springer, New York, 2004), 9th ed.[K. Seeger, Semiconductor Physics (Mir, Moscow, 1977) (in Russian)]

20. L.I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968) [L.I. Schiff,Quantum Mechanics (IL, Moscow, 1957) (in Russian)]

21. C. Kittel, Quantum Theory of Solids. (Wiley, New York, 1987), 2nd ed.[C. Kittel, Quantum Theory of Solids (Nauka, Moscow, 1980) (in Russian)]

22. R.E. Peierls, Quantum Theory of Solids (Clarendon Press, Oxford, 2001) [R.E.Peierls, Quantum Theory of Solids (IL, Moscow, 2002) (in Russian)]

23. A.I. Anselm, B.M. Askerov, Thermomagnetic phenomena in semimetals instrong magnetic field. Sov. Phys.-Solid State 2 2060 (1960) Chemical poten-tial and criterion of degeneracy. Soviet Phys.-Solid State 2 2571 (1960). FizikaTverdogo Tela (ru) 2(9), 2310 (1960); (11), 2821 (1960)

24. A.I. Anselm, B.M. Askerov, Quantum theory of Nernst-Ettingshausen insemiconductors. Fizika Tverdogo Tela (ru) 9 31 (1967)

25. B.M. Askerov, M. Cankurtaran, Isobaric specific heat and thermal expansion ofsolids in the debye approximation. Physica Status Solidi (b) 185 341–348 (1994)

26. M. Cankurtaran, B.M. Askerov, Equation of state, isobaric specific heat andthermal expansion of solids with polyatomic basis in the Einstein–Debyeapproximation. Physica Status Solidi (b) 194 499–507 (1996)

27. B.M. Askerov, M.M. Machmudov, Kh.A. Gasanov, Equation of state of an elec-tron gas and theory of the thermal voltage in a quantizing magnetic field.J Semicond. 32(3), 261 (1998). Fizika i Tekhnika Poluprovodnikov (ru) 32290 (1998)

Index

Absolute temperature, 4, 6, 12, 35–38,40, 96, 150, 214, 298

Acoustic phonons, 291, 348–351, 353,355

Adiabatic process, 34, 51, 55, 59, 73, 86,112, 288

Band theory of solids, 265Bohr magneton, 219, 258, 300Boltzmann distribution, 99, 142, 214,

217, 219, 221, 230, 232, 237, 249,254, 286

Boltzmann equation, 322–330, 332, 333,336, 338–340, 344, 347, 359, 360

Bose–Einstein condensation, 232, 277,280, 281

Bose–Einstein distribution, 213, 222,231–233, 237, 283, 292

Boyle temperature, 163, 167

Caloric equation of state, 5, 49, 68, 69,111, 138, 139, 163, 191, 236–238,312

Carnot cycle, 51–54, 78Characteristic temperature, 123, 126,

129, 131, 132, 197, 199, 254Classical systems, 6, 7, 12, 13, 21, 22,

24, 29, 56, 93, 94, 97, 99, 100, 102,104, 106, 213, 214

Clausius postulate, 51, 54–56Conductivity tensors, 340Configurational integral, 158, 159Conwell and Weisskopf method, 346Current density, 322, 340, 341, 359, 362

Debye function, 199–202, 207–209, 211,295

Debye model, 190, 194, 197, 199, 206,293

Debye screening, 172Deformation potential method, 348, 351Degenerate electron gas, 255, 260, 261,

263–265, 304, 307, 308, 310–314,317, 318, 326, 333–336, 347, 350

Density of states, 30, 98, 236, 263, 279,303, 304, 308, 309, 358

Diamagnetic susceptibility, 298, 315,317

Distribution function, 13–21, 23–30, 32,33, 56, 93, 94, 96–100, 106, 107,142, 213, 214, 216, 229–232, 240,241, 244, 245, 249, 259, 268, 269,284, 305, 306, 313, 315, 322, 324,325, 328, 331, 333, 336, 346, 352

Disturbing force, 326, 331, 332, 336–339,361

Dulong–Petit law, 193, 195

Effective mass approximation, 298Einstein model, 196, 197Equilibrium distribution function, 321Ergodic hypothesis, 14

Fermi boundary energy, 230, 231, 242,243, 246, 308, 311, 316, 319

Fermi–Dirac distribution, 213, 222,230–232, 237, 249, 252, 259, 268,321, 328

374 Index

Galvanomagnetic effects, 342Gibbs canonical distribution, 19, 93, 97,

99, 101, 103, 108, 213, 215, 229Gibbs paradox, 101, 112, 113, 115Gibbs statistical ensemble, 16Gruneisen relationship, 194, 210

Harmonic oscillator, 8–11, 128, 129,187, 196, 204, 205, 290, 299, 300

Impurity semiconductors, 267, 272Intrinsic semiconductors, 267–270, 275Irreversible processes, 31, 34, 53, 56, 61,

62, 113

Kelvin postulate, 51, 54–56Kinetic coefficients, 322, 340, 342–344

Landau diamagnetism, 298, 314, 315Langevin function, 136, 139, 143, 146,

316Le Chatelier–Brown principle, 73Liouville theorem, 13, 16, 18, 24, 25,

323

Magnetic length, 299, 301, 316Magnetocaloric effect, 83, 86Microcanonical distribution, 19–21, 26,

27, 32, 94, 95, 98, 102–104, 107,108

Mixture of ideal gases, 112, 113

Negative absolute temperature, 150,151, 154, 155

Nernst principle, 74–78, 83, 86, 195,197, 210, 282, 288

Non-degenerate electron gas, 247, 255,260, 268, 306, 307, 310, 311, 313,315, 316, 318, 319, 326, 333, 336,347, 350, 353, 357

Nonequilibrium distribution function,321, 323–325, 330, 332

Nonpolar optical phonons, 332, 348,351–354, 358

Paramagnetic susceptibility, 219, 258,260, 261, 298, 317, 318

Pauli paramagnetism, 219, 220, 258Planck formula, 196, 220, 221, 231, 285,

286Polar optical phonons, 348, 354–357

Quantum limit, 302, 307–309, 311, 312,314, 317–319

Quantum systems, 6, 9, 11, 22, 24, 26,27, 56, 93–95, 97, 99, 102, 104,213, 214

Quasi-classical approximation, 11, 12,29, 146, 301, 306, 311, 314–316,318, 321

Relaxation time, 3, 15, 31, 327–332, 338,344–346, 348, 350–353, 355–361

Reversible processes, 31, 34, 35, 51–53

Scattering mechanisms, 325, 332, 344,345, 354, 357–359

Schrodinger equation, 9, 213, 223, 224,226, 265, 298, 299, 344

Statistical matrix, 22–25, 57Statistical weights, 1, 27–30, 50, 75, 95

Temperature of degeneracy, 240, 243,244, 247, 250, 251, 255–257, 261,264, 279, 280, 326, 350

Thermal equation of state, 5, 49, 50,60, 61, 68, 102, 110, 117, 120,138, 139, 162, 166, 191, 194, 204,208, 209, 211, 234–238, 262, 295,310–312

Thermodynamic inequalities, 68–71, 73,75, 101

Thermomagnetic effects, 342Transport phenomena, 338, 344

Ultra-relativistic electron gas, 262–265

Van der Waals gas, 72, 163–165, 167,169

Virial coefficient, 162–166, 170