Statistical Mechanics Pathria Beale Solutions Manual

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Problems from Pathria & Beale's Statistical Mechanics (3rd Edition). Read the disclaimer before use.

Chapter 1

Pathria (R.K.P.) and the publishers agreed it was time for a third edition to incorporate the important changes that had occurred in the field since the publication of the second edition and invited Paul B. Beale (P.D.B.) to join as coauthor. R K Pathria Solutions. Below are Chegg supported textbooks by R K Pathria. Select a textbook to see worked-out Solutions. Books by R K Pathria with Solutions. Author(s) Statistical Mechanics 3rd Edition 209 Problems solved: Paul D Beale, R K Pathria: Statistical Mechanics 3rd Edition 209 Problems solved: Paul D Beale, R K Pathria: Join.

Solution: Pathria 1.3: Two systems A and B, of identical composition, are brought together and allowed to exchange both energy and particles, keeping volumes V_A and V_B constant. Show that the minimum value of the quantity (d E_A / d N_A) is given by

dfrac{mu_{A} T_{B} -mu_{B} T_{A}}{T_{B} - T_{A}} ,

where the mu's and the T's are the respective chemical potentials and temperatures.

Solution: Pathria 1.8: Consider a system of quasiparticles whose energy eigenvalues are given by

varepsilon(n) = n h nu; quad n=0,1,2,dots

Obtain an asymptotic expression for the number Omega of this system for a given number N of the quasiparticles and a given total energy E. Determine the temperature T of the system as a function of E/N and h nu, and examine the situation for which E/(N h nu) gg 1.

Solution: Pathria 1.15: We have seen that the (P, V)-relationship during a reversible adiabatic process in an ideal gas is governed by the exponent gamma, such that

P V^gamma = text{const.}

Consider a mixture of two ideal gases, with mole fractions f_1 and f_2 and respective exponents gamma_1 and gamma_2. Show that the effective exponent for the mixture is given by

dfrac{1}{gamma -1} = dfrac{f_1}{gamma_1 - 1} + dfrac{f_2}{gamma_2 - 1} .

Chapter 2

Solution: Pathria 2.7: Derive (i) an asymptotic expression for the number of ways in which a given energy E can be distributed among a set of N one-dimensional harmonic oscillators, the energy eigenvalues of the oscillators being (n+frac{1}{2})hbar omega;,n = 0, 1, 2, dots, and (ii) the corresponding expression for the 'volume' of the relevant region of the phase space of this system. Establish the correspondence between the two results, showing that the conversion factor omega_{0} is precisely h^N.

Solution: Pathria 2.8: Following the method of Appendix C, replacing equation (C.4) by the integral

intlimits_0^infty e^{-r} r^2 dr =2 ,

show that

V_{3N}=intlimits_{0leqsumlimits_{i=1}^{N}r_{i}leq R}^{}dotsintprodlimits_{i=1}^{N}(4pi r_i^2 dr_i)=(8pi R^{3})^{N}/(3N)! .

Using this result, compute the 'volume' of the relevant region of the phase space of an extreme relativistic gas (varepsilon = pc ) of N particles moving in three dimensions. Hence, derive expressions for the various thermodynamic properties of this system and compare your results with those of Problem 1.7.

Solution: Pathria 2.9:
(a) Solve the integral

intlimits_{0leqsumlimits_{i=1}^{3N}|x_{i}|leq R}^{}dotsint(dx_1 dots dx_{3N})

and use it to determine the 'volume' of the relevant region of the phase space of an extreme relativistic gas (varepsilon = pc) of 3N particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy E among this system of particles and show that, asymptotically, omega_0 = h^{3N}.

(b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8.

(Note: Watch out — I made a mistake and did not receive full credit)

Chapter 3

Solution: Pathria 3.7: Prove that, quite generally,

C_P - C_V = -kcfrac{Bigg[cfrac{partial}{partial T}Bigg{TBigg(cfrac{partial ln Q}{partial V}Bigg)_T Bigg} Bigg]^2 _V }{Bigg( cfrac{partial^2 ln Q}{partial V^2} Bigg)_T}>0 .

Verify that the value of this quantity for a classical ideal classical gas is Nk.(oops: the 3rd line should begin with, 'use 1.3.17 and 1.3.18.')

Solution: Pathria 3.18: Show that for a system in the canonical ensemble

langle(Delta E)^3 rangle = k^2 Bigg {T^4 left( cfrac{partial C_V}{partial T} right)_V + 2T^3 C_V Bigg }.

Verify that for an ideal gas

Bigg langle bigg( dfrac{Delta E}{U} bigg)^2 Bigg rangle = dfrac{2}{3N} qquad text{and} qquad Bigg langle bigg( dfrac{Delta E}{U} bigg)^3 Bigg rangle = dfrac{8}{9N^2}.

(Note: I did not verify the ideal gas simplifications)

Solution: Pathria 3.26: The energy eigenvalues of an s-dimensional harmonic oscillator can be written as

varepsilon_{j}=(j+s/2)hbaromega;,j=0,1,2,dots

Show that the jth energy level has a multiplicity (j + s - 1)!/(j!(s - 1)!). Evaluate the partition function, and the major thermodynamic properties, of a system of N such oscillators, and compare your results with a corresponding system of sN one-dimensional oscillators. Show, in particular, that the chemical potential mu_{s}=smu_{1}.

Solution: Pathria 3.31: Study, along the lines of Section 3.8, the statistical mechanics of a system of N 'Fermi oscillators,' which are characterized by only two eigenvalues, namely 0 and varepsilon.

Chapter 4

Solution: Pathria 4.4: The probability that a system in the grand canonical ensemble has exactlyN particles is given by

p(N) = cfrac{z^{N} Q_{N}(V,T)}{mathbb{Q}(z,V,T)}.

Verify this statement and show that in the case of a classical, ideal gas the distribution of particles among the members of a grand canonical ensemble is identically a Poisson distribution. Calculate the root-mean-square value of Delta N for this system both from the general formula (4.5.3) and from the Poisson distribution, and show that the two results are the same. (mathbb{Q} is the grand partition function.)

Solution: Pathria 4.7: Consider a classical system of noninteracting, diatomic molecules enclosed in a box of volume V at temperature T . The Hamiltonian of a single molecule is given by

H(mathbf{r}_1 , mathbf{r}_2 , mathbf{p}_1 , mathbf{p}_2) = cfrac{1}{2m} (p_{1}^2 + p_{2}^2) + cfrac{1}{2} K |mathbf{r}_1 - mathbf{r}_2 |^2.

Study the thermodynamics of this system, including the dependence of the quantity langle r_{12}^2 rangleon T.

Chapter 5

Solution: Pathria 5.1: Evaluate the density matrix rho_{mn} of an electron spin in the representation that makes hat{sigma}_x diagonal. Next, show that the value of langle sigma_z rangle , resulting from this representation, is precisely the same as the one obtained in Section 5.3.

Solution: Pathria 5.5: Show that in the first approximation the partition function of a system of N noninteracting, indistinguishable particles is given by

Q_{N}(V,T) = dfrac{1}{N! lambda^{3N}}Z_{N}(V,T),

where

Z_{N}(V,T) = int exp{bigg{-beta sum_{i

nu_{s}(r) being the statistical potential (5.5.28). Hence evaluate the first-order correction to the equation of state of this system.

Chapter 6

Solution: Pathria 6.3: Refer to Section 6.2 and show that, if the occupation number n_varepsilon of an energy level varepsilon is restricted to the values 0, 1, . . . ,l, then the mean occupation number of that level is given by

langle n_varepsilon rangle = dfrac{1}{z^{-1} e^{beta varepsilon}-1} - dfrac{l+1}{(z^{-1}e^{beta varepsilon})^{l+1}-1}.

Check that while l = 1 leads to langle n_varepsilon rangle _{F.D.}, l rightarrow infty leads to langle n_varepsilon rangle _{B.E.}.

Solution: Pathria 6.8:An ideal classical gas composed of N particles, each of mass m, is enclosed in a vertical cylinder of height L placed in a uniform gravitational field (of acceleration g) and is in thermal equilibrium; ultimately, both N and N rightarrow infty. Evaluate the partition function of the gas and derive expressions for its major thermodynamic properties. Explain why the specific heat of this system is larger than that of a corresponding system in free space.

Solution: Pathria 6.11:
(a) Show that the momentum distribution of particles in a relativistic Boltzmannian gas, with varepsilon = c(p^2 + m_{0}^2 c^2)^{1/2} is given by

f(mathbf{p})dmathbf{p} = C e^{-beta c (p^2 + m_{0}^2 c^2)^{1/2}} p^2 dp,

with the normalization constant

C = dfrac{beta}{m_{0}^2 c K_2 (beta m_0 c^2)},

K_nu (z) being a modified Bessel function.

(b) Check that in the nonrelativistic limit (kT ll m_{0}c^2) we recover the Maxwellian distribution,

f(mathbf{p})dmathbf{p} =bigg(dfrac{beta}{2pi m_0}bigg)^{3/2}e^{-beta p^2/2m_0}(4pi p^2 dp),

while in the extreme relativistic limit (kT gg m_{0}c^2) we obtain

f(mathbf{p})dmathbf{p} =dfrac{(beta c)^3}{8pi}e^{-beta p c}(4pi p^2 dp).

(c) Verify that, quite generally,

langle pu rangle=3kT.

Solution: Pathria 6.19: What is the probability that two molecules picked at random from a Maxwellian gas will have a total energy between E and E+dE? Verify that langle E rangle =3kT.

Chapter 7

Solution: Pathria 7.7: Evaluate the quantities(partial^2 P/ partial T^2)_{nu}, (partial^2 mu / partial T^2)_{nu}, and (partial^2 mu / partial T^2)_{P} for an ideal Bose gas and check that your results satisfy the thermodynamic relationships

C_{V} = VT bigg( cfrac{partial^2 P}{partial T^2}bigg)_{nu} - NT bigg( cfrac{partial^2 mu}{partial T^2}bigg)_{nu},

and

C_{P} = -NT bigg( cfrac{partial^2 mu}{partial T^2}bigg)_{P}.

Examine the behavior of these quantities as T rightarrow T_c from above and from below.

Solution: Pathria 7.14: Consider an n-dimensional Bose gas whose single-particle energy spectrum is given by varepsilon propto p^{s}, where s is some positive number. Discuss the onset of Bose–Einstein condensation in this system, especially its dependence on the numbers n and s. Study the thermodynamic behavior of this system and show that,

P=cfrac{s}{n}cfrac{U}{V}, quad C_V (T rightarrow infty)=cfrac{n}{s}Nk, quad text{and} quad C_{P}(Trightarrow infty)=bigg(cfrac{n}{s}+1bigg)Nk.

(Note: My derivation of the density of states here is flawed. Problem 8.10 is basically the same, but for Fermi-Dirac statistics, and that solution has a much better derivation of the density of states. See below.)

Solution: Pathria 7.20: The (canonical) partition function of the blackbody radiation may be written as

Q(V,T)=displaystyleprodlimits_{omega} Q_1 (omega, T),

so that

ln Q(V,T) = displaystylesumlimits_{omega} ln Q_1 (omega , T) approx intlimits_0^infty ln Q_1 (omega, T) g(omega) domega ;

here, Q_1 (omega , T) is the single-oscillator partition function given by equation (3.8.14) and g(omega) is the density of states given by equation (7.3.2). Using this information, evaluate the Helmholtz free energy of the system and derive other thermodynamic properties such as the pressure P and the (thermal) energy density U/V. Compare your results with the ones derived in Section 7.3 from the q-potential of the system.

Chapter 8

Solution: Pathria 8.10: Consider an ideal Fermi gas, with energy spectrum varepsilon propto p^s, contained in a box of 'volume' V in a space of n dimensions. Show that, for this system,

(a)PV = dfrac{s}{n}U;

(b)dfrac{C_V}{Nk}=dfrac{n}{s}bigg( dfrac{n}{s}+1 bigg) dfrac{f_{(n/s)+1}(z)}{f_{n/s}(z)}-bigg( dfrac{n}{s} bigg)^2 dfrac{f_{n/s}(z)}{f_{(n/s)-1}(z)};

(c) dfrac{C_P - C_V}{Nk}=bigg( dfrac{sC_V}{nNk} bigg)^2 dfrac{f_{(n/s)-1}(z)}{f_{n/s}(z)};

(d) the equation of an adiabat is PV^{1+(s/n)}=text{const.}, and

(e) the index (1+(s/n)) in the foregoing equation agrees with the ratio (C_p/C_V) of the gas only when T gg T_F. On the other hand, when T ll T_F, the ratio (C_p/C_V) simeq 1+(pi^2/3)(kT/varepsilon_F)^2, irrespective of the values of s and n.

(Note: I did not know how to do parts c or e.)

Solution: Pathria 8.12: Show that, in two dimensions, the specific heat C_V(N,T) of an ideal Fermi gas is identical to the specific heat of an ideal Bose gas, for allN and T.
[Hint: It will suffice to show that, for given N and T, the thermal energies of the two systems differ at most by a constant. For this, first show that the fugacities, z_F and z_B, of the two systems are mutually related:

(1+z_F)(1-z_B)=1,quadtext{i.e.,}quad z_B=z_F/(1+z_F).

Next, show that the functions f_2(z_F) and g_2(z_B) are also related:

f_2(z_F) = displaystyleintlimits_0^{z_F}dfrac{ln(1+z_F)}{z}dz

=g_2 bigg( dfrac{z_F}{1+z_F} bigg)+dfrac{1}{2} ln^2(1+z_F).

It is now straightforward to show that

E_F(N,T)=E_B(N,T) + text{const.},

the constant being E_F(N,0).]

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Solution: Pathria 8.13: Show that, quite generally, the low-temperature behavior of the chemical potential, the specific heat, and the entropy of an ideal Fermi gas is given by

mu simeq varepsilon_F bigg[ 1- dfrac{pi^2}{6}bigg( dfrac{partial ln a(varepsilon)}{partial ln varepsilon} bigg)_{varepsilon=varepsilon_F} bigg(dfrac{kT}{varepsilon_F}bigg)^2 bigg] ,

and

C_V simeq S simeq dfrac{pi^2}{3}k^2 T a(varepsilon_F),

where a(varepsilon) is the density of (the single-particle) states in the system. Examine these results for a gas with energy spectrum varepsilon propto p^s, confined to a space of n dimensions, and discuss the special cases: s=1 and 2, with n=2 and 3.
[Hint: Use equation (E.18) from Appendix E.]

Chapter 10

Solution: Pathria 10.3:
(a) Show that for a gas obeying van der Waals equation of state (10.3.9),

C_P-C_V=Nkbigg{ 1-dfrac{2a}{kTnu^3}(nu-b)^2 bigg}^{-1}.

(b) Also show that, for a van der Waals gas with constant specific heat C_V, an adiabatic process conforms to the equation

(nu - b)T^{C_V/Nk}=text{const};

compare with equation (1.4.30).

(c) Further show that the temperature change resulting from an expansion of the gas (into vacuum) from volume V_1 to volume V_2 is given by

T_2-T_1=dfrac{N^2 a}{C_V}bigg( dfrac{1}{V_2} - dfrac{1}{V_1} bigg).

Solution: Pathria 10.5: Show that the first-order Joule-Thomson coefficient of a gas is given by the formula

bigg( dfrac{partial T}{partial P}bigg)_H = dfrac{N}{C_P} bigg(T dfrac{partial (a_2 lambda^3 )}{partial T} -a_2 lambda^3 bigg),

where a_2(T) is the second virial coefficient of the gas and H its enthalpy; see equation (10.2.1). Derive an explicit expression for the Joule-Thomson coefficient in the case of a gas with interparticle interaction

u(r) = Bigg{ begin{array}{cl} +infty & mathrm{for}~0

and discuss the temperature dependence of this coefficient.

(Note: I forgot to 'discuss the temperature dependence of this coefficient')

Chapter 12

Solution: Pathria 12.3: Consider a nonideal gas obeying a modified van der Waals equation of state

(P + a / nu^{n})(nu - b) = RT quad (n>1) .

Examine how the critical constants P_c, nu_c, and T_c, and the critical exponents beta, gamma, gamma^{prime}, and delta of this system depend on the number n.

(Note: I did not attempt the critical exponent part of the problem)

Solution: Pathria 12.20: Consider a system with a modified expression for the Landau free energy, namely

psi_h(t,m) = -hm +q(t)+r(t)m^2 + s(t)m^4 + u(t) m^6 ,

with u(t) a fixed positive constant. Minimize psi with respect to the variable m and examine the spontaneous magnetization m_0 as a function of the parameters r and s. In particular, show the following:

(a) For r>0 and s>-(3ur)^{1/2},~m_0=0 is the only real solution.

(b) For r>0 and -(4ur)^{1/2} or pm m_1, where m_1^2 = frac{sqrt{(s^2-3ur)-s}}{3u}. However, the minimum of psi at m_0 = 0 is lower than the minima at m_0 = pm m_1, so the ultimate equilibrium value of m_0 is 0.

(c) For r>0 and s = -(4ur)^{1/2},~m_0=0 or pm(r/u)^{1/4}. Now the minimum of psi at m_0 = 0 is of the same height as the ones at m_0 = pm (r/u)^{1/4}, so a nonzero spontaneous magnetization is as likely to occur as the zero one.

(d) For r>0 and s < -(4ur)^{1/2},~m_0= pm m_1 — which implies a first-order phase transition (becuase the two possible states availible here differ by a finite amount in m). The line s=-(4ur)^{1/2}, with r positive, is generally referred to as a 'line of first-order phase transitions.'

(e) For r=0 and s<0, m_0 = pm(2|s|/3u)^{1/2}.

(f) For r<0, m_0 = pm m_1 for all s. As rrightarrow 0, m_1 rightarrow 0 if s is positive.

(g) For r=0 and s>0, m_0=0 is only solution. [sic] Combining this result with (f), we conclude that the line r=0, with s positive, is a 'line of second-order phase transitions,' for the two states available here differ by a vanishing amount in m.

The lines of first-order phase transitions and second-order phase transitions meet at the point (r=0,s=0), which is commonly referred to as a tricritical point (Griffiths, 1970).

Chapter 15

Solution: Pathria 15.1:Making use of expressions (15.1.11) and (15.1.12) for Delta S and Delta P, and expressions (15.1.14) for overline{(Delta T)^2}, overline{(Delta V)^2}, and overline{(Delta T Delta V)}, show that

(a)overline{(Delta T Delta S)} = kT;

(b)overline{(Delta P Delta V)} = -kT;

(c)overline{(Delta S Delta V)} = kT(partial V/partial T)_P;

(d)overline{(Delta P Delta T)} = kT^2C_V^{-1}(partial P/partial T)_V.

Chapter 2

Solution: Pathria 2.7: Derive (i) an asymptotic expression for the number of ways in which a given energy E can be distributed among a set of N one-dimensional harmonic oscillators, the energy eigenvalues of the oscillators being (n+frac{1}{2})hbar omega;,n = 0, 1, 2, dots, and (ii) the corresponding expression for the 'volume' of the relevant region of the phase space of this system. Establish the correspondence between the two results, showing that the conversion factor omega_{0} is precisely h^N.

Solution: Pathria 2.8: Following the method of Appendix C, replacing equation (C.4) by the integral

intlimits_0^infty e^{-r} r^2 dr =2 ,

show that

V_{3N}=intlimits_{0leqsumlimits_{i=1}^{N}r_{i}leq R}^{}dotsintprodlimits_{i=1}^{N}(4pi r_i^2 dr_i)=(8pi R^{3})^{N}/(3N)! .

Using this result, compute the 'volume' of the relevant region of the phase space of an extreme relativistic gas (varepsilon = pc ) of N particles moving in three dimensions. Hence, derive expressions for the various thermodynamic properties of this system and compare your results with those of Problem 1.7.

Solution: Pathria 2.9:
(a) Solve the integral

intlimits_{0leqsumlimits_{i=1}^{3N}|x_{i}|leq R}^{}dotsint(dx_1 dots dx_{3N})

and use it to determine the 'volume' of the relevant region of the phase space of an extreme relativistic gas (varepsilon = pc) of 3N particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy E among this system of particles and show that, asymptotically, omega_0 = h^{3N}.

(b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8.

(Note: Watch out — I made a mistake and did not receive full credit)

Chapter 3

Solution: Pathria 3.7: Prove that, quite generally,

C_P - C_V = -kcfrac{Bigg[cfrac{partial}{partial T}Bigg{TBigg(cfrac{partial ln Q}{partial V}Bigg)_T Bigg} Bigg]^2 _V }{Bigg( cfrac{partial^2 ln Q}{partial V^2} Bigg)_T}>0 .

Verify that the value of this quantity for a classical ideal classical gas is Nk.(oops: the 3rd line should begin with, 'use 1.3.17 and 1.3.18.')

Solution: Pathria 3.18: Show that for a system in the canonical ensemble

langle(Delta E)^3 rangle = k^2 Bigg {T^4 left( cfrac{partial C_V}{partial T} right)_V + 2T^3 C_V Bigg }.

Verify that for an ideal gas

Bigg langle bigg( dfrac{Delta E}{U} bigg)^2 Bigg rangle = dfrac{2}{3N} qquad text{and} qquad Bigg langle bigg( dfrac{Delta E}{U} bigg)^3 Bigg rangle = dfrac{8}{9N^2}.

(Note: I did not verify the ideal gas simplifications)

Solution: Pathria 3.26: The energy eigenvalues of an s-dimensional harmonic oscillator can be written as

varepsilon_{j}=(j+s/2)hbaromega;,j=0,1,2,dots

Show that the jth energy level has a multiplicity (j + s - 1)!/(j!(s - 1)!). Evaluate the partition function, and the major thermodynamic properties, of a system of N such oscillators, and compare your results with a corresponding system of sN one-dimensional oscillators. Show, in particular, that the chemical potential mu_{s}=smu_{1}.

Solution: Pathria 3.31: Study, along the lines of Section 3.8, the statistical mechanics of a system of N 'Fermi oscillators,' which are characterized by only two eigenvalues, namely 0 and varepsilon.

Chapter 4

Solution: Pathria 4.4: The probability that a system in the grand canonical ensemble has exactlyN particles is given by

p(N) = cfrac{z^{N} Q_{N}(V,T)}{mathbb{Q}(z,V,T)}.

Verify this statement and show that in the case of a classical, ideal gas the distribution of particles among the members of a grand canonical ensemble is identically a Poisson distribution. Calculate the root-mean-square value of Delta N for this system both from the general formula (4.5.3) and from the Poisson distribution, and show that the two results are the same. (mathbb{Q} is the grand partition function.)

Solution: Pathria 4.7: Consider a classical system of noninteracting, diatomic molecules enclosed in a box of volume V at temperature T . The Hamiltonian of a single molecule is given by

H(mathbf{r}_1 , mathbf{r}_2 , mathbf{p}_1 , mathbf{p}_2) = cfrac{1}{2m} (p_{1}^2 + p_{2}^2) + cfrac{1}{2} K |mathbf{r}_1 - mathbf{r}_2 |^2.

Study the thermodynamics of this system, including the dependence of the quantity langle r_{12}^2 rangleon T.

Chapter 5

Solution: Pathria 5.1: Evaluate the density matrix rho_{mn} of an electron spin in the representation that makes hat{sigma}_x diagonal. Next, show that the value of langle sigma_z rangle , resulting from this representation, is precisely the same as the one obtained in Section 5.3.

Solution: Pathria 5.5: Show that in the first approximation the partition function of a system of N noninteracting, indistinguishable particles is given by

Q_{N}(V,T) = dfrac{1}{N! lambda^{3N}}Z_{N}(V,T),

where

Z_{N}(V,T) = int exp{bigg{-beta sum_{i

nu_{s}(r) being the statistical potential (5.5.28). Hence evaluate the first-order correction to the equation of state of this system.

Chapter 6

Solution: Pathria 6.3: Refer to Section 6.2 and show that, if the occupation number n_varepsilon of an energy level varepsilon is restricted to the values 0, 1, . . . ,l, then the mean occupation number of that level is given by

langle n_varepsilon rangle = dfrac{1}{z^{-1} e^{beta varepsilon}-1} - dfrac{l+1}{(z^{-1}e^{beta varepsilon})^{l+1}-1}.

Check that while l = 1 leads to langle n_varepsilon rangle _{F.D.}, l rightarrow infty leads to langle n_varepsilon rangle _{B.E.}.

Solution: Pathria 6.8:An ideal classical gas composed of N particles, each of mass m, is enclosed in a vertical cylinder of height L placed in a uniform gravitational field (of acceleration g) and is in thermal equilibrium; ultimately, both N and N rightarrow infty. Evaluate the partition function of the gas and derive expressions for its major thermodynamic properties. Explain why the specific heat of this system is larger than that of a corresponding system in free space.

Solution: Pathria 6.11:
(a) Show that the momentum distribution of particles in a relativistic Boltzmannian gas, with varepsilon = c(p^2 + m_{0}^2 c^2)^{1/2} is given by

f(mathbf{p})dmathbf{p} = C e^{-beta c (p^2 + m_{0}^2 c^2)^{1/2}} p^2 dp,

with the normalization constant

C = dfrac{beta}{m_{0}^2 c K_2 (beta m_0 c^2)},

K_nu (z) being a modified Bessel function.

(b) Check that in the nonrelativistic limit (kT ll m_{0}c^2) we recover the Maxwellian distribution,

f(mathbf{p})dmathbf{p} =bigg(dfrac{beta}{2pi m_0}bigg)^{3/2}e^{-beta p^2/2m_0}(4pi p^2 dp),

while in the extreme relativistic limit (kT gg m_{0}c^2) we obtain

f(mathbf{p})dmathbf{p} =dfrac{(beta c)^3}{8pi}e^{-beta p c}(4pi p^2 dp).

(c) Verify that, quite generally,

langle pu rangle=3kT.

Solution: Pathria 6.19: What is the probability that two molecules picked at random from a Maxwellian gas will have a total energy between E and E+dE? Verify that langle E rangle =3kT.

Chapter 7

Solution: Pathria 7.7: Evaluate the quantities(partial^2 P/ partial T^2)_{nu}, (partial^2 mu / partial T^2)_{nu}, and (partial^2 mu / partial T^2)_{P} for an ideal Bose gas and check that your results satisfy the thermodynamic relationships

C_{V} = VT bigg( cfrac{partial^2 P}{partial T^2}bigg)_{nu} - NT bigg( cfrac{partial^2 mu}{partial T^2}bigg)_{nu},

and

C_{P} = -NT bigg( cfrac{partial^2 mu}{partial T^2}bigg)_{P}.

Examine the behavior of these quantities as T rightarrow T_c from above and from below.

Solution: Pathria 7.14: Consider an n-dimensional Bose gas whose single-particle energy spectrum is given by varepsilon propto p^{s}, where s is some positive number. Discuss the onset of Bose–Einstein condensation in this system, especially its dependence on the numbers n and s. Study the thermodynamic behavior of this system and show that,

P=cfrac{s}{n}cfrac{U}{V}, quad C_V (T rightarrow infty)=cfrac{n}{s}Nk, quad text{and} quad C_{P}(Trightarrow infty)=bigg(cfrac{n}{s}+1bigg)Nk.

(Note: My derivation of the density of states here is flawed. Problem 8.10 is basically the same, but for Fermi-Dirac statistics, and that solution has a much better derivation of the density of states. See below.)

Solution: Pathria 7.20: The (canonical) partition function of the blackbody radiation may be written as

Q(V,T)=displaystyleprodlimits_{omega} Q_1 (omega, T),

so that

ln Q(V,T) = displaystylesumlimits_{omega} ln Q_1 (omega , T) approx intlimits_0^infty ln Q_1 (omega, T) g(omega) domega ;

here, Q_1 (omega , T) is the single-oscillator partition function given by equation (3.8.14) and g(omega) is the density of states given by equation (7.3.2). Using this information, evaluate the Helmholtz free energy of the system and derive other thermodynamic properties such as the pressure P and the (thermal) energy density U/V. Compare your results with the ones derived in Section 7.3 from the q-potential of the system.

Chapter 8

Solution: Pathria 8.10: Consider an ideal Fermi gas, with energy spectrum varepsilon propto p^s, contained in a box of 'volume' V in a space of n dimensions. Show that, for this system,

(a)PV = dfrac{s}{n}U;

(b)dfrac{C_V}{Nk}=dfrac{n}{s}bigg( dfrac{n}{s}+1 bigg) dfrac{f_{(n/s)+1}(z)}{f_{n/s}(z)}-bigg( dfrac{n}{s} bigg)^2 dfrac{f_{n/s}(z)}{f_{(n/s)-1}(z)};

(c) dfrac{C_P - C_V}{Nk}=bigg( dfrac{sC_V}{nNk} bigg)^2 dfrac{f_{(n/s)-1}(z)}{f_{n/s}(z)};

(d) the equation of an adiabat is PV^{1+(s/n)}=text{const.}, and

(e) the index (1+(s/n)) in the foregoing equation agrees with the ratio (C_p/C_V) of the gas only when T gg T_F. On the other hand, when T ll T_F, the ratio (C_p/C_V) simeq 1+(pi^2/3)(kT/varepsilon_F)^2, irrespective of the values of s and n.

(Note: I did not know how to do parts c or e.)

Solution: Pathria 8.12: Show that, in two dimensions, the specific heat C_V(N,T) of an ideal Fermi gas is identical to the specific heat of an ideal Bose gas, for allN and T.
[Hint: It will suffice to show that, for given N and T, the thermal energies of the two systems differ at most by a constant. For this, first show that the fugacities, z_F and z_B, of the two systems are mutually related:

(1+z_F)(1-z_B)=1,quadtext{i.e.,}quad z_B=z_F/(1+z_F).

Next, show that the functions f_2(z_F) and g_2(z_B) are also related:

f_2(z_F) = displaystyleintlimits_0^{z_F}dfrac{ln(1+z_F)}{z}dz

=g_2 bigg( dfrac{z_F}{1+z_F} bigg)+dfrac{1}{2} ln^2(1+z_F).

It is now straightforward to show that

E_F(N,T)=E_B(N,T) + text{const.},

the constant being E_F(N,0).]

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Solution: Pathria 8.13: Show that, quite generally, the low-temperature behavior of the chemical potential, the specific heat, and the entropy of an ideal Fermi gas is given by

mu simeq varepsilon_F bigg[ 1- dfrac{pi^2}{6}bigg( dfrac{partial ln a(varepsilon)}{partial ln varepsilon} bigg)_{varepsilon=varepsilon_F} bigg(dfrac{kT}{varepsilon_F}bigg)^2 bigg] ,

and

C_V simeq S simeq dfrac{pi^2}{3}k^2 T a(varepsilon_F),

where a(varepsilon) is the density of (the single-particle) states in the system. Examine these results for a gas with energy spectrum varepsilon propto p^s, confined to a space of n dimensions, and discuss the special cases: s=1 and 2, with n=2 and 3.
[Hint: Use equation (E.18) from Appendix E.]

Chapter 10

Solution: Pathria 10.3:
(a) Show that for a gas obeying van der Waals equation of state (10.3.9),

C_P-C_V=Nkbigg{ 1-dfrac{2a}{kTnu^3}(nu-b)^2 bigg}^{-1}.

(b) Also show that, for a van der Waals gas with constant specific heat C_V, an adiabatic process conforms to the equation

(nu - b)T^{C_V/Nk}=text{const};

compare with equation (1.4.30).

(c) Further show that the temperature change resulting from an expansion of the gas (into vacuum) from volume V_1 to volume V_2 is given by

T_2-T_1=dfrac{N^2 a}{C_V}bigg( dfrac{1}{V_2} - dfrac{1}{V_1} bigg).

Solution: Pathria 10.5: Show that the first-order Joule-Thomson coefficient of a gas is given by the formula

bigg( dfrac{partial T}{partial P}bigg)_H = dfrac{N}{C_P} bigg(T dfrac{partial (a_2 lambda^3 )}{partial T} -a_2 lambda^3 bigg),

where a_2(T) is the second virial coefficient of the gas and H its enthalpy; see equation (10.2.1). Derive an explicit expression for the Joule-Thomson coefficient in the case of a gas with interparticle interaction

u(r) = Bigg{ begin{array}{cl} +infty & mathrm{for}~0

and discuss the temperature dependence of this coefficient.

(Note: I forgot to 'discuss the temperature dependence of this coefficient')

Chapter 12

Solution: Pathria 12.3: Consider a nonideal gas obeying a modified van der Waals equation of state

(P + a / nu^{n})(nu - b) = RT quad (n>1) .

Examine how the critical constants P_c, nu_c, and T_c, and the critical exponents beta, gamma, gamma^{prime}, and delta of this system depend on the number n.

(Note: I did not attempt the critical exponent part of the problem)

Solution: Pathria 12.20: Consider a system with a modified expression for the Landau free energy, namely

psi_h(t,m) = -hm +q(t)+r(t)m^2 + s(t)m^4 + u(t) m^6 ,

with u(t) a fixed positive constant. Minimize psi with respect to the variable m and examine the spontaneous magnetization m_0 as a function of the parameters r and s. In particular, show the following:

(a) For r>0 and s>-(3ur)^{1/2},~m_0=0 is the only real solution.

(b) For r>0 and -(4ur)^{1/2} or pm m_1, where m_1^2 = frac{sqrt{(s^2-3ur)-s}}{3u}. However, the minimum of psi at m_0 = 0 is lower than the minima at m_0 = pm m_1, so the ultimate equilibrium value of m_0 is 0.

(c) For r>0 and s = -(4ur)^{1/2},~m_0=0 or pm(r/u)^{1/4}. Now the minimum of psi at m_0 = 0 is of the same height as the ones at m_0 = pm (r/u)^{1/4}, so a nonzero spontaneous magnetization is as likely to occur as the zero one.

(d) For r>0 and s < -(4ur)^{1/2},~m_0= pm m_1 — which implies a first-order phase transition (becuase the two possible states availible here differ by a finite amount in m). The line s=-(4ur)^{1/2}, with r positive, is generally referred to as a 'line of first-order phase transitions.'

(e) For r=0 and s<0, m_0 = pm(2|s|/3u)^{1/2}.

(f) For r<0, m_0 = pm m_1 for all s. As rrightarrow 0, m_1 rightarrow 0 if s is positive.

(g) For r=0 and s>0, m_0=0 is only solution. [sic] Combining this result with (f), we conclude that the line r=0, with s positive, is a 'line of second-order phase transitions,' for the two states available here differ by a vanishing amount in m.

The lines of first-order phase transitions and second-order phase transitions meet at the point (r=0,s=0), which is commonly referred to as a tricritical point (Griffiths, 1970).

Chapter 15

Solution: Pathria 15.1:Making use of expressions (15.1.11) and (15.1.12) for Delta S and Delta P, and expressions (15.1.14) for overline{(Delta T)^2}, overline{(Delta V)^2}, and overline{(Delta T Delta V)}, show that

(a)overline{(Delta T Delta S)} = kT;

(b)overline{(Delta P Delta V)} = -kT;

(c)overline{(Delta S Delta V)} = kT(partial V/partial T)_P;

(d)overline{(Delta P Delta T)} = kT^2C_V^{-1}(partial P/partial T)_V.

Pathria And Beale

Statistical Mechanics Pathria Pdf

Statistical Mechanics Pathria Solution